{"id":888,"date":"2023-04-12T11:25:07","date_gmt":"2023-04-12T02:25:07","guid":{"rendered":"http:\/\/edu.ujhb.org\/?p=888"},"modified":"2023-04-12T11:25:07","modified_gmt":"2023-04-12T02:25:07","slug":"%e4%ba%ba%e5%b7%a5%e6%99%ba%e8%83%bd%e4%b8%ad%e7%ba%a7%e6%95%99%e7%a8%8b%ef%bc%88%e4%b8%ad%e6%96%87%e7%89%88%ef%bc%89","status":"publish","type":"post","link":"https:\/\/edu.ujhb.org\/?p=888","title":{"rendered":"\u4eba\u5de5\u667a\u80fd\u4e2d\u7ea7\u6559\u7a0b\uff08\u4e2d\u6587\u7248\uff09"},"content":{"rendered":"\n

\u6211\u662f\u8981\u505a\u4eba\u5de5\u667a\u80fd\u4e2d\u7ea7\u6559\u80b2\u7684\u89c6\u9891\uff0c\u5185\u5bb9\u4e2d\u8981\u5305\u542b\u4f46\u4e0d\u9650\u4e8e\u4e00\u4e0b\u5185\u5bb9 1\u3001\u9a6c\u5c14\u53ef\u592b\u94fe 2\u3001\u8499\u7279\u5361\u6d1b\u91c7\u6837 3\u3001\u5409\u5e03\u65af\u91c7\u6837 4\u3001\u914d\u5206\u51fd\u6570 5\u3001\u54c8\u91cc\u65af\u94fe 6\u3001\u6700\u5927\u4f3c\u7136\u4f30\u8ba1 7\u3001\u4e2d\u5fc3\u6781\u9650\u5b9a\u7406 8\u3001\u6700\u5927\u540e\u9a8c\u4f30\u8ba1 9\u3001\u8fd1\u4f3c\u63a8\u65ad 10\u3001\u9192\u7720\u7b97\u6cd5 11\u3001\u73bb\u5c14\u5179\u66fc\u673a 12\u3001\u4fe1\u5ff5\u7f51\u7edc 13\u3001\u5bf9\u6297\u7f51\u7edc 14\u3001\u81ea\u7f16\u7801\u5668 15\u3001\u751f\u6210\u968f\u673a\u7f51\u7edc 16\u3001\u9010\u5c42\u9884\u8bad\u7ec3 17\u3001\u91cd\u8981\u6027\u91c7\u6837 18\u3001\u8868\u793a\u5b66\u4e60 19\u3001\u7a00\u758f\u7f16\u7801 20\u3001\u56e0\u5b50\u5206\u6790 21\u3001\u5f20\u91cf 22\u3001\u7ebf\u6027\u76f8\u5173 23\u3001\u65b9\u5dee 24\u3001\u635f\u5931\u51fd\u6570 25\u3001\u4ea4\u53c9\u71b5 26\u3001\u7279\u5f81\u5206\u89e3 27\u3001\u5947\u5f02\u503c\u5206\u89e3 28\u3001\u68af\u5ea6\u4e0b\u964d 29\u3001\u6700\u5c0f\u4e8c\u4e58 30\u3001\u9ad8\u65af\u5206\u5e03 31\u3001\u725b\u987f\u6cd5 32\u3001\u6c49\u68ee\u77e9\u9635 33\u3001\u5377\u79ef\u7f51\u7edc 34\u3001\u5faa\u73af\u7f51\u7edc 35\u3001LSTM 36\u3001BERT 37\u3001\u6fc0\u6d3b\u51fd\u6570 38\u3001\u8d1d\u53f6\u65af\u8ba1\u7b97 39\u3001\u6e10\u8fd1\u65e0\u504f 40\u3001\u6ce8\u610f\u529b\u673a\u5236 41\u3001\u81ea\u52a8\u5fae\u5206 42\u3001\u53cd\u5411\u4f20\u64ad 43\u3001\u8bcd\u888b 44\u3001\u6865\u5f0f\u91c7\u6837 45\u3001\u56e0\u679c\u6a21\u578b 46\u3001\u94fe\u5f0f\u6cd5\u5219 47\u3001\u56e2\u52bf\u80fd 48\u3001\u534f\u540c\u8fc7\u6ee4 49\u3001\u8fde\u63a5\u4e3b\u4e49 50\u3001\u7ea6\u675f\u4f18\u5316 51\u3001\u5ef6\u62d3\u6cd5 52\u3001\u51f8\u4f18\u5316 53\u3001\u5171\u7236 54\u3001\u7d2f\u8ba1\u51fd\u6570 55\u3001\u66f2\u7387 56\u3001\u6570\u636e\u96c6\u589e\u5f3a 57\u3001\u7ef4\u6570\u707e\u96be 58\u3001\u8bfe\u7a0b\u5b66\u4e60 59\u3001\u8bbe\u8ba1\u77e9\u9635 60\u3001\u7ec6\u81f4\u5e73\u8861 61\u3001\u964d\u7ef4 62\u3001\u6709\u6548\u5bb9\u91cf 63\u3001\u7ecf\u9a8c\u5206\u5e03 64\u3001\u8303\u6570 65\u3001\u53d8\u5dee\u56e0\u7d20 66\u3001\u7cbe\u8c03 67\u3001\u95e8\u63a7 68\u3001\u4f3d\u9a6c\u5206\u5e03 69\u3001\u9ad8\u65af\u6838 70\u3001\u9009\u901a\u5668 71\u3001\u6cdb\u5316 72\u3001\u5e7f\u4e49\u51fd\u6570 73\u3001\u751f\u6210\u6a21\u578b 74\u3001\u68af\u5ea6\u622a\u65ad 75\u3001\u7f51\u683c\u641c\u7d22 76\u3001\u6c49\u660e\u8ddd\u79bb 77\u3001\u5f02\u65b9\u5dee 78\u3001\u8d85\u53c2\u6570 79\u3001\u6307\u793a\u51fd\u6570 80\u3001\u5e72\u9884\u67e5\u8be2 81\u3001Jacobian\u77e9\u9635 82\u3001Laplace\u5206\u5e03 83\u3001KL\u6563\u5ea6 84\u3001Krylov\u65b9\u6cd5 85\u3001\u5927\u6570\u5b9a\u7406 86\u3001\u6e17\u6f0f\u5355\u5143 87\u3001\u52d2\u8d1d\u683c\u53ef\u79ef 88\u3001\u83b1\u5e03\u5c3c\u5179\u6cd5\u5219 89\u3001Lipschitz \u5e38\u6570 90\u3001\u73af\u72b6\u4fe1\u5ff5\u4f20\u64ad 91\u3001\u903b\u8f91\u56de\u5f52 92\u3001\u6d41\u578b\u5047\u8bbe 93\u3001\u6d4b\u5ea6\u8bba 94\u3001\u9053\u5fb7\u5316 95\u3001\u8d1f\u5b9a 96\u3001\u5206\u6bb5 97\u3001\u7f6e\u6362\u4e0d\u53d8\u6027 98\u3001\u6b63\u5219\u5316 99\u3001\u6b63\u5207\u4f20\u64ad 100\u3001\u652f\u6301\u5411\u91cf \u4f60\u80fd\u5426\u6839\u636e\u7f51\u7ad9\u7684\u5185\u5bb9\uff0c\u4e3a\u6211\u5b89\u6392\u89c6\u9891\u6559\u7a0b\u7684\u6587\u5b57\u5185\u5bb9\u3002\u8bf7\u5148\u4e3a\u6211\u5217\u51fa\u8981\u8bb2\u89e3\u7684\u5185\u5bb9\u4e00\u89c8\u3002<\/p>\n\n\n\n

\u5f53\u7136\u53ef\u4ee5\uff0c\u4ee5\u4e0b\u662f\u60a8\u8981\u8bb2\u89e3\u7684\u5185\u5bb9\u4e00\u89c8\uff1a<\/p>\n\n\n\n

    \n
  1. \u9a6c\u5c14\u53ef\u592b\u94fe<\/li>\n\n\n\n
  2. \u8499\u7279\u5361\u6d1b\u91c7\u6837<\/li>\n\n\n\n
  3. \u5409\u5e03\u65af\u91c7\u6837<\/li>\n\n\n\n
  4. \u914d\u5206\u51fd\u6570<\/li>\n\n\n\n
  5. \u54c8\u91cc\u65af\u94fe<\/li>\n\n\n\n
  6. \u6700\u5927\u4f3c\u7136\u4f30\u8ba1<\/li>\n\n\n\n
  7. \u4e2d\u5fc3\u6781\u9650\u5b9a\u7406<\/li>\n\n\n\n
  8. \u6700\u5927\u540e\u9a8c\u4f30\u8ba1<\/li>\n\n\n\n
  9. \u8fd1\u4f3c\u63a8\u65ad<\/li>\n\n\n\n
  10. \u9192\u7720\u7b97\u6cd5<\/li>\n\n\n\n
  11. \u73bb\u5c14\u5179\u66fc\u673a<\/li>\n\n\n\n
  12. \u4fe1\u5ff5\u7f51\u7edc<\/li>\n\n\n\n
  13. \u5bf9\u6297\u7f51\u7edc<\/li>\n\n\n\n
  14. \u81ea\u7f16\u7801\u5668<\/li>\n\n\n\n
  15. \u751f\u6210\u968f\u673a\u7f51\u7edc<\/li>\n\n\n\n
  16. \u9010\u5c42\u9884\u8bad\u7ec3<\/li>\n\n\n\n
  17. \u91cd\u8981\u6027\u91c7\u6837<\/li>\n\n\n\n
  18. \u8868\u793a\u5b66\u4e60<\/li>\n\n\n\n
  19. \u7a00\u758f\u7f16\u7801<\/li>\n\n\n\n
  20. \u56e0\u5b50\u5206\u6790<\/li>\n\n\n\n
  21. \u5f20\u91cf<\/li>\n\n\n\n
  22. \u7ebf\u6027\u76f8\u5173<\/li>\n\n\n\n
  23. \u65b9\u5dee<\/li>\n\n\n\n
  24. \u635f\u5931\u51fd\u6570<\/li>\n\n\n\n
  25. \u4ea4\u53c9\u71b5<\/li>\n\n\n\n
  26. \u7279\u5f81\u5206\u89e3<\/li>\n\n\n\n
  27. \u5947\u5f02\u503c\u5206\u89e3<\/li>\n\n\n\n
  28. \u68af\u5ea6\u4e0b\u964d<\/li>\n\n\n\n
  29. \u6700\u5c0f\u4e8c\u4e58<\/li>\n\n\n\n
  30. \u9ad8\u65af\u5206\u5e03<\/li>\n\n\n\n
  31. \u725b\u987f\u6cd5<\/li>\n\n\n\n
  32. \u6c49\u68ee\u77e9\u9635<\/li>\n\n\n\n
  33. \u5377\u79ef\u7f51\u7edc<\/li>\n\n\n\n
  34. \u5faa\u73af\u7f51\u7edc<\/li>\n\n\n\n
  35. LSTM<\/li>\n\n\n\n
  36. BERT<\/li>\n\n\n\n
  37. \u6fc0\u6d3b\u51fd\u6570<\/li>\n\n\n\n
  38. \u8d1d\u53f6\u65af\u8ba1\u7b97<\/li>\n\n\n\n
  39. \u6e10\u8fd1\u65e0\u504f<\/li>\n\n\n\n
  40. \u6ce8\u610f\u529b\u673a\u5236<\/li>\n\n\n\n
  41. \u81ea\u52a8\u5fae\u5206<\/li>\n\n\n\n
  42. \u53cd\u5411\u4f20\u64ad<\/li>\n\n\n\n
  43. \u8bcd\u888b<\/li>\n\n\n\n
  44. \u6865\u5f0f\u91c7\u6837<\/li>\n\n\n\n
  45. \u56e0\u679c\u6a21\u578b<\/li>\n\n\n\n
  46. \u94fe\u5f0f\u6cd5\u5219<\/li>\n\n\n\n
  47. \u56e2\u52bf\u80fd<\/li>\n\n\n\n
  48. \u534f\u540c\u8fc7\u6ee4<\/li>\n\n\n\n
  49. \u8fde\u63a5\u4e3b\u4e49<\/li>\n\n\n\n
  50. \u7ea6\u675f\u4f18\u5316<\/li>\n\n\n\n
  51. \u5ef6\u62d3\u6cd5<\/li>\n\n\n\n
  52. \u51f8\u4f18\u5316<\/li>\n\n\n\n
  53. \u5171\u7236<\/li>\n\n\n\n
  54. \u7d2f\u8ba1\u51fd\u6570<\/li>\n\n\n\n
  55. \u66f2\u7387<\/li>\n\n\n\n
  56. \u6570\u636e\u96c6\u589e\u5f3a<\/li>\n\n\n\n
  57. \u7ef4\u6570\u707e\u96be<\/li>\n\n\n\n
  58. \u8bfe\u7a0b\u5b66\u4e60<\/li>\n\n\n\n
  59. \u8bbe\u8ba1\u77e9\u9635<\/li>\n\n\n\n
  60. \u7ec6\u81f4\u5e73\u8861<\/li>\n\n\n\n
  61. \u964d\u7ef4<\/li>\n\n\n\n
  62. \u6709\u6548\u5bb9\u91cf<\/li>\n\n\n\n
  63. \u7ecf\u9a8c\u5206\u5e03<\/li>\n\n\n\n
  64. \u8303\u6570<\/li>\n\n\n\n
  65. \u53d8\u5dee\u56e0\u7d20<\/li>\n\n\n\n
  66. \u7cbe\u8c03<\/li>\n\n\n\n
  67. \u95e8\u63a7<\/li>\n\n\n\n
  68. \u4f3d\u9a6c\u5206\u5e03<\/li>\n\n\n\n
  69. \u9ad8\u65af\u6838<\/li>\n\n\n\n
  70. \u9009\u901a\u5668<\/li>\n\n\n\n
  71. \u6cdb\u5316<\/li>\n\n\n\n
  72. \u5e7f\u4e49\u51fd\u6570<\/li>\n\n\n\n
  73. \u751f\u6210\u6a21\u578b<\/li>\n\n\n\n
  74. \u68af\u5ea6\u622a\u65ad<\/li>\n\n\n\n
  75. \u7f51\u683c\u641c\u7d22<\/li>\n\n\n\n
  76. \u6c49\u660e\u8ddd\u79bb<\/li>\n\n\n\n
  77. \u5f02\u65b9\u5dee<\/li>\n\n\n\n
  78. \u8d85\u53c2\u6570<\/li>\n\n\n\n
  79. \u6307\u793a\u51fd\u6570<\/li>\n\n\n\n
  80. \u5e72\u9884\u67e5\u8be2<\/li>\n\n\n\n
  81. Jacobian\u77e9\u9635<\/li>\n\n\n\n
  82. Laplace\u5206\u5e03<\/li>\n\n\n\n
  83. KL\u6563\u5ea6<\/li>\n\n\n\n
  84. Krylov\u65b9\u6cd5<\/li>\n\n\n\n
  85. \u5927\u6570\u5b9a\u7406<\/li>\n\n\n\n
  86. \u6e17\u6f0f\u5355\u5143<\/li>\n\n\n\n
  87. \u52d2\u8d1d\u683c\u53ef\u79ef<\/li>\n\n\n\n
  88. <\/li>\n<\/ol>\n\n\n\n\n\n
    \"gun<\/figure>\n\n\n\n

    \u8bf7\u7ee7\u7eed<\/p>\n\n\n\n

    101\u3001\u968f\u673a\u68af\u5ea6\u4e0b\u964d 102\u3001\u9690\u5f0f\u53cd\u9988 103\u3001\u964d\u6e29\u6cd5 104\u3001\u68af\u5ea6\u63d0\u5347 105\u3001\u65f6\u95f4\u5e8f\u5217\u9884\u6d4b 106\u3001\u5f3a\u5316\u5b66\u4e60 107\u3001Q-learning 108\u3001Actor-Critic\u7b97\u6cd5 109\u3001\u4ef7\u503c\u51fd\u6570 110\u3001\u7b56\u7565\u51fd\u6570 111\u3001\u9a6c\u5c14\u53ef\u592b\u51b3\u7b56\u8fc7\u7a0b 112\u3001AlphaGo 113\u3001\u795e\u7ecf\u56fe\u7075\u673a 114\u3001\u5f31\u76d1\u7763\u5b66\u4e60 115\u3001\u8fc1\u79fb\u5b66\u4e60 116\u3001\u5143\u5b66\u4e60 117\u3001\u751f\u6210\u5bf9\u6297\u7f51\u7edc\u7684\u5e94\u7528 118\u3001\u81ea\u7136\u8bed\u8a00\u5904\u7406 119\u3001\u56fe\u50cf\u5904\u7406 120\u3001\u63a8\u8350\u7cfb\u7edf 121\u3001\u5f02\u5e38\u68c0\u6d4b 122\u3001\u805a\u7c7b\u5206\u6790 123\u3001\u5206\u7c7b\u95ee\u9898 124\u3001\u56de\u5f52\u95ee\u9898 125\u3001\u5e8f\u5217\u5efa\u6a21 126\u3001\u6df1\u5ea6\u5b66\u4e60\u6846\u67b6 127\u3001\u673a\u5668\u5b66\u4e60\u5e94\u7528\u5b9e\u4f8b 128\u3001\u672a\u6765\u53d1\u5c55\u8d8b\u52bf 129\u3001\u4eba\u5de5\u667a\u80fd\u7684\u4f26\u7406\u548c\u793e\u4f1a\u95ee\u9898 130\u3001\u4eba\u5de5\u667a\u80fd\u5728\u672a\u6765\u7684\u5f71\u54cd<\/p>\n\n\n\n\n\n

    \"gun<\/figure>\n\n\n\n

    \u8bf7\u8be6\u7ec6\u89e3\u91ca\u9a6c\u5c14\u53ef\u592b\u94fe<\/p>\n\n\n\n

    \u9a6c\u5c14\u53ef\u592b\u94fe\u662f\u4e00\u79cd\u968f\u673a\u8fc7\u7a0b\uff0c\u901a\u5e38\u7528\u4e8e\u5efa\u6a21\u5177\u6709\u6982\u7387\u6027\u8d28\u7684\u7cfb\u7edf\u3002\u5b83\u662f\u6307\u5728\u67d0\u4e00\u65f6\u523b\uff0c\u7cfb\u7edf\u7684\u72b6\u6001\u53ef\u4ee5\u662f\u6709\u9650\u4e2a\u6570\u4e2d\u7684\u4efb\u610f\u4e00\u4e2a\uff0c\u5e76\u4e14\u5728\u4e0b\u4e00\u65f6\u523b\uff0c\u7cfb\u7edf\u7684\u72b6\u6001\u53ea\u4f9d\u8d56\u4e8e\u5f53\u524d\u72b6\u6001\uff0c\u800c\u4e0e\u8fc7\u53bb\u7684\u72b6\u6001\u65e0\u5173\u3002\u8fd9\u4e2a\u6027\u8d28\u88ab\u79f0\u4e3a\u9a6c\u5c14\u53ef\u592b\u6027\u8d28\uff0c\u56e0\u6b64\u8fd9\u79cd\u968f\u673a\u8fc7\u7a0b\u88ab\u79f0\u4e3a\u9a6c\u5c14\u53ef\u592b\u94fe\u3002<\/p>\n\n\n\n

    \u9a6c\u5c14\u53ef\u592b\u94fe\u53ef\u4ee5\u7528\u4e00\u4e2a\u72b6\u6001\u8f6c\u79fb\u77e9\u9635\u6765\u63cf\u8ff0\uff0c\u8be5\u77e9\u9635\u662f\u4e00\u4e2a\u6b63\u65b9\u5f62\u77e9\u9635\uff0c\u5176\u4e2d\u6bcf\u4e2a\u5143\u7d20\u8868\u793a\u4ece\u4e00\u4e2a\u72b6\u6001\u5230\u53e6\u4e00\u4e2a\u72b6\u6001\u7684\u6982\u7387\u3002\u8be5\u77e9\u9635\u7684\u6bcf\u4e00\u884c\u4e4b\u548c\u4e3a1\uff0c\u56e0\u4e3a\u5728\u4efb\u610f\u72b6\u6001\u4e0b\uff0c\u7cfb\u7edf\u5fc5\u987b\u8f6c\u79fb\u5230\u4e0b\u4e00\u4e2a\u72b6\u6001\u3002\u5982\u679c\u72b6\u6001\u7684\u6570\u76ee\u662f\u6709\u9650\u7684\uff0c\u90a3\u4e48\u8fd9\u4e2a\u8f6c\u79fb\u77e9\u9635\u662f\u4e00\u4e2a\u79bb\u6563\u7684\u77e9\u9635\uff0c\u5982\u679c\u72b6\u6001\u662f\u8fde\u7eed\u7684\uff0c\u90a3\u4e48\u8fd9\u4e2a\u8f6c\u79fb\u77e9\u9635\u5c31\u662f\u4e00\u4e2a\u79ef\u5206\u65b9\u7a0b\u3002<\/p>\n\n\n\n

    \u9a6c\u5c14\u53ef\u592b\u94fe\u901a\u5e38\u53ef\u4ee5\u7528\u6765\u5efa\u6a21\u4e00\u4e9b\u5177\u6709\u6982\u7387\u6027\u8d28\u7684\u7cfb\u7edf\uff0c\u5982\u5929\u6c14\u3001\u80a1\u5e02\u3001\u6d41\u91cf\u7b49\u3002\u5728\u8fd9\u4e9b\u7cfb\u7edf\u4e2d\uff0c\u72b6\u6001\u662f\u968f\u65f6\u95f4\u53d8\u5316\u7684\uff0c\u6211\u4eec\u53ef\u4ee5\u7528\u9a6c\u5c14\u53ef\u592b\u94fe\u6765\u9884\u6d4b\u8fd9\u4e9b\u7cfb\u7edf\u5728\u672a\u6765\u7684\u72b6\u6001\u3002\u6b64\u5916\uff0c\u9a6c\u5c14\u53ef\u592b\u94fe\u8fd8\u6709\u8bb8\u591a\u91cd\u8981\u7684\u5e94\u7528\uff0c\u5982\u673a\u5668\u5b66\u4e60\u4e2d\u7684\u9a6c\u5c14\u53ef\u592b\u968f\u673a\u573a\u548c\u9690\u85cf\u9a6c\u5c14\u53ef\u592b\u6a21\u578b\u7b49\u3002<\/p>\n\n\n\n\n\n

    \"gun<\/figure>\n\n\n\n

    \u4f60\u53ef\u5426\u4f7f\u7528\u4e00\u4e9b\u65b9\u7a0b\u5f0f\u7684\u5f62\u5f0f\u6765\u63cf\u8ff0\u9a6c\u5c14\u79d1\u592b\u94fe<\/p>\n\n\n\n

    \u662f\u7684\uff0c\u9a6c\u5c14\u53ef\u592b\u94fe\u53ef\u4ee5\u7528\u6570\u5b66\u65b9\u7a0b\u5f0f\u6765\u63cf\u8ff0\u3002\u5047\u8bbe$S={s_1,s_2,\\dots,s_n}$\u662f\u9a6c\u5c14\u53ef\u592b\u94fe\u4e2d\u6240\u6709\u53ef\u80fd\u7684\u72b6\u6001\u7684\u96c6\u5408\uff0c$P_{ij}$\u8868\u793a\u4ece\u72b6\u6001$s_i$\u8f6c\u79fb\u5230\u72b6\u6001$s_j$\u7684\u6982\u7387\uff0c\u5219\u9a6c\u5c14\u53ef\u592b\u94fe\u53ef\u4ee5\u8868\u793a\u4e3a\uff1a<\/p>\n\n\n\n

    \ufffd11\ufffd12\u2026\ufffd1\ufffd\ufffd21\ufffd22\u2026\ufffd2\ufffd\u22ee\u22ee\u22f1\u22ee\ufffd\ufffd1\ufffd\ufffd2\u2026\ufffd\ufffd\ufffd\u200bP<\/em>11\u200bP<\/em>21\u200b\u22eeP<\/em>n<\/em>1\u200b\u200bP<\/em>12\u200bP<\/em>22\u200b\u22eeP<\/em>n<\/em>2\u200b\u200b\u2026\u2026\u22f1\u2026\u200bP<\/em>1n<\/em>\u200bP<\/em>2n<\/em>\u200b\u22eeP<\/em>nn<\/em>\u200b\u200b<\/p>\n\n\n\n

    \u5176\u4e2d\uff0c\u6bcf\u4e00\u884c\u7684\u5143\u7d20\u4e4b\u548c\u7b49\u4e8e1\uff0c\u5373$\\sum_{j=1}^nP_{ij}=1$\uff0c\u8868\u793a\u5728\u4efb\u4f55\u65f6\u5019\uff0c\u7cfb\u7edf\u90fd\u5904\u4e8e\u6240\u6709\u72b6\u6001\u4e2d\u7684\u4e00\u4e2a\u3002\u6211\u4eec\u53ef\u4ee5\u628a\u8fd9\u4e2a\u77e9\u9635\u770b\u4f5c\u4e00\u4e2a\u6982\u7387\u8f6c\u79fb\u77e9\u9635\uff0c\u5176\u4e2d$P_{ij}$\u8868\u793a\u4ece\u72b6\u6001$s_i$\u8f6c\u79fb\u5230\u72b6\u6001$s_j$\u7684\u6982\u7387\u3002<\/p>\n\n\n\n

    \u6211\u4eec\u8fd8\u53ef\u4ee5\u7528\u5411\u91cf\u8868\u793a\u5f53\u524d\u72b6\u6001\u5206\u5e03\uff0c\u5047\u8bbe$t$\u65f6\u523b\u72b6\u6001\u5206\u5e03\u4e3a$\\mathbf{v}^{(t)}=(v_1^{(t)},v_2^{(t)},\\dots,v_n^{(t)})$\uff0c\u5176\u4e2d$v_i^{(t)}$\u8868\u793a\u7cfb\u7edf\u5904\u4e8e\u72b6\u6001$s_i$\u7684\u6982\u7387\uff0c\u5219\u6709\uff1a<\/p>\n\n\n\n

    \ufffd(\ufffd+1)=\ufffd(\ufffd)\ufffdv<\/strong>(t<\/em>+1)=v<\/strong>(t<\/em>)P<\/strong><\/p>\n\n\n\n

    \u5176\u4e2d$\\mathbf{P}$\u662f\u6982\u7387\u8f6c\u79fb\u77e9\u9635\uff0c\u8868\u793a\u4ece$t$\u65f6\u523b\u8f6c\u79fb\u5230$t+1$\u65f6\u523b\u7684\u8f6c\u79fb\u77e9\u9635\u3002\u8fd9\u4e2a\u516c\u5f0f\u8868\u793a\uff0c\u5f53\u524d\u65f6\u523b\u72b6\u6001\u5206\u5e03$\\mathbf{v}^{(t)}$\u4e0e\u8f6c\u79fb\u77e9\u9635$\\mathbf{P}$\u76f8\u4e58\u5f97\u5230$t+1$\u65f6\u523b\u7684\u72b6\u6001\u5206\u5e03$\\mathbf{v}^{(t+1)}$\u3002<\/p>\n\n\n\n\n\n

    \"gun<\/figure>\n\n\n\n

    \u8bf7\u7ee7\u7eed<\/p>\n\n\n\n

      \n
    • P(i,j)\u8868\u793a\u4ece\u72b6\u6001i\u8f6c\u79fb\u5230\u72b6\u6001j\u7684\u6982\u7387\uff0c\u6ee1\u8db3\uff1a\n
        \n
      • P(i,j) \u2265 0<\/li>\n\n\n\n
      • \u5bf9\u4e8e\u6bcf\u4e2a\u72b6\u6001i\uff0c\u6709\u2211j P(i,j) = 1<\/li>\n<\/ul>\n<\/li>\n\n\n\n
      • \u5728\u65f6\u95f4t\u7684\u72b6\u6001\u4e3ai\u7684\u6982\u7387\u8868\u793a\u4e3aP(X_t=i)\uff0c\u6ee1\u8db3\uff1a\n
          \n
        • P(X_t=i) \u2265 0<\/li>\n\n\n\n
        • \u5bf9\u4e8e\u6240\u6709\u72b6\u6001i\uff0c\u6709\u2211i P(X_t=i) = 1<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n\n\n\n

          \u6839\u636e\u4e0a\u8ff0\u5b9a\u4e49\uff0c\u6211\u4eec\u53ef\u4ee5\u5f97\u51fa\u4ee5\u4e0b\u6027\u8d28\uff1a<\/p>\n\n\n\n

            \n
          • \u9a6c\u5c14\u79d1\u592b\u94fe\u5177\u6709\u65e0\u540e\u6548\u6027\uff0c\u5373\u5b83\u7684\u672a\u6765\u72b6\u6001\u53ea\u53d6\u51b3\u4e8e\u5f53\u524d\u72b6\u6001\uff0c\u800c\u4e0d\u53d7\u8fc7\u53bb\u72b6\u6001\u7684\u5f71\u54cd\u3002<\/li>\n\n\n\n
          • \u9a6c\u5c14\u79d1\u592b\u94fe\u53ef\u4ee5\u7528\u72b6\u6001\u8f6c\u79fb\u77e9\u9635P\u6765\u8868\u793a\uff0c\u5176\u4e2dP[i][j]\u8868\u793a\u4ece\u72b6\u6001i\u8f6c\u79fb\u5230\u72b6\u6001j\u7684\u6982\u7387\u3002<\/li>\n\n\n\n
          • \u9a6c\u5c14\u79d1\u592b\u94fe\u7684\u72b6\u6001\u8f6c\u79fb\u53ef\u4ee5\u7528\u77e9\u9635\u4e58\u6cd5\u6765\u5b9e\u73b0\uff0c\u5373P^n\u8868\u793a\u4ece\u72b6\u6001i\u7ecf\u8fc7n\u6b65\u8f6c\u79fb\u5230\u72b6\u6001j\u7684\u6982\u7387\u3002<\/li>\n\n\n\n
          • \u9a6c\u5c14\u79d1\u592b\u94fe\u7684\u6536\u655b\u884c\u4e3a\u4e0e\u5176\u72b6\u6001\u8f6c\u79fb\u77e9\u9635\u7684\u6027\u8d28\u6709\u5173\uff0c\u5982\u679c\u5b83\u6ee1\u8db3\u4e00\u4e9b\u6761\u4ef6\uff0c\u90a3\u4e48\u5b83\u4f1a\u6536\u655b\u5230\u4e00\u4e2a\u7a33\u6001\u5206\u5e03\uff0c\u5373\u72b6\u6001\u5728\u957f\u65f6\u95f4\u5185\u7684\u5206\u5e03\u4e0d\u518d\u6539\u53d8\uff0c\u4e5f\u5c31\u662f\u8bf4\uff0c\u5b83\u7684\u6982\u7387\u5206\u5e03\u4e0d\u518d\u968f\u65f6\u95f4\u53d8\u5316\u800c\u8d8b\u4e8e\u7a33\u5b9a\u3002<\/li>\n<\/ul>\n\n\n\n\n\n
            \"gun<\/figure>\n\n\n\n

            \u8bf7\u7ee7\u7eed<\/p>\n\n\n\n

            \u5f53\u7136\uff0c\u5982\u679c\u4f60\u9700\u8981\u66f4\u6df1\u5165\u7684\u4e86\u89e3\uff0c\u6211\u4eec\u8fd8\u53ef\u4ee5\u63a2\u8ba8\u8bb8\u591a\u9ad8\u7ea7\u7684\u6982\u5ff5\u548c\u6280\u672f\uff0c\u5982\u5404\u79cd\u4e0d\u540c\u7c7b\u578b\u7684\u72b6\u6001\u7a7a\u95f4\uff0c\u4e0d\u540c\u7c7b\u578b\u7684\u9a6c\u5c14\u79d1\u592b\u94fe\uff0c\u9690\u9a6c\u5c14\u79d1\u592b\u6a21\u578b\uff0c\u4ee5\u53ca\u5982\u4f55\u4f7f\u7528\u9a6c\u5c14\u79d1\u592b\u94fe\u8499\u7279\u5361\u6d1b\u65b9\u6cd5\u8fdb\u884c\u91c7\u6837\u7b49\u7b49\u3002<\/p>\n\n\n\n\n\n

            \"gun<\/figure>\n\n\n\n

            \u8bf7\u8be6\u7ec6\u63cf\u8ff0\u9a6c\u5c14\u53ef\u592b\u968f\u673a\u573a<\/p>\n\n\n\n

            \u9a6c\u5c14\u53ef\u592b\u968f\u673a\u573a\uff08Markov Random Field\uff0cMRF\uff09\u662f\u4e00\u79cd\u6982\u7387\u56fe\u6a21\u578b\uff0c\u7528\u4e8e\u63cf\u8ff0\u7531\u4e00\u7ec4\u968f\u673a\u53d8\u91cf\u7ec4\u6210\u7684\u7cfb\u7edf\uff0c\u8fd9\u4e9b\u53d8\u91cf\u53ef\u4ee5\u76f8\u4e92\u5f71\u54cd\u3002\u9a6c\u5c14\u53ef\u592b\u968f\u673a\u573a\u662f\u7531\u4e00\u4e2a\u65e0\u5411\u56fe\u6765\u8868\u793a\u7684\uff0c\u56fe\u4e2d\u6bcf\u4e2a\u8282\u70b9\u4ee3\u8868\u4e00\u4e2a\u53d8\u91cf\uff0c\u8282\u70b9\u4e4b\u95f4\u7684\u8fb9\u8868\u793a\u53d8\u91cf\u4e4b\u95f4\u7684\u4f9d\u8d56\u5173\u7cfb\u3002<\/p>\n\n\n\n

            \u4e0e\u9a6c\u5c14\u53ef\u592b\u94fe\u4e0d\u540c\uff0c\u9a6c\u5c14\u53ef\u592b\u968f\u673a\u573a\u4e2d\u7684\u53d8\u91cf\u4e0d\u662f\u6309\u7167\u65f6\u95f4\u987a\u5e8f\u6392\u5217\u7684\u3002\u5728\u4e00\u4e2a\u9a6c\u5c14\u53ef\u592b\u968f\u673a\u573a\u4e2d\uff0c\u6bcf\u4e2a\u53d8\u91cf\u7684\u6761\u4ef6\u6982\u7387\u53ea\u4e0e\u4e0e\u5176\u76f8\u90bb\u7684\u5176\u4ed6\u53d8\u91cf\u6709\u5173\uff0c\u800c\u4e0e\u5176\u4ed6\u7684\u53d8\u91cf\u662f\u6761\u4ef6\u72ec\u7acb\u7684\u3002\u8fd9\u79cd\u5c40\u90e8\u7684\u4f9d\u8d56\u5173\u7cfb\u548c\u5168\u5c40\u7684\u72ec\u7acb\u6027\u662f\u9a6c\u5c14\u53ef\u592b\u968f\u673a\u573a\u7684\u6838\u5fc3\u7279\u70b9\u3002<\/p>\n\n\n\n

            \u9a6c\u5c14\u53ef\u592b\u968f\u673a\u573a\u901a\u5e38\u7528\u4e00\u4e2a\u65e0\u5411\u56fe $G=(V,E)$ \u6765\u8868\u793a\uff0c\u5176\u4e2d $V$ \u8868\u793a\u53d8\u91cf\u96c6\u5408\uff0c$E$ \u8868\u793a\u53d8\u91cf\u4e4b\u95f4\u7684\u4f9d\u8d56\u5173\u7cfb\u3002\u56fe\u4e2d\u7684\u6bcf\u4e2a\u8282\u70b9\u8868\u793a\u4e00\u4e2a\u53d8\u91cf\uff0c\u8fb9\u8868\u793a\u53d8\u91cf\u4e4b\u95f4\u7684\u4f9d\u8d56\u5173\u7cfb\u3002\u5982\u679c\u4e24\u4e2a\u53d8\u91cf\u4e4b\u95f4\u6709\u8fb9\u76f8\u8fde\uff0c\u90a3\u4e48\u5b83\u4eec\u4e4b\u95f4\u5c31\u5b58\u5728\u6761\u4ef6\u6982\u7387\u5206\u5e03\u7684\u4f9d\u8d56\u5173\u7cfb\u3002<\/p>\n\n\n\n

            \u9a6c\u5c14\u53ef\u592b\u968f\u673a\u573a\u53ef\u4ee5\u7528\u8054\u5408\u6982\u7387\u5206\u5e03 $P(X)$ \u6765\u63cf\u8ff0\uff0c\u5176\u4e2d $X$ \u8868\u793a\u53d8\u91cf\u96c6\u5408\u3002\u5982\u679c\u56fe\u4e2d\u6240\u6709\u7684\u53d8\u91cf\u90fd\u662f\u79bb\u6563\u578b\u7684\uff0c\u90a3\u4e48\u8054\u5408\u6982\u7387\u5206\u5e03\u53ef\u4ee5\u8868\u793a\u4e3a\uff1a<\/p>\n\n\n\n

            \ufffd(\ufffd)=1\ufffdexp\u2061(\u2212\u2211\ufffd\u2208\ufffd\ufffd\ufffd(\ufffd\ufffd))P<\/em>(X<\/em>)=Z<\/em>1\u200bexp(\u2212\u2211c<\/em>\u2208C<\/em>\u200bV<\/em>c<\/em>\u200b(X<\/em>c<\/em>\u200b))<\/p>\n\n\n\n

            \u5176\u4e2d\uff0c$C$ \u8868\u793a\u56fe\u4e2d\u6240\u6709\u7684\u56e2\uff0c$X_c$ \u8868\u793a\u56e2 $c$ \u4e2d\u7684\u53d8\u91cf\uff0c$V_c(X_c)$ \u8868\u793a\u5728\u56e2 $c$ \u4e2d\u6ee1\u8db3 $X_c$ \u7684\u6982\u7387\u3002$Z$ \u662f\u89c4\u8303\u5316\u5e38\u6570\uff0c\u79f0\u4e3a\u914d\u5206\u51fd\u6570\uff0c\u5b83\u7684\u8ba1\u7b97\u9700\u8981\u5bf9\u6574\u4e2a\u53d8\u91cf\u96c6\u5408\u8fdb\u884c\u679a\u4e3e\uff0c\u56e0\u6b64\u901a\u5e38\u662f\u4e0d\u53ef\u884c\u7684\u3002<\/p>\n\n\n\n

            \u5982\u679c\u56fe\u4e2d\u7684\u53d8\u91cf\u90fd\u662f\u8fde\u7eed\u578b\u7684\uff0c\u90a3\u4e48\u8054\u5408\u6982\u7387\u5206\u5e03\u53ef\u4ee5\u8868\u793a\u4e3a\uff1a<\/p>\n\n\n\n

            \ufffd(\ufffd)=1\ufffdexp\u2061(\u2212\u2211\ufffd\u2208\ufffd\ufffd\ufffd(\ufffd\ufffd))P<\/em>(X<\/em>)=Z<\/em>1\u200bexp(\u2212\u2211c<\/em>\u2208C<\/em>\u200bE<\/em>c<\/em>\u200b(X<\/em>c<\/em>\u200b))<\/p>\n\n\n\n

            \u5176\u4e2d\uff0c$E_c(X_c)$ \u8868\u793a\u5728\u56e2 $c$ \u4e2d\u6ee1\u8db3 $X_c$ \u7684\u80fd\u91cf\u503c\uff0c$Z$ \u662f\u89c4\u8303\u5316\u5e38\u6570\u3002\u548c\u79bb\u6563\u578b\u7684\u60c5\u51b5\u7c7b\u4f3c\uff0c\u8ba1\u7b97\u914d\u5206\u51fd\u6570\u4e5f\u9700\u8981\u5bf9\u6574\u4e2a\u53d8\u91cf\u96c6\u5408\u8fdb\u884c\u679a\u4e3e\uff0c\u901a\u5e38\u662f\u4e0d\u53ef\u884c\u7684\u3002<\/p>\n\n\n\n

            \u9a6c\u5c14\u53ef\u592b\u968f\u673a\u573a\u5728\u56fe\u50cf\u5904\u7406\u3001\u8ba1\u7b97\u673a\u89c6\u89c9\u3001\u81ea\u7136\u8bed\u8a00\u5904\u7406\u7b49\u9886\u57df\u4e2d\u5f97\u5230\u4e86\u5e7f\u6cdb\u5e94\u7528\uff0c\u4f8b\u5982\u56fe\u50cf\u5206\u5272\u3001\u7269\u4f53\u8bc6\u522b<\/p>\n\n\n\n\n\n

            \"gun<\/figure>\n\n\n\n

            \u8bf7\u7ee7\u7eed<\/p>\n\n\n\n

            \u5728\u9a6c\u5c14\u53ef\u592b\u968f\u673a\u573a\u4e2d\uff0c\u6211\u4eec\u5b9a\u4e49\u4e00\u4e2a\u968f\u673a\u53d8\u91cf\u96c6\u5408$X={X_1,X_2,\\ldots,X_n}$\uff0c\u5176\u4e2d\u6bcf\u4e2a\u53d8\u91cf$X_i$\u53ef\u4ee5\u53d6\u503c\u4e8e\u4e00\u4e2a\u6709\u9650\u7684\u72b6\u6001\u7a7a\u95f4$\\mathcal{S}_i$\u3002\u9a6c\u5c14\u53ef\u592b\u968f\u673a\u573a\u8fd8\u6d89\u53ca\u90bb\u63a5\u77e9\u9635$\\mathbf{G}$\uff0c\u5b83\u7528\u4e8e\u63cf\u8ff0\u968f\u673a\u53d8\u91cf\u4e4b\u95f4\u7684\u4f9d\u8d56\u5173\u7cfb\u3002<\/p>\n\n\n\n

            \u5047\u8bbe\u968f\u673a\u53d8\u91cf$X_i$\u548c$X_j$\u662f\u90bb\u5c45\uff0c\u90a3\u4e48\u5b83\u4eec\u4e4b\u95f4\u5b58\u5728\u4e00\u6761\u8fb9$(i,j)\\in E$\u3002\u6211\u4eec\u53ef\u4ee5\u5c06\u90bb\u63a5\u77e9\u9635$\\mathbf{G}$\u5b9a\u4e49\u4e3a\u4e00\u4e2a$n\\times n$\u7684\u77e9\u9635\uff0c\u5176\u4e2d$G_{ij}=1$\u5f53$(i,j)\\in E$\u65f6\uff0c$G_{ij}=0$\u5f53$(i,j)\\notin E$\u65f6\u3002<\/p>\n\n\n\n

            \u73b0\u5728\uff0c\u6211\u4eec\u8003\u8651\u4e00\u4e2a\u72b6\u6001\u5206\u914d$\\mathbf{x}={x_1,x_2,\\ldots,x_n}$\uff0c\u5176\u4e2d\u6bcf\u4e2a$x_i\\in\\mathcal{S}_i$\u3002\u5bf9\u4e8e\u9a6c\u5c14\u53ef\u592b\u968f\u673a\u573a\uff0c\u8054\u5408\u6982\u7387\u5206\u5e03\u53ef\u4ee5\u5b9a\u4e49\u4e3a\uff1a<\/p>\n\n\n\n

            \ufffd(\ufffd)=1\ufffd\u220f\ufffd\u2208\ufffd\ufffd\ufffd(\ufffd\ufffd)p<\/em>(x<\/strong>)=Z<\/em>1\u200b\u220fC<\/em>\u2208C\u200b\u03c8<\/em>C<\/em>\u200b(x<\/strong>C<\/em>\u200b)<\/p>\n\n\n\n

            \u5176\u4e2d$Z$\u662f\u914d\u5206\u51fd\u6570\uff0c$\\mathcal{C}$\u662f\u56e2\u7684\u96c6\u5408\uff0c$\\mathbf{x}_C$\u8868\u793a\u56e2$C$\u4e2d\u7684\u53d8\u91cf\u53d6\u503c\uff0c$\\psi_C(\\mathbf{x}_C)$\u662f\u52bf\u51fd\u6570\u3002<\/p>\n\n\n\n

            \u52bf\u51fd\u6570$\\psi_C(\\mathbf{x}_C)$\u662f\u4e00\u4e2a\u975e\u8d1f\u5b9e\u503c\u51fd\u6570\uff0c\u5b9a\u4e49\u5728\u56e2$\\mathbf{C}$\u7684\u53d8\u91cf\u7ec4\u5408\u4e0a\u3002\u5bf9\u4e8e\u9a6c\u5c14\u53ef\u592b\u968f\u673a\u573a\uff0c\u52bf\u51fd\u6570\u53ef\u4ee5\u5b9a\u4e49\u4e3a\uff1a<\/p>\n\n\n\n

            \ufffd\ufffd(\ufffd\ufffd)=exp\u2061{\u2211\ufffd,\ufffd\u2208\ufffd\ufffd\ufffd\ufffd[\u2009\u2063[\ufffd\ufffd=\ufffd\ufffd]\u2009\u2063]}\u03c8<\/em>C<\/em>\u200b(x<\/strong>C<\/em>\u200b)=exp{\u2211i<\/em>,j<\/em>\u2208C<\/em>\u200bw<\/em>ij<\/em>\u200b[[x<\/em>i<\/em>\u200b=x<\/em>j<\/em>\u200b]]}<\/p>\n\n\n\n

            \u5176\u4e2d\uff0c$w_{ij}$\u662f\u76f8\u90bb\u53d8\u91cf$X_i$\u548c$X_j$\u4e4b\u95f4\u7684\u6743\u91cd\uff0c$[![x_i=x_j]!]$\u662f\u6307\u793a\u51fd\u6570\uff0c\u5f53$x_i=x_j$\u65f6\u53d6\u503c\u4e3a1\uff0c\u5426\u5219\u4e3a0\u3002<\/p>\n\n\n\n

            \u6700\u540e\uff0c\u914d\u5206\u51fd\u6570\u53ef\u4ee5\u5b9a\u4e49\u4e3a\uff1a<\/p>\n\n\n\n

            \ufffd=\u2211\ufffd\u2208\ufffd\u220f\ufffd\u2208\ufffd\ufffd\ufffd(\ufffd\ufffd)Z<\/em>=\u2211x<\/strong>\u2208S\u200b\u220fC<\/em>\u2208C\u200b\u03c8<\/em>C<\/em>\u200b(x<\/strong>C<\/em>\u200b)<\/p>\n\n\n\n

            \u5176\u4e2d$\\mathcal{S}$\u662f\u6240\u6709\u72b6\u6001\u5206\u914d\u7684\u96c6\u5408\u3002<\/p>\n\n\n\n

            \u603b\u4e4b\uff0c\u9a6c\u5c14\u53ef\u592b\u968f\u673a\u573a\u63d0\u4f9b\u4e86\u4e00\u79cd\u7528\u4e8e\u5efa\u6a21\u5177\u6709\u968f\u673a\u6027\u548c\u7ed3\u6784\u6027\u8d28\u7684\u968f\u673a\u53d8\u91cf\u7684\u5f3a\u5927\u5de5\u5177\u3002<\/p>\n\n\n\n\n\n

            \"gun<\/figure>\n\n\n\n

            \u8bf7\u8be6\u7ec6\u89e3\u91ca\u9690\u85cf\u9a6c\u5c14\u53ef\u592b\u6a21\u578b<\/p>\n\n\n\n

            \u9690\u85cf\u9a6c\u5c14\u53ef\u592b\u6a21\u578b\uff08Hidden Markov Model\uff0cHMM\uff09\u662f\u4e00\u79cd\u57fa\u4e8e\u9a6c\u5c14\u53ef\u592b\u8fc7\u7a0b\u7684\u6982\u7387\u6a21\u578b\uff0c\u5e38\u7528\u4e8e\u5e8f\u5217\u6570\u636e\u7684\u5efa\u6a21\uff0c\u5982\u8bed\u97f3\u8bc6\u522b\u3001\u81ea\u7136\u8bed\u8a00\u5904\u7406\u3001\u751f\u7269\u4fe1\u606f\u5b66\u7b49\u9886\u57df\u3002<\/p>\n\n\n\n

            HMM\u7531\u4e24\u90e8\u5206\u7ec4\u6210\uff1a\u89c2\u6d4b\u5e8f\u5217\u548c\u9690\u85cf\u72b6\u6001\u5e8f\u5217\u3002\u89c2\u6d4b\u5e8f\u5217\u662f\u5df2\u77e5\u7684\uff0c\u800c\u9690\u85cf\u72b6\u6001\u5e8f\u5217\u5219\u662f\u4e0d\u53ef\u89c2\u6d4b\u7684\u3002\u6bcf\u4e2a\u65f6\u523b\u90fd\u6709\u4e00\u4e2a\u9690\u85cf\u72b6\u6001\u4e0e\u4e4b\u5bf9\u5e94\uff0c\u800c\u6bcf\u4e2a\u9690\u85cf\u72b6\u6001\u4ea7\u751f\u4e00\u4e2a\u89c2\u6d4b\u503c\u3002\u9690\u85cf\u72b6\u6001\u5e8f\u5217\u548c\u89c2\u6d4b\u5e8f\u5217\u7684\u5173\u7cfb\u7531\u8f6c\u79fb\u6982\u7387\u548c\u53d1\u5c04\u6982\u7387\u6765\u51b3\u5b9a\u3002<\/p>\n\n\n\n

            \u5177\u4f53\u5730\uff0c\u8bbe\u9690\u85cf\u72b6\u6001\u96c6\u5408\u4e3a$S$\uff0c\u89c2\u6d4b\u503c\u96c6\u5408\u4e3a$O$\uff0c\u9690\u85cf\u72b6\u6001\u5e8f\u5217\u4e3a$Q$\uff0c\u89c2\u6d4b\u5e8f\u5217\u4e3a$V$\uff0c\u957f\u5ea6\u4e3a$T$\u3002\u8bbe$A$\u4e3a\u5927\u5c0f\u4e3a$|S|\\times|S|$\u7684\u8f6c\u79fb\u77e9\u9635\uff0c\u5176\u4e2d$A_{i,j}$\u8868\u793a\u4ece\u72b6\u6001$i$\u8f6c\u79fb\u5230\u72b6\u6001$j$\u7684\u6982\u7387\u3002\u8bbe$B$\u4e3a\u5927\u5c0f\u4e3a$|S|\\times|O|$\u7684\u53d1\u5c04\u77e9\u9635\uff0c\u5176\u4e2d$B_{i,j}$\u8868\u793a\u5728\u72b6\u6001$i$\u4e0b\u4ea7\u751f\u89c2\u6d4b\u503c$j$\u7684\u6982\u7387\u3002\u8bbe$\\pi$\u4e3a\u5927\u5c0f\u4e3a$|S|$\u7684\u521d\u59cb\u72b6\u6001\u5411\u91cf\uff0c\u5176\u4e2d$\\pi_i$\u8868\u793a\u6a21\u578b\u7684\u521d\u59cb\u72b6\u6001\u4e3a$i$\u7684\u6982\u7387\u3002<\/p>\n\n\n\n

            HMM\u7684\u751f\u6210\u8fc7\u7a0b\u5982\u4e0b\uff1a<\/p>\n\n\n\n

              \n
            1. \u9009\u62e9\u4e00\u4e2a\u521d\u59cb\u72b6\u6001$i$\uff0c\u6309\u7167$\\pi_i$\u7684\u5206\u5e03\u8fdb\u884c\u968f\u673a\u9009\u62e9\u3002<\/li>\n\n\n\n
            2. \u5bf9\u4e8e\u6bcf\u4e2a\u65f6\u523b$t=1,2,\\cdots,T$\uff0c\u6839\u636e\u5f53\u524d\u72b6\u6001$i$\uff0c\u6309\u7167$A_{i,j}$\u7684\u5206\u5e03\u8fdb\u884c\u72b6\u6001\u8f6c\u79fb\uff0c\u5f97\u5230\u72b6\u6001$j$\u3002<\/li>\n\n\n\n
            3. \u5bf9\u4e8e\u6bcf\u4e2a\u65f6\u523b$t=1,2,\\cdots,T$\uff0c\u6839\u636e\u5f53\u524d\u72b6\u6001$j$\uff0c\u6309\u7167$B_{j,v_t}$\u7684\u5206\u5e03\u4ea7\u751f\u89c2\u6d4b\u503c$v_t$\u3002<\/li>\n\n\n\n
            4. \u91cd\u590d\u6b65\u9aa42\u548c3\uff0c\u76f4\u5230\u4ea7\u751f\u5b8c\u6574\u7684\u89c2\u6d4b\u5e8f\u5217$V$\u3002<\/li>\n<\/ol>\n\n\n\n

              HMM\u7684\u5b66\u4e60\u95ee\u9898\u5c31\u662f\u7ed9\u5b9a\u89c2\u6d4b\u5e8f\u5217$V$\uff0c\u4f30\u8ba1\u6a21\u578b\u7684\u53c2\u6570$A,B,\\pi$\u3002\u5e38\u7528\u7684\u65b9\u6cd5\u662fBaum-Welch\u7b97\u6cd5\uff0c\u4e5f\u79f0\u4e3a\u524d\u5411-\u540e\u5411\u7b97\u6cd5\u3002Baum-Welch\u7b97\u6cd5\u662f\u4e00\u79cd\u57fa\u4e8e\u671f\u671b\u6700\u5927\u5316\uff08Expectation Maximization\uff0cEM\uff09\u7b97\u6cd5\u7684\u8fed\u4ee3\u4f18\u5316\u7b97\u6cd5\uff0c\u53ef\u4ee5\u5728\u4e0d\u77e5\u9053\u9690\u85cf\u72b6\u6001\u5e8f\u5217\u7684\u60c5\u51b5\u4e0b\u4f30\u8ba1\u6a21\u578b\u53c2\u6570\u3002<\/p>\n\n\n\n

              HMM\u7684\u9884\u6d4b\u95ee\u9898\u5c31\u662f\u7ed9\u5b9a\u89c2\u6d4b\u5e8f\u5217$V$\u548c\u6a21\u578b\u53c2\u6570$A,B,\\pi$\uff0c\u6c42\u89e3\u6700\u53ef\u80fd\u7684\u9690\u85cf\u72b6\u6001\u5e8f\u5217$Q$\u3002\u5e38\u7528\u7684\u65b9\u6cd5\u662f\u7ef4\u7279\u6bd4\u7b97\u6cd5\uff0c\u5b83\u5229\u7528\u52a8\u6001\u89c4\u5212\u7684\u601d\u60f3\uff0c\u53ef\u4ee5\u9ad8\u6548\u5730\u6c42\u89e3\u6700\u53ef\u80fd\u7684\u9690\u85cf\u72b6\u6001\u5e8f\u5217\u3002<\/p>\n\n\n\n\n\n

              \"gun<\/figure>\n\n\n\n

              \u8bf7\u8be6\u7ec6\u89e3\u91ca\u8499\u7279\u5361\u6d1b\u91c7\u6837<\/p>\n\n\n\n

              \u8499\u7279\u5361\u6d1b\u91c7\u6837\u662f\u4e00\u79cd\u7528\u4e8e\u4f30\u8ba1\u6982\u7387\u5206\u5e03\u51fd\u6570\u6216\u51fd\u6570\u79ef\u5206\u7684\u65b9\u6cd5\uff0c\u901a\u5e38\u7528\u4e8e\u65e0\u6cd5\u7cbe\u786e\u8ba1\u7b97\u7684\u95ee\u9898\u3002\u5176\u57fa\u672c\u601d\u60f3\u662f\u901a\u8fc7\u4ece\u6982\u7387\u5206\u5e03\u4e2d\u62bd\u53d6\u4e00\u7cfb\u5217\u6837\u672c\u6765\u8fd1\u4f3c\u8ba1\u7b97\u671f\u671b\u503c\u548c\u79ef\u5206\u503c\u3002<\/p>\n\n\n\n

              \u8499\u7279\u5361\u6d1b\u91c7\u6837\u7684\u6838\u5fc3\u662f\u5229\u7528\u968f\u673a\u6837\u672c\u6765\u6a21\u62df\u6982\u7387\u5206\u5e03\u7684\u6027\u8d28\u3002\u5177\u4f53\u6765\u8bf4\uff0c\u8499\u7279\u5361\u6d1b\u91c7\u6837\u7b97\u6cd5\u5305\u62ec\u4ee5\u4e0b\u6b65\u9aa4\uff1a<\/p>\n\n\n\n

                \n
              1. \u4ece\u6982\u7387\u5206\u5e03\u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u7ec4\u6837\u672c\u3002<\/li>\n\n\n\n
              2. \u6839\u636e\u6837\u672c\u8ba1\u7b97\u9700\u8981\u4f30\u8ba1\u7684\u91cf\uff0c\u4f8b\u5982\u671f\u671b\u503c\u3001\u65b9\u5dee\u7b49\u3002<\/li>\n\n\n\n
              3. \u91cd\u590d\u4e0a\u8ff0\u8fc7\u7a0b\u591a\u6b21\uff0c\u4ee5\u83b7\u5f97\u66f4\u51c6\u786e\u7684\u4f30\u8ba1\u7ed3\u679c\u3002<\/li>\n<\/ol>\n\n\n\n

                \u8499\u7279\u5361\u6d1b\u91c7\u6837\u7684\u4f18\u70b9\u662f\u53ef\u4ee5\u7528\u4e8e\u89e3\u51b3\u590d\u6742\u7684\u95ee\u9898\uff0c\u65e0\u8bba\u6982\u7387\u5206\u5e03\u662f\u5426\u5177\u6709\u660e\u786e\u7684\u89e3\u6790\u5f62\u5f0f\u3002\u7f3a\u70b9\u662f\u9700\u8981\u91c7\u6837\u6570\u76ee\u8db3\u591f\u591a\u624d\u80fd\u83b7\u5f97\u51c6\u786e\u7684\u7ed3\u679c\uff0c\u56e0\u6b64\u8ba1\u7b97\u6548\u7387\u53ef\u80fd\u8f83\u4f4e\u3002<\/p>\n\n\n\n

                \u5728\u673a\u5668\u5b66\u4e60\u4e2d\uff0c\u8499\u7279\u5361\u6d1b\u91c7\u6837\u7ecf\u5e38\u7528\u4e8e\u8ba1\u7b97\u540e\u9a8c\u5206\u5e03\u3001\u671f\u671b\u503c\u548c\u79ef\u5206\u7b49\u95ee\u9898\u3002\u5728\u9a6c\u5c14\u53ef\u592b\u94fe\u8499\u7279\u5361\u6d1b\uff08MCMC\uff09\u91c7\u6837\u4e2d\uff0c\u8499\u7279\u5361\u6d1b\u65b9\u6cd5\u88ab\u7528\u4e8e\u4ece\u6982\u7387\u5206\u5e03\u4e2d\u62bd\u53d6\u6837\u672c\uff0c\u800c\u9a6c\u5c14\u53ef\u592b\u94fe\u5219\u7528\u4e8e\u751f\u6210\u6837\u672c\u5e8f\u5217\uff0c\u4ee5\u83b7\u5f97\u76ee\u6807\u5206\u5e03\u7684\u4f30\u8ba1\u3002<\/p>\n\n\n\n\n\n

                \"gun<\/figure>\n\n\n\n

                \u8bf7\u8be6\u7ec6\u89e3\u91ca\u6709\u54ea\u4e9b\u6982\u7387\u5206\u5e03\u51fd\u6570<\/p>\n\n\n\n

                \u6982\u7387\u5206\u5e03\u51fd\u6570\u662f\u63cf\u8ff0\u968f\u673a\u53d8\u91cf\u5206\u5e03\u7684\u51fd\u6570\u3002\u4e0b\u9762\u662f\u4e00\u4e9b\u5e38\u89c1\u7684\u6982\u7387\u5206\u5e03\u51fd\u6570\uff1a<\/p>\n\n\n\n

                  \n
                1. \u4f2f\u52aa\u5229\u5206\u5e03\uff08Bernoulli Distribution\uff09\uff1a\u4f2f\u52aa\u5229\u5206\u5e03\u662f\u6307\u5728\u4e00\u4e2a\u53ea\u6709\u4e24\u79cd\u53ef\u80fd\u7ed3\u679c\u7684\u968f\u673a\u8bd5\u9a8c\u4e2d\uff0c\u67d0\u4e00\u79cd\u7ed3\u679c\u51fa\u73b0\u7684\u6982\u7387\u4e3ap\uff0c\u53e6\u4e00\u79cd\u7ed3\u679c\u51fa\u73b0\u7684\u6982\u7387\u4e3a1-p\u3002\u4f2f\u52aa\u5229\u5206\u5e03\u7684\u6982\u7387\u8d28\u91cf\u51fd\u6570\u4e3a\uff1a P(X=k) = p^k * (1-p)^(1-k) (k=0,1)<\/li>\n\n\n\n
                2. \u4e8c\u9879\u5206\u5e03\uff08Binomial Distribution\uff09\uff1a\u4e8c\u9879\u5206\u5e03\u662f\u6307\u5728n\u4e2a\u72ec\u7acb\u7684\u662f\/\u975e\u8bd5\u9a8c\u4e2d\u6210\u529f\u6b21\u6570\u7684\u6982\u7387\u5206\u5e03\uff0c\u5176\u4e2d\u6bcf\u6b21\u8bd5\u9a8c\u6210\u529f\u7684\u6982\u7387\u4e3ap\u3002\u4e8c\u9879\u5206\u5e03\u7684\u6982\u7387\u8d28\u91cf\u51fd\u6570\u4e3a\uff1a P(X=k) = C(n,k) * p^k * (1-p)^(n-k) (k=0,1,…,n)<\/li>\n\n\n\n
                3. \u6cca\u677e\u5206\u5e03\uff08Poisson Distribution\uff09\uff1a\u6cca\u677e\u5206\u5e03\u662f\u6307\u5728\u5355\u4f4d\u65f6\u95f4\uff08\u6216\u5355\u4f4d\u9762\u79ef\uff09\u5185\u67d0\u4e8b\u4ef6\u53d1\u751f\u7684\u6b21\u6570\u670d\u4ece\u7684\u6982\u7387\u5206\u5e03\uff0c\u5047\u8bbe\u4e8b\u4ef6\u5728\u4e00\u4e2a\u56fa\u5b9a\u7684\u65f6\u95f4\u95f4\u9694\u5185\u53d1\u751f\u7684\u5e73\u5747\u6b21\u6570\u4e3a\u03bb\uff0c\u90a3\u4e48\u6cca\u677e\u5206\u5e03\u7684\u6982\u7387\u8d28\u91cf\u51fd\u6570\u4e3a\uff1a P(X=k) = e^(-\u03bb) * \u03bb^k \/ k!<\/li>\n\n\n\n
                4. \u5747\u5300\u5206\u5e03\uff08Uniform Distribution\uff09\uff1a\u5747\u5300\u5206\u5e03\u662f\u6307\u5728\u4e00\u6bb5\u533a\u95f4\u5185\u4efb\u4f55\u4e00\u4e2a\u503c\u51fa\u73b0\u7684\u6982\u7387\u662f\u76f8\u7b49\u7684\u3002\u5747\u5300\u5206\u5e03\u7684\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\u4e3a\uff1a f(x) = 1\/(b-a) (a<=x<=b)<\/li>\n\n\n\n
                5. \u6b63\u6001\u5206\u5e03\uff08Normal Distribution\uff09\uff1a\u6b63\u6001\u5206\u5e03\u662f\u4e00\u79cd\u5e38\u89c1\u7684\u8fde\u7eed\u6982\u7387\u5206\u5e03\uff0c\u4e5f\u79f0\u4e3a\u9ad8\u65af\u5206\u5e03\u3002\u5728\u81ea\u7136\u754c\u4e2d\uff0c\u8bb8\u591a\u968f\u673a\u53d8\u91cf\u90fd\u5177\u6709\u6b63\u6001\u5206\u5e03\uff0c\u4f8b\u5982\u8eab\u9ad8\u3001\u4f53\u91cd\u7b49\u3002\u6b63\u6001\u5206\u5e03\u7684\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\u4e3a\uff1a f(x) = 1\/(\u03c3\u221a(2\u03c0)) * e^(-(x-\u03bc)^2\/(2\u03c3^2))<\/li>\n\n\n\n
                6. \u6307\u6570\u5206\u5e03\uff08Exponential Distribution\uff09\uff1a\u6307\u6570\u5206\u5e03\u662f\u63cf\u8ff0\u7b49\u5f85\u65f6\u95f4\u7684\u6982\u7387\u5206\u5e03\uff0c\u4f8b\u5982\u7b49\u5f85\u4e0b\u4e00\u6b21\u5730\u9707\u7684\u65f6\u95f4\uff0c\u7b49\u5f85\u4e0b\u4e00\u6b21\u7535\u8bdd\u7684\u65f6\u95f4\u7b49\u3002\u6307\u6570\u5206\u5e03\u7684\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\u4e3a\uff1a f(x) = \u03bb * e^(-\u03bbx) (x>=0)<\/li>\n<\/ol>\n\n\n\n

                  \u4ee5\u4e0a\u662f\u5e38\u89c1\u7684\u51e0\u79cd\u6982\u7387\u5206\u5e03\u51fd\u6570\uff0c\u4e0d\u540c\u7684\u6982\u7387\u5206\u5e03\u51fd\u6570\u9002\u7528\u4e8e\u4e0d\u540c\u7684\u968f\u673a\u53d8\u91cf\u3002<\/p>\n\n\n\n\n\n

                  \"gun<\/figure>\n\n\n\n

                  \u8bf7\u8be6\u7ec6\u89e3\u91ca\u4ec0\u4e48\u662f\u540e\u9a8c\u5206\u5e03\uff0c\u5982\u4f55\u7528\u8499\u7279\u5361\u6d1b\u91c7\u6837\u8ba1\u7b97\u540e\u9a8c\u5206\u5e03<\/p>\n\n\n\n

                  \u540e\u9a8c\u5206\u5e03\u662f\u6307\u7ed9\u5b9a\u89c2\u5bdf\u6570\u636e\u548c\u6a21\u578b\u53c2\u6570\u7684\u60c5\u51b5\u4e0b\uff0c\u6a21\u578b\u53c2\u6570\u7684\u6982\u7387\u5206\u5e03\u3002\u5728\u8d1d\u53f6\u65af\u7edf\u8ba1\u4e2d\uff0c\u540e\u9a8c\u5206\u5e03\u662f\u8d1d\u53f6\u65af\u5b9a\u7406\u4e2d\u7684\u5206\u5b50\uff0c\u8868\u793a\u6a21\u578b\u53c2\u6570\u7684\u4e0d\u786e\u5b9a\u6027\u3002\u901a\u5e38\u60c5\u51b5\u4e0b\uff0c\u6211\u4eec\u9700\u8981\u5bf9\u540e\u9a8c\u5206\u5e03\u8fdb\u884c\u8ba1\u7b97\u548c\u5206\u6790\uff0c\u4ee5\u4fbf\u4e8e\u6211\u4eec\u5bf9\u6a21\u578b\u8fdb\u884c\u63a8\u65ad\u548c\u9884\u6d4b\u3002<\/p>\n\n\n\n

                  \u8499\u7279\u5361\u6d1b\u91c7\u6837\u662f\u4e00\u79cd\u7528\u4e8e\u4f30\u8ba1\u540e\u9a8c\u5206\u5e03\u7684\u65b9\u6cd5\u3002\u8be5\u65b9\u6cd5\u7684\u57fa\u672c\u601d\u60f3\u662f\u6839\u636e\u67d0\u4e2a\u6982\u7387\u5206\u5e03\u51fd\u6570\uff0c\u4ee5\u968f\u673a\u91c7\u6837\u7684\u65b9\u5f0f\u6765\u751f\u6210\u4e00\u7ec4\u6837\u672c\u3002\u5728\u8ba1\u7b97\u540e\u9a8c\u5206\u5e03\u65f6\uff0c\u53ef\u4ee5\u901a\u8fc7\u8fd9\u4e9b\u6837\u672c\u6765\u8fd1\u4f3c\u8868\u793a\u540e\u9a8c\u5206\u5e03\u3002\u901a\u5e38\u60c5\u51b5\u4e0b\uff0c\u6211\u4eec\u4f7f\u7528\u9a6c\u5c14\u53ef\u592b\u94fe\u8499\u7279\u5361\u6d1b(MCMC)\u65b9\u6cd5\u6765\u8fdb\u884c\u91c7\u6837\u3002\u8be5\u65b9\u6cd5\u5229\u7528\u9a6c\u5c14\u53ef\u592b\u94fe\u7684\u6536\u655b\u6027\u8d28\uff0c\u4ece\u67d0\u4e2a\u8d77\u59cb\u72b6\u6001\u51fa\u53d1\uff0c\u901a\u8fc7\u591a\u6b21\u8fed\u4ee3\uff0c\u4f7f\u5f97\u9a6c\u5c14\u53ef\u592b\u94fe\u9010\u6e10\u6536\u655b\u4e8e\u540e\u9a8c\u5206\u5e03\u3002\u901a\u8fc7\u8fd9\u79cd\u65b9\u5f0f\uff0c\u6211\u4eec\u53ef\u4ee5\u5f97\u5230\u4e00\u7ec4\u8fd1\u4f3c\u4e8e\u540e\u9a8c\u5206\u5e03\u7684\u6837\u672c\u3002<\/p>\n\n\n\n

                  \u5728\u4f7f\u7528MCMC\u65b9\u6cd5\u8fdb\u884c\u91c7\u6837\u65f6\uff0c\u6211\u4eec\u9700\u8981\u5b9a\u4e49\u4e00\u4e2a\u9a6c\u5c14\u53ef\u592b\u94fe\uff0c\u4f7f\u5176\u5177\u6709\u6240\u9700\u7684\u5e73\u7a33\u5206\u5e03\uff0c\u5373\u540e\u9a8c\u5206\u5e03\u3002\u901a\u5e38\u60c5\u51b5\u4e0b\uff0c\u6211\u4eec\u9009\u62e9Metropolis-Hastings\u7b97\u6cd5\uff0c\u8be5\u7b97\u6cd5\u662f\u4e00\u79cd\u8457\u540d\u7684MCMC\u7b97\u6cd5\u3002\u8be5\u7b97\u6cd5\u6839\u636e\u67d0\u4e2a\u8f6c\u79fb\u6982\u7387\u5206\u5e03\uff0c\u4ece\u5f53\u524d\u72b6\u6001\u8fdb\u884c\u62bd\u6837\uff0c\u751f\u6210\u4e00\u4e2a\u65b0\u7684\u72b6\u6001\u3002\u7136\u540e\u901a\u8fc7\u63a5\u53d7\u6216\u8005\u62d2\u7edd\u7684\u65b9\u5f0f\u6765\u51b3\u5b9a\u662f\u5426\u63a5\u53d7\u65b0\u72b6\u6001\u3002\u901a\u8fc7\u591a\u6b21\u8fed\u4ee3\uff0c\u6211\u4eec\u53ef\u4ee5\u5f97\u5230\u4e00\u7ec4\u8fd1\u4f3c\u4e8e\u540e\u9a8c\u5206\u5e03\u7684\u6837\u672c\uff0c\u4ece\u800c\u53ef\u4ee5\u5bf9\u540e\u9a8c\u5206\u5e03\u8fdb\u884c\u8fd1\u4f3c\u8ba1\u7b97\u3002<\/p>\n\n\n\n\n\n

                  \"gun<\/figure>\n\n\n\n

                  \u8bf7\u7ed9\u51fa\u8be6\u7ec6\u7684\u4ee3\u7801\u793a\u4f8b<\/p>\n\n\n\n

                  \u4ee5\u4e0b\u662f\u4e00\u4e2a\u4f7f\u7528\u8499\u7279\u5361\u6d1b\u91c7\u6837\u65b9\u6cd5\u6765\u4f30\u8ba1\u67d0\u4e2a\u5206\u5e03\u7684\u540e\u9a8c\u5206\u5e03\u7684 Python \u4ee3\u7801\u793a\u4f8b\uff1a<\/p>\n\n\n\n

                  pythonCopy codeimport numpy as np\n\n# \u5047\u8bbe\u6211\u4eec\u8981\u4f30\u8ba1\u7684\u662f\u4e00\u4e2a\u4e8c\u5143\u9ad8\u65af\u5206\u5e03\u7684\u540e\u9a8c\u5206\u5e03\uff0c\u5373 p(x1,x2|D)\n# \u5176\u4e2d D \u662f\u89c2\u6d4b\u6570\u636e\uff0c\u6211\u4eec\u5047\u8bbe\u5148\u9a8c\u5206\u5e03 p(x1,x2) \u662f\u4e00\u4e2a\u6807\u51c6\u6b63\u6001\u5206\u5e03\n\n# \u5b9a\u4e49\u6a21\u578b\u53c2\u6570\nmu = np.array([0, 0])\nsigma = np.array([[1, 0.8], [0.8, 1]])  # \u534f\u65b9\u5dee\u77e9\u9635\n\n# \u5b9a\u4e49\u89c2\u6d4b\u6570\u636e\nD = np.array([[1, 2], [2, 1], [2, 3], [3, 2]])\n\n# \u5b9a\u4e49\u8499\u7279\u5361\u6d1b\u91c7\u6837\u51fd\u6570\ndef monte_carlo_sampling(pdf, num_samples):\n    # \u91c7\u6837 num_samples \u4e2a\u6837\u672c\n    samples = np.zeros((num_samples, pdf.ndim))\n    for i in range(num_samples):\n        x = np.random.normal(size=pdf.ndim)\n        while pdf(x) == 0:\n            x = np.random.normal(size=pdf.ndim)\n        samples[i] = x\n    return samples\n\n# \u5b9a\u4e49\u4e8c\u5143\u9ad8\u65af\u5206\u5e03\u7684\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\ndef gaussian_pdf(x):\n    return 1 \/ np.sqrt(np.linalg.det(2 * np.pi * sigma)) * np.exp(-0.5 * np.dot(x - mu, np.linalg.solve(sigma, x - mu)))\n\n# \u5b9a\u4e49\u540e\u9a8c\u5206\u5e03\u7684\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\ndef posterior_pdf(x):\n    return gaussian_pdf(x) * np.prod([gaussian_pdf(xi) for xi in D])\n\n# \u4f7f\u7528\u8499\u7279\u5361\u6d1b\u91c7\u6837\u6765\u4f30\u8ba1\u540e\u9a8c\u5206\u5e03\nnum_samples = 100000\nsamples = monte_carlo_sampling(posterior_pdf, num_samples)\n\n# \u7edf\u8ba1\u6837\u672c\u7684\u5206\u5e03\u60c5\u51b5\nx1_samples = samples[:, 0]\nx2_samples = samples[:, 1]\nmean = np.mean(samples, axis=0)\ncovariance = np.cov(samples, rowvar=False)\n\n# \u8f93\u51fa\u7ed3\u679c\nprint(\"\u540e\u9a8c\u5206\u5e03\u7684\u5747\u503c\uff1a\", mean)\nprint(\"\u540e\u9a8c\u5206\u5e03\u7684\u534f\u65b9\u5dee\u77e9\u9635\uff1a\", covariance)\n<\/code><\/pre>\n\n\n\n
                  \u5728\u8fd9\u4e2a\u4ee3\u7801\u793a\u4f8b\u4e2d\uff0c\u6211\u4eec\u5b9a\u4e49\u4e86\u4e00\u4e2a\u4e8c\u5143\u9ad8\u65af\u5206\u5e03\u4f5c\u4e3a\u5148\u9a8c\u5206\u5e03\uff0c\u540c\u65f6\u7ed9\u51fa\u4e86\u4e00\u4e9b\u89c2\u6d4b\u6570\u636e D\u3002\u6211\u4eec\u4f7f\u7528\u8499\u7279\u5361\u6d1b\u91c7\u6837\u65b9\u6cd5\u6765\u4f30\u8ba1\u8fd9\u4e2a\u5148\u9a8c\u5206\u5e03\u548c\u89c2\u6d4b\u6570\u636e\u7684\u540e\u9a8c\u5206\u5e03\u3002\u5728 monte_carlo_sampling<\/code> \u51fd\u6570\u4e2d\uff0c\u6211\u4eec\u4f7f\u7528\u6807\u51c6\u6b63\u6001\u5206\u5e03\u6765\u751f\u6210\u91c7\u6837\u6837\u672c\u3002\u5728 gaussian_pdf<\/code> \u51fd\u6570\u4e2d\uff0c\u6211\u4eec\u5b9a\u4e49\u4e86\u4e8c\u5143\u9ad8\u65af\u5206\u5e03\u7684\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\uff0c\u5b83\u5c06\u88ab\u7528\u6765\u8ba1\u7b97\u5148\u9a8c\u5206\u5e03\u548c\u540e\u9a8c\u5206\u5e03\u3002\u5728 posterior_pdf<\/code> \u51fd\u6570\u4e2d\uff0c\u6211\u4eec\u6839\u636e\u8d1d\u53f6\u65af\u516c\u5f0f\u5b9a\u4e49\u4e86\u540e\u9a8c\u5206\u5e03\u7684\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\u3002\u6700\u540e\uff0c\u6211\u4eec\u4f7f\u7528 monte_carlo_sampling<\/code> \u51fd\u6570\u6765\u91c7\u6837\u4e00\u4e9b\u6837\u672c\uff0c\u5e76\u8ba1\u7b97\u6837\u672c<\/p>\n\n\n\n\n\n
                  \"gun<\/figure>\n\n\n\n
                  \u5982\u4f55\u5b9e\u73b0\u9a6c\u5c14\u53ef\u592b\u94fe\uff0c\u8bf7\u7ed9\u51fa\u8be6\u7ec6\u7684\u4ee3\u7801<\/p>\n\n\n\n
                  \u9a6c\u5c14\u53ef\u592b\u94fe\u662f\u4e00\u79cd\u6982\u7387\u6a21\u578b\uff0c\u7528\u4e8e\u63cf\u8ff0\u968f\u673a\u8fc7\u7a0b\uff0c\u5176\u4e2d\u6bcf\u4e2a\u72b6\u6001\u53ea\u4f9d\u8d56\u4e8e\u524d\u4e00\u4e2a\u72b6\u6001\u3002\u5b9e\u73b0\u9a6c\u5c14\u53ef\u592b\u94fe\u9700\u8981\u5b9a\u4e49\u72b6\u6001\u7a7a\u95f4\uff0c\u8f6c\u79fb\u6982\u7387\u77e9\u9635\u548c\u521d\u59cb\u72b6\u6001\u5206\u5e03\u3002\u4ee5\u4e0b\u662f\u4e00\u4e2aPython\u5b9e\u73b0\u7684\u7b80\u5355\u793a\u4f8b\uff1a<\/p>\n\n\n\n
                  pythonCopy codeimport numpy as np\n\n# \u5b9a\u4e49\u72b6\u6001\u7a7a\u95f4\nstates = [\"A\", \"B\", \"C\", \"D\", \"E\"]\n\n# \u5b9a\u4e49\u8f6c\u79fb\u6982\u7387\u77e9\u9635\ntransition_matrix = np.array([\n    [0, 0.5, 0.5, 0, 0],\n    [0.25, 0, 0.25, 0.25, 0.25],\n    [0.5, 0, 0, 0.5, 0],\n    [0, 0.25, 0.25, 0, 0.5],\n    [0, 0, 0.5, 0.5, 0]\n])\n\n# \u5b9a\u4e49\u521d\u59cb\u72b6\u6001\u5206\u5e03\ninitial_distribution = np.array([0.2, 0.2, 0.2, 0.2, 0.2])\n\n# \u5b9e\u73b0\u9a6c\u5c14\u53ef\u592b\u94fe\ndef markov_chain(states, transition_matrix, initial_distribution, num_steps):\n    current_state = np.random.choice(states, p=initial_distribution)\n    states_visited = [current_state]\n    for i in range(num_steps):\n        current_state = np.random.choice(states, p=transition_matrix[states.index(current_state)])\n        states_visited.append(current_state)\n    return states_visited\n\n# \u751f\u6210\u4e00\u4e2a\u9a6c\u5c14\u53ef\u592b\u94fe\uff0c\u5305\u62ec10\u4e2a\u72b6\u6001\nchain = markov_chain(states, transition_matrix, initial_distribution, 10)\nprint(chain)\n<\/code><\/pre>\n\n\n\n
                  \u5728\u8fd9\u4e2a\u793a\u4f8b\u4e2d\uff0c\u6211\u4eec\u9996\u5148\u5b9a\u4e49\u4e86\u72b6\u6001\u7a7a\u95f4\u3001\u8f6c\u79fb\u6982\u7387\u77e9\u9635\u548c\u521d\u59cb\u72b6\u6001\u5206\u5e03\u3002\u7136\u540e\uff0c\u6211\u4eec\u5b9e\u73b0\u4e86\u4e00\u4e2amarkov_chain<\/code>\u51fd\u6570\u6765\u751f\u6210\u9a6c\u5c14\u53ef\u592b\u94fe\u3002\u5728\u51fd\u6570\u4e2d\uff0c\u6211\u4eec\u4ece\u521d\u59cb\u72b6\u6001\u5f00\u59cb\uff0c\u4f9d\u636e\u8f6c\u79fb\u6982\u7387\u77e9\u9635\u8fdb\u884c\u72b6\u6001\u8f6c\u79fb\uff0c\u751f\u6210\u5305\u542b\u6307\u5b9a\u6570\u91cf\u72b6\u6001\u7684\u9a6c\u5c14\u53ef\u592b\u94fe\u3002\u6700\u540e\uff0c\u6211\u4eec\u8c03\u7528markov_chain<\/code>\u51fd\u6570\u751f\u6210\u4e86\u4e00\u4e2a\u5305\u62ec10\u4e2a\u72b6\u6001\u7684\u9a6c\u5c14\u53ef\u592b\u94fe\u5e76\u6253\u5370\u8f93\u51fa\u3002<\/p>\n\n\n\n\n\n
                  \"gun<\/figure>\n\n\n\n
                  \u5982\u4f55\u5b9e\u73b0\u9a6c\u5c14\u53ef\u592b\u968f\u673a\u573a\uff0c\u8bf7\u7ed9\u51fa\u8be6\u7ec6\u7684\u4ee3\u7801\u793a\u4f8b<\/p>\n\n\n\n
                  \u9a6c\u5c14\u53ef\u592b\u968f\u673a\u573a\u53ef\u4ee5\u7528\u6982\u7387\u56fe\u6a21\u578b\u6765\u8868\u793a\uff0c\u53ef\u4ee5\u4f7f\u7528Python\u7684\u7b2c\u4e09\u65b9\u5e93\u5982PyMC3\u3001Stan\u7b49\u6765\u6784\u5efa\u548c\u63a8\u65ad\u9a6c\u5c14\u53ef\u592b\u968f\u673a\u573a\u3002\u4e0b\u9762\u4ee5PyMC3\u4e3a\u4f8b\uff0c\u7ed9\u51fa\u4e00\u4e2a\u7b80\u5355\u7684\u4ee3\u7801\u793a\u4f8b\u3002<\/p>\n\n\n\n
                  \u5047\u8bbe\u67095\u4e2a\u4e8c\u5143\u53d8\u91cfX1,X2,X3,X4,X5\uff0c\u5176\u72b6\u6001\u7a7a\u95f4\u4e3a{0,1}\uff0c\u9700\u8981\u6784\u5efa\u4e00\u4e2a\u9a6c\u5c14\u53ef\u592b\u968f\u673a\u573a\u6765\u63cf\u8ff0\u8fd9\u4e9b\u53d8\u91cf\u4e4b\u95f4\u7684\u5173\u7cfb\u3002\u5177\u4f53\u5b9e\u73b0\u6b65\u9aa4\u5982\u4e0b\uff1a<\/p>\n\n\n\n
                  1.\u5bfc\u5165\u5fc5\u8981\u7684\u5e93<\/p>\n\n\n\n
                  javascriptCopy codeimport numpy as np\nimport pymc3 as pm\n<\/code><\/pre>\n\n\n\n
                  2.\u5b9a\u4e49\u72b6\u6001\u7a7a\u95f4\u548c\u6982\u7387\u5206\u5e03<\/p>\n\n\n\n
                  pythonCopy code# \u5b9a\u4e49\u72b6\u6001\u7a7a\u95f4\nstates = [0, 1]\n\n# \u5b9a\u4e49\u6982\u7387\u5206\u5e03\ndef p(x1, x2, x3, x4, x5):\n    energy = 0\n    energy += 2*x1 - x2 - x3\n    energy += -x2 + x3 - x4\n    energy += -x3 - x5\n    return np.exp(energy)\n<\/code><\/pre>\n\n\n\n
                  3.\u6784\u5efa\u6a21\u578b<\/p>\n\n\n\n
                  scssCopy codewith pm.Model() as model:\n    # \u5b9a\u4e49\u6a21\u578b\u53c2\u6570\n    x1 = pm.Categorical('x1', p=[0.5, 0.5])\n    x2 = pm.Categorical('x2', p=[0.5, 0.5])\n    x3 = pm.Categorical('x3', p=[0.5, 0.5])\n    x4 = pm.Categorical('x4', p=[0.5, 0.5])\n    x5 = pm.Categorical('x5', p=[0.5, 0.5])\n\n    # \u5b9a\u4e49\u52bf\u80fd\u51fd\u6570\n    energy = pm.math.sum([\n        2*x1 - x2 - x3,\n        -x2 + x3 - x4,\n        -x3 - x5\n    ])\n\n    # \u5b9a\u4e49\u4f3c\u7136\u51fd\u6570\n    pm.Potential('potential', energy)\n\n    # \u5b9a\u4e49\u540e\u9a8c\u5206\u5e03\n    trace = pm.sample(5000)\n<\/code><\/pre>\n\n\n\n
                  \u5728\u4e0a\u9762\u7684\u4ee3\u7801\u4e2d\uff0c\u4f7f\u7528Categorical\u5206\u5e03\u6765\u5b9a\u4e49\u6bcf\u4e2a\u53d8\u91cf\u7684\u72b6\u6001\uff0c\u5e76\u4f7f\u7528Potential\u51fd\u6570\u5b9a\u4e49\u52bf\u80fd\u51fd\u6570\u3002\u7136\u540e\u4f7f\u7528PyMC3\u7684sample\u51fd\u6570\u8fdb\u884c\u9a6c\u5c14\u53ef\u592b\u94fe\u8499\u7279\u5361\u6d1b\u91c7\u6837\uff0c\u5f97\u5230\u540e\u9a8c\u5206\u5e03trace\u3002<\/p>\n\n\n\n
                  \u901a\u8fc7\u8fd9\u4e2a\u4f8b\u5b50\uff0c\u53ef\u4ee5\u770b\u5230PyMC3\u63d0\u4f9b\u4e86\u65b9\u4fbf\u6613\u7528\u7684API\uff0c\u53ef\u4ee5\u8f7b\u677e\u6784\u5efa\u548c\u63a8\u65ad\u590d\u6742\u7684\u6982\u7387\u6a21\u578b\u3002<\/p>\n\n\n\n\n\n
                  \"gun<\/figure>\n\n\n\n
                  \u5982\u4f55\u5b9e\u73b0\u9690\u85cf\u9a6c\u5c14\u53ef\u592b\u94fe\uff0c\u8bf7\u7ed9\u51fa\u8be6\u7ec6\u7684\u4ee3\u7801\u793a\u4f8b<\/p>\n\n\n\n
                  \u4e0b\u9762\u662f\u4e00\u4e2a\u7b80\u5355\u7684Python\u4ee3\u7801\u793a\u4f8b\uff0c\u7528\u4e8e\u5b9e\u73b0\u4e00\u4e2a\u7b80\u5355\u7684\u9690\u9a6c\u5c14\u53ef\u592b\u6a21\u578b\uff1a<\/p>\n\n\n\n
                  pythonCopy codeimport numpy as np\n\nclass HiddenMarkovModel:\n    def __init__(self, A, B, pi):\n        self.A = A  # \u8f6c\u79fb\u77e9\u9635\n        self.B = B  # \u89c2\u6d4b\u77e9\u9635\n        self.pi = pi  # \u521d\u59cb\u6982\u7387\n\n    def generate_observation_sequence(self, n):\n        # \u751f\u6210\u957f\u5ea6\u4e3an\u7684\u89c2\u6d4b\u5e8f\u5217\n        state = np.random.choice(len(self.pi), p=self.pi)\n        obs = []\n        for i in range(n):\n            obs.append(np.random.choice(len(self.B[state]), p=self.B[state]))\n            state = np.random.choice(len(self.A[state]), p=self.A[state])\n        return obs\n<\/code><\/pre>\n\n\n\n
                  \u5728\u8fd9\u4e2a\u4ee3\u7801\u793a\u4f8b\u4e2d\uff0cHiddenMarkovModel<\/code>\u7c7b\u8868\u793a\u4e00\u4e2a\u9690\u9a6c\u5c14\u53ef\u592b\u6a21\u578b\u3002\u5b83\u6709\u4e09\u4e2a\u53c2\u6570\uff1a\u8f6c\u79fb\u77e9\u9635A<\/code>\u3001\u89c2\u6d4b\u77e9\u9635B<\/code>\u548c\u521d\u59cb\u6982\u7387pi<\/code>\u3002\u5176\u4e2d\uff0cA<\/code>\u662f\u4e00\u4e2a\u4e8c\u7ef4\u6570\u7ec4\uff0c\u8868\u793a\u4ece\u4e00\u4e2a\u72b6\u6001\u8f6c\u79fb\u5230\u53e6\u4e00\u4e2a\u72b6\u6001\u7684\u6982\u7387\u3002B<\/code>\u4e5f\u662f\u4e00\u4e2a\u4e8c\u7ef4\u6570\u7ec4\uff0c\u8868\u793a\u5728\u6bcf\u4e2a\u72b6\u6001\u4e0b\uff0c\u751f\u6210\u4e0d\u540c\u89c2\u6d4b\u7684\u6982\u7387\u3002pi<\/code>\u662f\u4e00\u4e2a\u4e00\u7ef4\u6570\u7ec4\uff0c\u8868\u793a\u6a21\u578b\u7684\u521d\u59cb\u72b6\u6001\u6982\u7387\u5206\u5e03\u3002<\/p>\n\n\n\n
                  \u5728\u8fd9\u4e2a\u7c7b\u4e2d\uff0cgenerate_observation_sequence<\/code>\u65b9\u6cd5\u7528\u4e8e\u751f\u6210\u4e00\u4e2a\u7ed9\u5b9a\u957f\u5ea6\u7684\u89c2\u6d4b\u5e8f\u5217\u3002\u5b83\u4ece\u521d\u59cb\u72b6\u6001\u5f00\u59cb\uff0c\u6839\u636e\u8f6c\u79fb\u6982\u7387\u548c\u89c2\u6d4b\u6982\u7387\u751f\u6210\u4e00\u4e2a\u72b6\u6001\u548c\u89c2\u6d4b\uff0c\u5e76\u6839\u636e\u72b6\u6001\u7ee7\u7eed\u751f\u6210\u4e0b\u4e00\u4e2a\u72b6\u6001\u548c\u89c2\u6d4b\uff0c\u76f4\u5230\u751f\u6210\u4e00\u4e2a\u7ed9\u5b9a\u957f\u5ea6\u7684\u89c2\u6d4b\u5e8f\u5217\u3002<\/p>\n\n\n\n
                  \u5728\u5b9e\u9645\u5e94\u7528\u4e2d\uff0c\u6211\u4eec\u8fd8\u9700\u8981\u4f7f\u7528\u4e00\u4e9b\u7b97\u6cd5\u6765\u4ece\u89c2\u6d4b\u5e8f\u5217\u4e2d\u4f30\u8ba1\u6a21\u578b\u7684\u53c2\u6570\uff0c\u4f8b\u5982\u524d\u5411-\u540e\u5411\u7b97\u6cd5\u6216\u7ef4\u7279\u6bd4\u7b97\u6cd5\u3002<\/p>\n\n\n\n\n\n
                  \"gun<\/figure>\n\n\n\n
                  \u8bf7\u8be6\u7ec6\u63cf\u8ff0\u5409\u5e03\u65af\u91c7\u6837<\/p>\n\n\n\n
                  \u5409\u5e03\u65af\u91c7\u6837\uff08Gibbs sampling\uff09\u662f\u4e00\u79cd\u8499\u7279\u5361\u7f57\u9a6c\u5c14\u53ef\u592b\u94fe\u8499\u7279\u5361\u7f57\uff08MCMC\uff09\u65b9\u6cd5\uff0c\u7528\u4e8e\u4ece\u591a\u7ef4\u6982\u7387\u5206\u5e03\u4e2d\u91c7\u6837\u3002\u5409\u5e03\u65af\u91c7\u6837\u7684\u601d\u60f3\u662f\u901a\u8fc7\u5728\u7ed9\u5b9a\u5176\u4ed6\u53d8\u91cf\u7684\u60c5\u51b5\u4e0b\u4f9d\u6b21\u91c7\u6837\u6bcf\u4e2a\u53d8\u91cf\u6765\u751f\u6210\u6837\u672c\u3002<\/p>\n\n\n\n
                  \u5177\u4f53\u6765\u8bf4\uff0c\u5bf9\u4e8e\u4e00\u4e2a\u591a\u7ef4\u8054\u5408\u5206\u5e03 $P(x_1, x_2, …, x_n)$\uff0c\u5409\u5e03\u65af\u91c7\u6837\u5c06\u6bcf\u4e2a\u53d8\u91cf $x_i$ \u89c6\u4e3a\u4e00\u4e2a\u8282\u70b9\uff0c\u6839\u636e\u7ed9\u5b9a\u5176\u4ed6\u53d8\u91cf\u7684\u6761\u4ef6\u5206\u5e03 $P(x_i \\mid x_{-i})$ \u4f9d\u6b21\u8fdb\u884c\u91c7\u6837\u3002\u5176\u4e2d $x_{-i}$ \u8868\u793a\u9664 $x_i$ \u5916\u7684\u5176\u4ed6\u53d8\u91cf\u3002<\/p>\n\n\n\n
                  \u5409\u5e03\u65af\u91c7\u6837\u7684\u6b65\u9aa4\u5982\u4e0b\uff1a<\/p>\n\n\n\n
                    \n
                  1. \u521d\u59cb\u5316\u6240\u6709\u53d8\u91cf\u7684\u503c\uff0c\u4f8b\u5982 $x_1^{(0)}, x_2^{(0)}, …, x_n^{(0)}$\u3002<\/li>\n\n\n\n
                  2. \u5bf9\u4e8e\u6bcf\u4e2a\u91c7\u6837\u8f6e\u6b21 $t=1,2,…,T$\uff1aa. \u5bf9\u4e8e\u6bcf\u4e2a\u53d8\u91cf $x_i$\uff0c\u6839\u636e\u6761\u4ef6\u5206\u5e03 $P(x_i \\mid x_{-i}^{(t)})$ \u91c7\u6837\u4e00\u4e2a\u65b0\u7684\u503c $x_i^{(t+1)}$\u3002b. \u5c06\u65b0\u91c7\u6837\u7684\u53d8\u91cf\u503c $x_1^{(t+1)}, x_2^{(t+1)}, …, x_n^{(t+1)}$ \u4f5c\u4e3a\u4e0b\u4e00\u8f6e\u7684\u521d\u59cb\u5316\u503c\u3002<\/li>\n\n\n\n
                  3. \u5bf9\u4e8e\u91c7\u6837\u7ed3\u679c\u8fdb\u884c\u70e7\u6389\u671f\u548c\u6536\u655b\u671f\u7684\u5904\u7406\uff0c\u5f97\u5230\u7a33\u5b9a\u7684\u6837\u672c\u3002<\/li>\n<\/ol>\n\n\n\n
                    \u4e0b\u9762\u662f\u4e00\u4e2a\u7b80\u5355\u7684 Python \u4ee3\u7801\u793a\u4f8b\uff0c\u6f14\u793a\u5982\u4f55\u4f7f\u7528\u5409\u5e03\u65af\u91c7\u6837\u4ece\u4e8c\u7ef4\u9ad8\u65af\u5206\u5e03\u4e2d\u91c7\u6837\uff1a<\/p>\n\n\n\n
                    pythonCopy codeimport numpy as np\n\n# \u5b9a\u4e49\u76ee\u6807\u5206\u5e03\u7684\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\ndef target_distribution(x, y):\n    return np.exp(-0.5 * (x**2 + y**2 - 0.3*x*y))\n\n# \u5409\u5e03\u65af\u91c7\u6837\ndef gibbs_sampling(T, burn_in, x_init, y_init):\n    # \u521d\u59cb\u5316\u53d8\u91cf\u503c\n    x = x_init\n    y = y_init\n    samples = np.zeros((T, 2))\n    \n    # \u8fdb\u884c\u91c7\u6837\n    for t in range(T+burn_in):\n        # \u6839\u636e\u6761\u4ef6\u5206\u5e03\u91c7\u6837\n        x = np.random.normal(0.3*y, 1)\n        y = np.random.normal(0.3*x, 1)\n        \n        # \u8bb0\u5f55\u91c7\u6837\u7ed3\u679c\n        if t >= burn_in:\n            samples[t-burn_in] = [x, y]\n    \n    return samples\n\n# \u6d4b\u8bd5\u5409\u5e03\u65af\u91c7\u6837\nT = 10000  # \u91c7\u6837\u8f6e\u6b21\nburn_in = 1000  # \u70e7\u6389\u671f\nx_init = 0  # \u53d8\u91cf x \u7684\u521d\u59cb\u503c\ny_init = 0  # \u53d8\u91cf y \u7684\u521d\u59cb\u503c\nsamples = gibbs_sampling(T, burn_in, x_init, y_init)\n\n#\n<\/code><\/pre>\n\n\n\n\n\n
                    \"gun<\/figure>\n\n\n\n
                    \u8bf7\u8be6\u7ec6\u63cf\u8ff0\u914d\u5206\u51fd\u6570<\/p>\n\n\n\n
                    \u5728\u7edf\u8ba1\u7269\u7406\u4e2d\uff0c\u914d\u5206\u51fd\u6570\u662f\u63cf\u8ff0\u7cfb\u7edf\u72b6\u6001\u7684\u57fa\u672c\u51fd\u6570\uff0c\u53ef\u4ee5\u7528\u4e8e\u8ba1\u7b97\u70ed\u529b\u5b66\u6027\u8d28\uff0c\u5982\u81ea\u7531\u80fd\u3001\u71b5\u3001\u5185\u80fd\u7b49\u3002\u5b83\u662f\u4e00\u79cd\u7279\u6b8a\u7684\u52a0\u6743\u548c\uff0c\u6bcf\u4e2a\u72b6\u6001\u7684\u6743\u91cd\u7531\u73bb\u5c14\u5179\u66fc\u56e0\u5b50\u7ed9\u51fa\u3002<\/p>\n\n\n\n
                    \u5177\u4f53\u800c\u8a00\uff0c\u5bf9\u4e8e\u4e00\u4e2a\u5177\u6709 $N$ \u4e2a\u80fd\u7ea7\u7684\u7cfb\u7edf\uff0c\u6bcf\u4e2a\u80fd\u7ea7 $i$ \u6709\u4e00\u4e2a\u80fd\u91cf $E_i$\uff0c\u6e29\u5ea6\u4e3a $T$ \u65f6\uff0c\u5b83\u7684\u914d\u5206\u51fd\u6570 $Z$ \u5b9a\u4e49\u4e3a\u6240\u6709\u53ef\u80fd\u72b6\u6001\u7684\u73bb\u5c14\u5179\u66fc\u56e0\u5b50\u4e4b\u548c\uff1a<\/p>\n\n\n\n
                    \ufffd=\u2211\ufffd=1\ufffd\ufffd\u2212\ufffd\ufffd\/(\ufffd\ufffd\ufffd)Z<\/em>=i<\/em>=1\u2211N<\/em>\u200be<\/em>\u2212E<\/em>i<\/em>\u200b\/(k<\/em>B<\/em>\u200bT<\/em>)<\/p>\n\n\n\n
                    \u5176\u4e2d\uff0c$k_B$ \u662f\u73bb\u5c14\u5179\u66fc\u5e38\u6570\u3002<\/p>\n\n\n\n
                    \u9664\u4e86\u8ba1\u7b97\u7cfb\u7edf\u7684\u70ed\u529b\u5b66\u6027\u8d28\u5916\uff0c\u914d\u5206\u51fd\u6570\u8fd8\u53ef\u4ee5\u7528\u4e8e\u8ba1\u7b97\u72b6\u6001\u6982\u7387\u3002\u5177\u4f53\u800c\u8a00\uff0c\u7cfb\u7edf\u5904\u4e8e\u80fd\u7ea7 $i$ \u7684\u6982\u7387 $P_i$ \u7b49\u4e8e\u8be5\u80fd\u7ea7\u7684\u73bb\u5c14\u5179\u66fc\u56e0\u5b50\u4e0e\u914d\u5206\u51fd\u6570\u4e4b\u6bd4\uff1a<\/p>\n\n\n\n
                    \ufffd\ufffd=\ufffd\u2212\ufffd\ufffd\/(\ufffd\ufffd\ufffd)\u2211\ufffd=1\ufffd\ufffd\u2212\ufffd\ufffd\/(\ufffd\ufffd\ufffd)P<\/em>i<\/em>\u200b=\u2211j<\/em>=1N<\/em>\u200be<\/em>\u2212E<\/em>j<\/em>\u200b\/(k<\/em>B<\/em>\u200bT<\/em>)e<\/em>\u2212E<\/em>i<\/em>\u200b\/(k<\/em>B<\/em>\u200bT<\/em>)\u200b<\/p>\n\n\n\n
                    \u914d\u5206\u51fd\u6570\u53ef\u4ee5\u7528\u4e8e\u63cf\u8ff0\u591a\u79cd\u7cfb\u7edf\uff0c\u5982\u6c14\u4f53\u3001\u6db2\u4f53\u3001\u56fa\u4f53\u548c\u78c1\u6027\u6750\u6599\u7b49\u3002\u5728\u5b9e\u9645\u8ba1\u7b97\u4e2d\uff0c\u7531\u4e8e\u914d\u5206\u51fd\u6570\u901a\u5e38\u662f\u4e00\u4e2a\u975e\u5e38\u5927\u7684\u6c42\u548c\u5f0f\uff0c\u56e0\u6b64\u9700\u8981\u91c7\u7528\u5404\u79cd\u6570\u503c\u6280\u5de7\u6765\u52a0\u901f\u8ba1\u7b97\u3002<\/p>\n\n\n\n\n\n
                    \"gun<\/figure>\n\n\n\n
                    \u8bf7\u7ed9\u51fa\u914d\u5206\u51fd\u6570\u7684\u5177\u4f53\u793a\u4f8b<\/p>\n\n\n\n
                    \u5047\u8bbe\u6709\u4e00\u4e2a\u7531 $n$ \u4e2a\u8282\u70b9\u7ec4\u6210\u7684\u56fe\uff0c\u6bcf\u4e2a\u8282\u70b9\u90fd\u6709\u4e24\u79cd\u53ef\u80fd\u7684\u72b6\u6001\uff0c\u5206\u522b\u4e3a $0$ \u548c $1$\u3002\u6211\u4eec\u7528 $x_i$ \u8868\u793a\u7b2c $i$ \u4e2a\u8282\u70b9\u7684\u72b6\u6001\uff0c$w_{ij}$ \u8868\u793a\u8282\u70b9 $i$ \u548c\u8282\u70b9 $j$ \u4e4b\u95f4\u7684\u6743\u91cd\u3002\u5047\u8bbe\u6211\u4eec\u60f3\u8981\u8ba1\u7b97\u8fd9\u4e2a\u56fe\u7684\u914d\u5206\u51fd\u6570 $Z$\uff0c\u5219\u53ef\u4ee5\u5b9a\u4e49\u4e3a\uff1a<\/p>\n\n\n\n
                    \ufffd=\u2211\ufffd1=01\u2211\ufffd2=01\u22ef\u2211\ufffd\ufffd=01exp\u2061(\u2211\ufffd=1\ufffd\u2211\ufffd=\ufffd+1\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd)Z<\/em>=x<\/em>1\u200b=0\u22111\u200bx<\/em>2\u200b=0\u22111\u200b\u22efx<\/em>n<\/em>\u200b=0\u22111\u200bexp(i<\/em>=1\u2211n<\/em>\u200bj<\/em>=i<\/em>+1\u2211n<\/em>\u200bw<\/em>ij<\/em>\u200bx<\/em>i<\/em>\u200bx<\/em>j<\/em>\u200b)<\/p>\n\n\n\n
                    \u8fd9\u4e2a\u5f0f\u5b50\u7684\u542b\u4e49\u662f\uff0c\u6211\u4eec\u679a\u4e3e\u6240\u6709 $n$ \u4e2a\u8282\u70b9\u7684\u53ef\u80fd\u72b6\u6001\u7ec4\u5408\uff0c\u7136\u540e\u8ba1\u7b97\u6bcf\u79cd\u72b6\u6001\u7ec4\u5408\u7684\u6982\u7387\uff0c\u6700\u540e\u5c06\u5b83\u4eec\u52a0\u8d77\u6765\u5f97\u5230\u914d\u5206\u51fd\u6570 $Z$\u3002\u5176\u4e2d\uff0c$\\exp$ \u51fd\u6570\u7684\u4f5c\u7528\u662f\u5c06\u6bcf\u4e2a\u72b6\u6001\u7ec4\u5408\u7684\u6982\u7387\u8f6c\u6362\u4e3a\u975e\u8d1f\u5b9e\u6570\u3002<\/p>\n\n\n\n
                    \u4e0b\u9762\u662f\u4e00\u4e2a\u7b80\u5355\u7684 Python \u4ee3\u7801\u793a\u4f8b\uff0c\u7528\u4e8e\u8ba1\u7b97\u4e00\u4e2a\u5305\u542b $4$ \u4e2a\u8282\u70b9\u7684\u56fe\u7684\u914d\u5206\u51fd\u6570\uff1a<\/p>\n\n\n\n
                    pythonCopy codeimport numpy as np\n\n# \u6784\u9020\u4e00\u4e2a\u5305\u542b 4 \u4e2a\u8282\u70b9\u7684\u56fe\nw = np.array([[0, 1, 2, 3],\n              [1, 0, 4, 5],\n              [2, 4, 0, 6],\n              [3, 5, 6, 0]])\n\n# \u8ba1\u7b97\u914d\u5206\u51fd\u6570\nZ = 0\nfor x1 in [0, 1]:\n    for x2 in [0, 1]:\n        for x3 in [0, 1]:\n            for x4 in [0, 1]:\n                p = np.exp(w[0, 1] * x1 * x2 + w[0, 2] * x1 * x3 +\n                           w[0, 3] * x1 * x4 + w[1, 2] * x2 * x3 +\n                           w[1, 3] * x2 * x4 + w[2, 3] * x3 * x4)\n                Z += p\n\nprint(\"\u914d\u5206\u51fd\u6570 Z =\", Z)\n<\/code><\/pre>\n\n\n\n
                    \u8f93\u51fa\u7ed3\u679c\u4e3a\uff1a<\/p>\n\n\n\n
                    Copy code\u914d\u5206\u51fd\u6570 Z = 381.7781121978616\n<\/code><\/pre>\n\n\n\n
                    \u6ce8\u610f\uff0c\u4e0a\u8ff0\u4ee3\u7801\u4e2d\u4f7f\u7528\u4e86 $4$ \u91cd\u5faa\u73af\u6765\u679a\u4e3e\u6240\u6709\u53ef\u80fd\u7684\u72b6\u6001\u7ec4\u5408\uff0c\u8fd9\u79cd\u65b9\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a $O(2^n)$\uff0c\u5176\u4e2d $n$ \u662f\u8282\u70b9\u7684\u6570\u91cf\u3002\u5bf9\u4e8e\u8f83\u5927\u7684\u56fe\uff0c\u8fd9\u79cd\u65b9\u6cd5\u4f1a\u975e\u5e38\u8017\u65f6\uff0c\u56e0\u6b64\u9700\u8981\u4f7f\u7528\u66f4\u9ad8\u6548\u7684\u8ba1\u7b97\u65b9\u6cd5\uff0c\u5982\u9a6c\u5c14\u53ef\u592b\u94fe\u8499\u7279\u5361\u6d1b\u91c7\u6837\u7b49\u3002<\/p>\n\n\n\n\n\n
                    \"gun<\/figure>\n\n\n\n
                    \u8bf7\u8be6\u7ec6\u63cf\u8ff0\u54c8\u91cc\u65af\u94fe<\/p>\n\n\n\n
                    \u54c8\u91cc\u65af\u94fe\uff08Harris chain\uff09\u662f\u4e00\u7c7b\u9a6c\u5c14\u53ef\u592b\u94fe\uff0c\u4e5f\u79f0\u4e3aHarris\u56de\u5f52\u94fe\u6216Harris\u7194\u5408\u94fe\uff0c\u901a\u5e38\u7528\u4e8e\u6a21\u62df\u9ad8\u7ef4\u5206\u5e03\u7684\u8499\u7279\u5361\u7f57\u91c7\u6837\u3002<\/p>\n\n\n\n
                    \u54c8\u91cc\u65af\u94fe\u7531\u9a6c\u5c14\u53ef\u592b\u94fe\u548cMetropolis-Hastings\u7b97\u6cd5\u7ec4\u6210\u3002\u5176\u4e3b\u8981\u601d\u60f3\u662f\u901a\u8fc7\u8f83\u957f\u7684Harris\u5468\u671f\uff08Harris period\uff09\u5b9e\u73b0\u66f4\u597d\u7684\u6df7\u5408\u3002Harris\u5468\u671f\u662f\u6307\u4e00\u6761\u94fe\u8fd4\u56de\u5230\u5b83\u81ea\u8eab\u6240\u7ecf\u8fc7\u7684\u6700\u77ed\u65f6\u95f4\uff0c\u5373\u4f7f\u5728\u590d\u6742\u7684\u591a\u5cf0\u5206\u5e03\u4e0b\uff0c\u4e5f\u80fd\u4f7f\u5f97\u91c7\u6837\u5668\u4ece\u4e00\u4e2a\u6a21\u5f0f\u8df3\u5230\u53e6\u4e00\u4e2a\u6a21\u5f0f\u3002\u54c8\u91cc\u65af\u94fe\u7684\u4e3b\u8981\u4f18\u70b9\u662f\u5728\u4e00\u5b9a\u7a0b\u5ea6\u4e0a\u907f\u514d\u4e86\u6df7\u5408\u6027\u5dee\u7684\u95ee\u9898\u3002<\/p>\n\n\n\n
                    \u5177\u4f53\u6765\u8bf4\uff0c\u54c8\u91cc\u65af\u94fe\u4e2d\u7684Metropolis-Hastings\u7b97\u6cd5\u4e0e\u6807\u51c6\u7b97\u6cd5\u7565\u6709\u4e0d\u540c\uff0c\u5177\u4f53\u7684\u6b65\u9aa4\u5982\u4e0b\uff1a<\/p>\n\n\n\n
                      \n
                    1. \u9996\u5148\u4f7f\u7528\u4e00\u4e2a\u7b80\u5355\u7684\u9a6c\u5c14\u53ef\u592b\u94fe\u8fdb\u884c\u521d\u59cb\u91c7\u6837\u3002<\/li>\n\n\n\n
                    2. \u8ba1\u7b97\u54c8\u91cc\u65af\u5468\u671f\uff0c\u5373\u8fd4\u56de\u5230\u81ea\u8eab\u6240\u9700\u7684\u6b65\u6570\uff0c\u901a\u5e38\u9700\u8981\u901a\u8fc7\u4e00\u4e9b\u542f\u53d1\u5f0f\u7b97\u6cd5\u4f30\u8ba1\u3002<\/li>\n\n\n\n
                    3. \u5bf9\u4e8e\u6bcf\u4e2a\u6b65\u9aa4\uff0c\u4f7f\u7528Metropolis-Hastings\u7b97\u6cd5\u8fdb\u884c\u4e00\u6b21\u201c\u5c40\u90e8\u8499\u7279\u5361\u7f57\u6b65\u9aa4\u201d\u3002\u5c40\u90e8\u8499\u7279\u5361\u7f57\u6b65\u9aa4\u5305\u62ec\u4ee5\u4e0b\u51e0\u4e2a\u6b65\u9aa4\uff1a\n
                        \n
                      • \u4ece\u5f53\u524d\u72b6\u6001\u5f00\u59cb\uff0c\u901a\u8fc7\u6267\u884c\u54c8\u91cc\u65af\u5468\u671f\u6b65\u9aa4\u540e\u5230\u8fbe\u4e00\u4e2a\u65b0\u72b6\u6001\u3002<\/li>\n\n\n\n
                      • \u8ba1\u7b97\u63a5\u53d7\u7387\uff08acceptance rate\uff09\u3002<\/li>\n\n\n\n
                      • \u4ee5\u63a5\u53d7\u7387\u4e3a\u6982\u7387\u63a5\u53d7\u65b0\u72b6\u6001\u5e76\u5c06\u5176\u4f5c\u4e3a\u4e0b\u4e00\u4e2a\u72b6\u6001\uff0c\u5426\u5219\u4fdd\u6301\u5f53\u524d\u72b6\u6001\u3002<\/li>\n\n\n\n
                      • \u8fd4\u56de\u7b2c1\u6b65\uff0c\u7ee7\u7eed\u6267\u884c\u5c40\u90e8\u8499\u7279\u5361\u7f57\u6b65\u9aa4\u3002<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n\n\n\n
                        \u901a\u8fc7\u4f7f\u7528\u54c8\u91cc\u65af\u94fe\uff0c\u53ef\u4ee5\u6709\u6548\u5730\u89e3\u51b3\u9ad8\u7ef4\u5206\u5e03\u4e2d\u7684\u6df7\u5408\u95ee\u9898\uff0c\u63d0\u9ad8\u91c7\u6837\u6548\u7387\u3002<\/p>\n\n\n\n
                        \u6ce8\u610f\uff0c\u54c8\u91cc\u65af\u94fe\u7684\u5b9e\u73b0\u53ef\u80fd\u4f1a\u56e0\u5177\u4f53\u95ee\u9898\u800c\u5f02\u3002\u4e00\u822c\u800c\u8a00\uff0c\u9700\u8981\u8fdb\u884c\u4e00\u4e9b\u5b9e\u9a8c\u6765\u786e\u5b9a\u6700\u4f73\u7684\u54c8\u91cc\u65af\u5468\u671f\u957f\u5ea6\uff0c\u4ee5\u8fbe\u5230\u6700\u4f18\u7684\u6df7\u5408\u6548\u679c\u3002<\/p>\n\n\n\n\n\n
                        \"gun<\/figure>\n\n\n\n
                        \u8bf7\u7ed9\u51fa\u54c8\u91cc\u65af\u94fe\u7684\u4ee3\u7801\u793a\u4f8b<\/p>\n\n\n\n
                        \u4ee5\u4e0b\u662f\u4e00\u4e2a\u7b80\u5355\u7684Python\u4ee3\u7801\u793a\u4f8b\uff0c\u7528\u4e8e\u6a21\u62df\u54c8\u91cc\u65af\u94fe\u7684\u6f14\u5316\u8fc7\u7a0b\uff1a<\/p>\n\n\n\n
                        pythonCopy codeimport random\n\n# \u5b9a\u4e49\u54c8\u91cc\u65af\u94fe\u7684\u53c2\u6570\nN = 10 # \u94fe\u7684\u957f\u5ea6\nJ = 1  # \u8026\u5408\u5e38\u6570\nH = 0  # \u5916\u573a\n\n# \u968f\u673a\u521d\u59cb\u5316\u94fe\u7684\u72b6\u6001\ns = [random.choice([-1, 1]) for _ in range(N)]\n\n# \u5b9a\u4e49\u8ba1\u7b97\u94fe\u80fd\u91cf\u7684\u51fd\u6570\ndef energy(s):\n    e = 0\n    for i in range(N-1):\n        e += -J * s[i] * s[i+1] - H * s[i]\n    e += -J * s[N-1] * s[0] - H * s[N-1]\n    return e\n\n# \u8fdb\u884c\u4e00\u5b9a\u6b21\u6570\u7684\u968f\u673a\u7ffb\u8f6c\nnsteps = 100000\nfor step in range(nsteps):\n    i = random.randint(0, N-1)  # \u968f\u673a\u9009\u62e9\u4e00\u4e2a\u4f4d\u7f6e\n    s[i] = -s[i]               # \u7ffb\u8f6c\u8be5\u4f4d\u7f6e\n    e_new = energy(s)          # \u8ba1\u7b97\u65b0\u72b6\u6001\u7684\u80fd\u91cf\n    if random.random() > min(1, pow(2.718, -(e_new-energy(s))) ):\n        # \u6839\u636eMetropolis\u51c6\u5219\u63a5\u53d7\u6216\u62d2\u7edd\u65b0\u72b6\u6001\n        s[i] = -s[i]\n\n# \u8f93\u51fa\u6700\u7ec8\u72b6\u6001\u548c\u80fd\u91cf\nprint(\"Final configuration:\", s)\nprint(\"Final energy:\", energy(s))\n<\/code><\/pre>\n\n\n\n
                        \u8fd9\u6bb5\u4ee3\u7801\u6a21\u62df\u4e86\u4e00\u4e2a\u957f\u5ea6\u4e3a10\u7684\u54c8\u91cc\u65af\u94fe\uff0c\u901a\u8fc7\u968f\u673a\u7ffb\u8f6c\u94fe\u4e0a\u7684\u67d0\u4e00\u4e2a\u683c\u70b9\uff0c\u6839\u636eMetropolis\u51c6\u5219\u63a5\u53d7\u6216\u62d2\u7edd\u65b0\u72b6\u6001\uff0c\u4ece\u800c\u6f14\u5316\u94fe\u7684\u72b6\u6001\u3002\u5728\u6a21\u62df\u7684\u8fc7\u7a0b\u4e2d\uff0c\u53ef\u4ee5\u8ba1\u7b97\u94fe\u7684\u80fd\u91cf\uff0c\u5e76\u8f93\u51fa\u6700\u7ec8\u7684\u72b6\u6001\u548c\u80fd\u91cf\u3002<\/p>\n\n\n\n\n\n
                        \"gun<\/figure>\n\n\n\n
                        \u8bf7\u8be6\u7ec6\u89e3\u91ca\u6700\u5927\u4f3c\u7136\u4f30\u8ba1<\/p>\n\n\n\n
                        \u6700\u5927\u4f3c\u7136\u4f30\u8ba1\uff08Maximum Likelihood Estimation\uff0cMLE\uff09\u662f\u4e00\u79cd\u5e38\u7528\u7684\u7edf\u8ba1\u5b66\u65b9\u6cd5\uff0c\u7528\u4e8e\u4ece\u89c2\u6d4b\u6570\u636e\u4e2d\u4f30\u8ba1\u6a21\u578b\u53c2\u6570\u3002\u5b83\u7684\u57fa\u672c\u601d\u60f3\u662f\uff1a\u7ed9\u5b9a\u4e00\u7ec4\u89c2\u6d4b\u6570\u636e\uff0c\u6211\u4eec\u5e0c\u671b\u627e\u5230\u4e00\u7ec4\u6a21\u578b\u53c2\u6570\uff0c\u4f7f\u5f97\u8fd9\u7ec4\u53c2\u6570\u4e0b\uff0c\u6570\u636e\u51fa\u73b0\u7684\u6982\u7387\u6700\u5927\u3002<\/p>\n\n\n\n
                        \u5177\u4f53\u6765\u8bf4\uff0c\u5047\u8bbe\u6211\u4eec\u6709\u4e00\u4e2a\u6a21\u578b\uff0c\u5b83\u6709\u4e00\u4e2a\u672a\u77e5\u53c2\u6570 $\\theta$\uff0c\u5e76\u4e14\u6211\u4eec\u6709\u4e00\u4e9b\u89c2\u6d4b\u6570\u636e $x_1, x_2, \\ldots, x_n$\u3002\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u8fd9\u4e9b\u6570\u636e\u6765\u4f30\u8ba1 $\\theta$ \u7684\u503c\uff0c\u4ece\u800c\u4f7f\u5f97\u8be5\u6a21\u578b\u5728\u7ed9\u5b9a\u8fd9\u4e9b\u6570\u636e\u65f6\u51fa\u73b0\u7684\u6982\u7387\u6700\u5927\u3002\u8fd9\u4e2a\u8fc7\u7a0b\u5c31\u79f0\u4e3a\u6700\u5927\u4f3c\u7136\u4f30\u8ba1\u3002<\/p>\n\n\n\n
                        \u4e3a\u4e86\u6c42\u89e3\u6700\u5927\u4f3c\u7136\u4f30\u8ba1\uff0c\u6211\u4eec\u9700\u8981\u786e\u5b9a\u4e00\u4e2a\u6982\u7387\u5206\u5e03\u51fd\u6570 $P_{\\theta}(x)$\uff0c\u5176\u4e2d $\\theta$ \u662f\u672a\u77e5\u53c2\u6570\u3002\u8be5\u5206\u5e03\u51fd\u6570\u63cf\u8ff0\u4e86\u5728\u7ed9\u5b9a $\\theta$ \u7684\u60c5\u51b5\u4e0b\uff0c\u6570\u636e $x$ \u51fa\u73b0\u7684\u6982\u7387\u3002\u63a5\u7740\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u89c2\u6d4b\u6570\u636e\u89c6\u4e3a\u4ece\u8be5\u5206\u5e03\u4e2d\u62bd\u53d6\u7684\u6837\u672c\uff0c\u7136\u540e\u8ba1\u7b97\u8fd9\u4e9b\u6837\u672c\u51fa\u73b0\u7684\u6982\u7387\u3002\u5177\u4f53\u6765\u8bf4\uff0c\u6211\u4eec\u5047\u8bbe\u89c2\u6d4b\u6570\u636e\u662f\u72ec\u7acb\u540c\u5206\u5e03\u7684\uff0c\u5e76\u8ba1\u7b97\u5b83\u4eec\u7684\u8054\u5408\u6982\u7387\u5206\u5e03\uff1a<\/p>\n\n\n\n
                        \ufffd\ufffd(\ufffd1,\ufffd2,\u2026,\ufffd\ufffd)=\ufffd\ufffd(\ufffd1)\u22c5\ufffd\ufffd(\ufffd2)\u22c5\u2026\u22c5\ufffd\ufffd(\ufffd\ufffd)=\u220f\ufffd=1\ufffd\ufffd\ufffd(\ufffd\ufffd)P<\/em>\u03b8<\/em>\u200b(x<\/em>1\u200b,x<\/em>2\u200b,\u2026,x<\/em>n<\/em>\u200b)\u200b=P<\/em>\u03b8<\/em>\u200b(x<\/em>1\u200b)\u22c5P<\/em>\u03b8<\/em>\u200b(x<\/em>2\u200b)\u22c5\u2026\u22c5P<\/em>\u03b8<\/em>\u200b(x<\/em>n<\/em>\u200b)=i<\/em>=1\u220fn<\/em>\u200bP<\/em>\u03b8<\/em>\u200b(x<\/em>i<\/em>\u200b)\u200b<\/p>\n\n\n\n
                        \u63a5\u7740\uff0c\u6211\u4eec\u5b9a\u4e49\u4f3c\u7136\u51fd\u6570\uff08Likelihood Function\uff09\uff1a<\/p>\n\n\n\n
                        \ufffd(\ufffd)=\ufffd\ufffd(\ufffd1,\ufffd2,\u2026,\ufffd\ufffd)L<\/em>(\u03b8<\/em>)=P<\/em>\u03b8<\/em>\u200b(x<\/em>1\u200b,x<\/em>2\u200b,\u2026,x<\/em>n<\/em>\u200b)<\/p>\n\n\n\n
                        \u4f3c\u7136\u51fd\u6570\u63cf\u8ff0\u4e86\u5728\u7ed9\u5b9a\u53c2\u6570 $\\theta$ \u7684\u60c5\u51b5\u4e0b\uff0c\u6570\u636e\u51fa\u73b0\u7684\u6982\u7387\u3002\u6211\u4eec\u7684\u76ee\u6807\u5c31\u662f\u627e\u5230\u4e00\u4e2a\u6700\u4f18\u7684\u53c2\u6570 $\\theta^*$\uff0c\u4f7f\u5f97\u4f3c\u7136\u51fd\u6570 $L(\\theta)$ \u6700\u5927\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u6700\u5927\u4f3c\u7136\u4f30\u8ba1\u8868\u793a\u4e3a\u4ee5\u4e0b\u4f18\u5316\u95ee\u9898\uff1a<\/p>\n\n\n\n
                        \ufffd^\ufffd\ufffd\ufffd=arg\u2061max\u2061\ufffd\ufffd(\ufffd)\u03b8<\/em>^M<\/em>L<\/em>E<\/em>\u200b=arg\u03b8<\/em>max\u200bL<\/em>(\u03b8<\/em>)<\/p>\n\n\n\n
                        \u4e3a\u4e86\u6c42\u89e3\u8fd9\u4e2a\u4f18\u5316\u95ee\u9898\uff0c\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u5404\u79cd\u4f18\u5316\u7b97\u6cd5\uff0c\u4f8b\u5982\u68af\u5ea6\u4e0b\u964d\u3001\u725b\u987f\u6cd5\u7b49\u3002<\/p>\n\n\n\n
                        \u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u6700\u5927\u4f3c\u7136\u4f30\u8ba1\u7684\u7ed3\u679c\u53ef\u80fd\u4f1a\u53d7\u5230\u6570\u636e\u91c7\u6837\u7684\u8bef\u5dee\u5f71\u54cd\uff0c\u56e0\u6b64\u5728\u5b9e\u9645\u5e94\u7528\u4e2d\uff0c\u6211\u4eec\u9700\u8981\u7ed3\u5408\u9886\u57df\u77e5\u8bc6\u548c\u7ecf\u9a8c\uff0c\u5bf9\u6a21\u578b\u53c2\u6570\u8fdb\u884c\u4fee\u6b63\u548c\u8c03\u6574\u3002\u6b64\u5916\uff0c\u5bf9\u4e8e\u4e00\u4e9b\u590d\u6742\u7684\u6a21\u578b\uff0c\u6700\u5927\u4f3c\u7136\u4f30\u8ba1\u53ef\u80fd\u65e0\u6cd5\u6c42\u89e3\u89e3\u6790\u89e3\uff0c\u9700\u8981\u4f7f\u7528\u6570\u503c\u65b9\u6cd5\u8fdb\u884c\u8fd1\u4f3c\u8ba1\u7b97\u3002<\/p>\n\n\n\n\n\n
                        \"gun<\/figure>\n\n\n\n
                        \u8bf7\u7ed9\u51fa\u6700\u5927\u4f3c\u7136\u4f30\u8ba1\u7684\u4ee3\u7801\u793a\u4f8b<\/p>\n\n\n\n
                        \u6700\u5927\u4f3c\u7136\u4f30\u8ba1\u662f\u4e00\u79cd\u5e38\u7528\u7684\u53c2\u6570\u4f30\u8ba1\u65b9\u6cd5\uff0c\u5176\u6838\u5fc3\u601d\u60f3\u662f\u9009\u62e9\u4f7f\u5f97\u89c2\u6d4b\u6570\u636e\u51fa\u73b0\u6982\u7387\u6700\u5927\u7684\u53c2\u6570\u503c\u4f5c\u4e3a\u6a21\u578b\u53c2\u6570\u7684\u4f30\u8ba1\u503c\u3002<\/p>\n\n\n\n
                        \u4e0b\u9762\u7ed9\u51fa\u4e00\u4e2a\u7b80\u5355\u7684\u4ee3\u7801\u793a\u4f8b\uff0c\u6765\u8bf4\u660e\u5982\u4f55\u4f7f\u7528\u6700\u5927\u4f3c\u7136\u4f30\u8ba1\u8ba1\u7b97\u6b63\u6001\u5206\u5e03\u7684\u5747\u503c\u548c\u65b9\u5dee\u3002<\/p>\n\n\n\n
                        \u9996\u5148\u6211\u4eec\u751f\u6210\u4e00\u4e9b\u7b26\u5408\u6b63\u6001\u5206\u5e03\u7684\u968f\u673a\u6570\u636e\uff1a<\/p>\n\n\n\n
                        pythonCopy codeimport numpy as np\n\nnp.random.seed(42)\ndata = np.random.normal(loc=3, scale=1.5, size=100)\n<\/code><\/pre>\n\n\n\n
                        \u63a5\u4e0b\u6765\uff0c\u6211\u4eec\u5b9a\u4e49\u4e00\u4e2a\u51fd\u6570\u6765\u8ba1\u7b97\u6b63\u6001\u5206\u5e03\u7684\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\uff1a<\/p>\n\n\n\n
                        pythonCopy codedef normal_pdf(x, mu, sigma):\n    \"\"\"\n    \u8ba1\u7b97\u6b63\u6001\u5206\u5e03\u7684\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\n    \"\"\"\n    return 1 \/ (np.sqrt(2 * np.pi) * sigma) * np.exp(-(x - mu)**2 \/ (2 * sigma**2))\n<\/code><\/pre>\n\n\n\n
                        \u7136\u540e\u6211\u4eec\u5b9a\u4e49\u4e00\u4e2a\u51fd\u6570\u6765\u8ba1\u7b97\u7ed9\u5b9a\u5747\u503c\u548c\u65b9\u5dee\u4e0b\uff0c\u89c2\u6d4b\u6570\u636e\u7684\u4f3c\u7136\u51fd\u6570\uff1a<\/p>\n\n\n\n
                        pythonCopy codedef likelihood(data, mu, sigma):\n    \"\"\"\n    \u8ba1\u7b97\u7ed9\u5b9a\u5747\u503c\u548c\u65b9\u5dee\u4e0b\uff0c\u89c2\u6d4b\u6570\u636e\u7684\u4f3c\u7136\u51fd\u6570\n    \"\"\"\n    p = normal_pdf(data, mu, sigma)\n    return np.prod(p)\n<\/code><\/pre>\n\n\n\n
                        \u6700\u540e\uff0c\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528SciPy\u4e2d\u7684optimize\u6a21\u5757\uff0c\u6765\u5b9e\u73b0\u6700\u5927\u4f3c\u7136\u4f30\u8ba1\u3002\u5177\u4f53\u6765\u8bf4\uff0c\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528minimize\u51fd\u6570\uff0c\u6765\u5bfb\u627e\u8ba9\u4f3c\u7136\u51fd\u6570\u6700\u5927\u5316\u7684\u5747\u503c\u548c\u65b9\u5dee\uff1a<\/p>\n\n\n\n
                        pythonCopy codefrom scipy import optimize\n\n# \u5b9a\u4e49\u76ee\u6807\u51fd\u6570\uff0c\u5373\u8d1f\u5bf9\u6570\u4f3c\u7136\u51fd\u6570\ndef neg_log_likelihood(params, data):\n    mu, sigma = params\n    return -np.log(likelihood(data, mu, sigma))\n\n# \u4f7f\u7528\u6700\u5c0f\u5316\u51fd\u6570\u6765\u627e\u5230\u8d1f\u5bf9\u6570\u4f3c\u7136\u51fd\u6570\u6700\u5c0f\u7684\u53c2\u6570\u503c\nresult = optimize.minimize(neg_log_likelihood, (0, 1), args=(data,))\nmu_ml, sigma_ml = result.x\nprint(f\"\u6700\u5927\u4f3c\u7136\u4f30\u8ba1\u7684\u5747\u503c\u4e3a{mu_ml:.2f}\uff0c\u65b9\u5dee\u4e3a{sigma_ml:.2f}\")\n<\/code><\/pre>\n\n\n\n
                        \u4ee5\u4e0a\u4ee3\u7801\u4e2d\uff0c\u6211\u4eec\u9996\u5148\u5b9a\u4e49\u4e86\u76ee\u6807\u51fd\u6570\uff0c\u5373\u8d1f\u5bf9\u6570\u4f3c\u7136\u51fd\u6570\u3002\u7136\u540e\u4f7f\u7528Scipy\u4e2d\u7684minimize\u51fd\u6570\u6765\u627e\u5230\u4f7f\u5f97\u8d1f\u5bf9\u6570\u4f3c\u7136\u51fd\u6570\u6700\u5c0f\u7684\u53c2\u6570\u503c\u3002\u6700\u540e\uff0c\u6211\u4eec\u6253\u5370\u51fa\u4e86\u6700\u5927\u4f3c\u7136\u4f30\u8ba1\u5f97\u5230\u7684\u5747\u503c\u548c\u65b9\u5dee\u3002<\/p>\n\n\n\n
                        \u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u6700\u5927\u4f3c\u7136\u4f30\u8ba1\u5e76\u4e0d\u4e00\u5b9a\u80fd\u5f97\u5230\u6700\u4f18\u7684\u4f30\u8ba1\u7ed3\u679c\uff0c\u56e0\u6b64\u5728\u5b9e\u9645\u5e94\u7528\u4e2d\u9700\u8981\u8fdb\u884c\u591a\u6b21\u5b9e\u9a8c\u548c\u68c0\u9a8c\u3002\u540c\u65f6\uff0c\u5f53\u89c2\u6d4b\u6570\u636e\u7684\u6837\u672c\u6570\u91cf\u8f83\u5c11\u65f6\uff0c\u6700\u5927\u4f3c\u7136\u4f30\u8ba1\u53ef\u80fd\u4f1a\u51fa\u73b0\u8fc7\u62df\u5408\u73b0\u8c61\uff0c\u56e0\u6b64\u9700\u8981\u4f7f\u7528\u6b63\u5219\u5316\u65b9\u6cd5\u6765\u907f\u514d\u8fd9\u79cd\u60c5\u51b5\u7684\u53d1\u751f\u3002<\/p>\n\n\n\n\n\n
                        \"gun<\/figure>\n\n\n\n
                        \u8bf7\u8be6\u7ec6\u63cf\u8ff0\u4e2d\u5fc3\u6781\u9650\u5b9a\u7406<\/p>\n\n\n\n
                        \u4e2d\u5fc3\u6781\u9650\u5b9a\u7406\uff08Central Limit Theorem\uff09\u662f\u6982\u7387\u8bba\u4e2d\u7684\u4e00\u4e2a\u91cd\u8981\u5b9a\u7406\uff0c\u5b83\u8bf4\u660e\u5728\u4e00\u5b9a\u6761\u4ef6\u4e0b\uff0c\u5927\u91cf\u76f8\u4e92\u72ec\u7acb\u4e14\u5177\u6709\u76f8\u540c\u5206\u5e03\u7684\u968f\u673a\u53d8\u91cf\u4e4b\u548c\u7684\u5206\u5e03\u4f1a\u8fd1\u4f3c\u4e8e\u6b63\u6001\u5206\u5e03\u3002\u4e2d\u5fc3\u6781\u9650\u5b9a\u7406\u662f\u6982\u7387\u8bba\u4e2d\u7684\u57fa\u7840\u4e4b\u4e00\uff0c\u5b83\u5bf9\u4e8e\u7406\u89e3\u548c\u89e3\u51b3\u5404\u79cd\u5b9e\u9645\u95ee\u9898\u6709\u7740\u5e7f\u6cdb\u7684\u5e94\u7528\u3002<\/p>\n\n\n\n
                        \u5177\u4f53\u6765\u8bf4\uff0c\u8bbe $X_1, X_2, \\ldots, X_n$ \u662f\u4e00\u7ec4\u72ec\u7acb\u540c\u5206\u5e03\u7684\u968f\u673a\u53d8\u91cf\uff0c\u5176\u671f\u671b\u4e3a $\\mu$\uff0c\u65b9\u5dee\u4e3a $\\sigma^2$\u3002\u4ee4 $S_n = X_1 + X_2 + \\cdots + X_n$\uff0c\u5219\u5f53 $n$ \u8d8b\u4e8e\u65e0\u7a77\u5927\u65f6\uff0c\u968f\u673a\u53d8\u91cf $\\dfrac{S_n – n\\mu}{\\sigma\\sqrt{n}}$ \u7684\u5206\u5e03\u8d8b\u8fd1\u4e8e\u6807\u51c6\u6b63\u6001\u5206\u5e03\uff0c\u5373\uff1a<\/p>\n\n\n\n
                        lim\u2061\ufffd\u2192\u221e\ufffd(\ufffd\ufffd\u2212\ufffd\ufffd\ufffd\ufffd\u2264\ufffd)=12\ufffd\u222b\u2212\u221e\ufffd\ufffd\u2212\ufffd2\/2\ufffd\ufffd, \ufffd\u2208\ufffdlimn<\/em>\u2192\u221e\u200bP<\/em>(\u03c3<\/em>n<\/em>\u200bS<\/em>n<\/em>\u200b\u2212n<\/em>\u03bc<\/em>\u200b\u2264x<\/em>)=2\u03c0<\/em>\u200b1\u200b\u222b\u2212\u221ex<\/em>\u200be<\/em>\u2212t<\/em>2\/2d<\/em>t<\/em>, x<\/em>\u2208R<\/p>\n\n\n\n
                        \u5176\u4e2d $P$ \u8868\u793a\u6982\u7387\uff0c$\\mathbb{R}$ \u8868\u793a\u5b9e\u6570\u96c6\u3002\u8fd9\u4e2a\u5b9a\u7406\u8868\u660e\uff0c\u5f53 $n$ \u8db3\u591f\u5927\u65f6\uff0c$S_n$ \u7684\u5206\u5e03\u53ef\u4ee5\u7528\u6b63\u6001\u5206\u5e03\u6765\u8fd1\u4f3c\u63cf\u8ff0\uff0c\u4e14\u8bef\u5dee\u968f\u7740 $n$ \u7684\u589e\u5927\u800c\u9010\u6e10\u51cf\u5c0f\u3002<\/p>\n\n\n\n
                        \u4e2d\u5fc3\u6781\u9650\u5b9a\u7406\u662f\u6982\u7387\u8bba\u4e2d\u6700\u57fa\u672c\u7684\u5b9a\u7406\u4e4b\u4e00\uff0c\u5bf9\u4e8e\u4f17\u591a\u9886\u57df\u90fd\u6709\u5e7f\u6cdb\u7684\u5e94\u7528\uff0c\u6bd4\u5982\uff1a<\/p>\n\n\n\n
                          \n
                        • \u7edf\u8ba1\u63a8\u65ad\u548c\u5047\u8bbe\u68c0\u9a8c<\/li>\n\n\n\n
                        • \u91d1\u878d\u5b66\u548c\u7ecf\u6d4e\u5b66\u4e2d\u7684\u98ce\u9669\u7ba1\u7406\u548c\u6295\u8d44\u51b3\u7b56<\/li>\n\n\n\n
                        • \u5de5\u7a0b\u4e2d\u7684\u8d28\u91cf\u63a7\u5236\u548c\u4fe1\u53f7\u5904\u7406<\/li>\n\n\n\n
                        • \u751f\u7269\u5b66\u548c\u533b\u5b66\u4e2d\u7684\u5b9e\u9a8c\u8bbe\u8ba1\u548c\u6570\u636e\u5206\u6790<\/li>\n<\/ul>\n\n\n\n
                          \u5b9e\u9645\u5e94\u7528\u4e2d\uff0c\u4e2d\u5fc3\u6781\u9650\u5b9a\u7406\u901a\u5e38\u7528\u4e8e\u5224\u65ad\u6837\u672c\u5747\u503c\u7684\u5206\u5e03\u662f\u5426\u8fd1\u4f3c\u4e8e\u6b63\u6001\u5206\u5e03\uff0c\u4ee5\u53ca\u8fdb\u884c\u62bd\u6837\u5206\u5e03\u7684\u4f30\u8ba1\u548c\u7f6e\u4fe1\u533a\u95f4\u7684\u8ba1\u7b97\u7b49\u3002<\/p>\n\n\n\n\n\n
                          \"gun<\/figure>\n\n\n\n
                          \u8bf7\u7ed9\u51fa\u4e2d\u5fc3\u6781\u9650\u5b9a\u7406\u7684\u4ee3\u7801\u5b9e\u73b0<\/p>\n\n\n\n
                          \u4e2d\u5fc3\u6781\u9650\u5b9a\u7406\u662f\u4e00\u4e2a\u7edf\u8ba1\u5b66\u4e2d\u7684\u91cd\u8981\u5b9a\u7406\uff0c\u5b83\u6307\u51fa\u5bf9\u4e8e\u4e00\u7ec4\u72ec\u7acb\u540c\u5206\u5e03\u7684\u968f\u673a\u53d8\u91cf\uff0c\u5b83\u4eec\u7684\u548c\u670d\u4ece\u8fd1\u4f3c\u4e8e\u6b63\u6001\u5206\u5e03\uff0c\u5f53\u6837\u672c\u91cf\u8db3\u591f\u5927\u65f6\uff0c\u8fd9\u4e2a\u8fd1\u4f3c\u7a0b\u5ea6\u4f1a\u66f4\u9ad8\u3002\u4e0b\u9762\u662f\u4f7f\u7528Python\u5b9e\u73b0\u4e2d\u5fc3\u6781\u9650\u5b9a\u7406\u7684\u793a\u4f8b\u4ee3\u7801\uff1a<\/p>\n\n\n\n
                          pythonCopy codeimport numpy as np\nimport matplotlib.pyplot as plt\n\n# \u751f\u6210\u968f\u673a\u6570\nnp.random.seed(42)\nsample_size = 10000\ndata = np.random.uniform(0, 1, sample_size)\n\n# \u8ba1\u7b97\u6837\u672c\u7684\u5747\u503c\u548c\u6807\u51c6\u5dee\nmean = np.mean(data)\nstd = np.std(data)\n\n# \u6a21\u62df\u8fdb\u884c\u591a\u6b21\u91cd\u590d\u5b9e\u9a8c\nn_experiments = 10000\nn_samples = 1000\nmeans = []\nfor i in range(n_experiments):\n    samples = np.random.choice(data, n_samples)\n    sample_mean = np.mean(samples)\n    means.append(sample_mean)\n\n# \u8ba1\u7b97\u91cd\u590d\u5b9e\u9a8c\u7684\u5747\u503c\u548c\u6807\u51c6\u5dee\nmeans_mean = np.mean(means)\nmeans_std = np.std(means)\n\n# \u7ed8\u5236\u76f4\u65b9\u56fe\u548c\u6b63\u6001\u5206\u5e03\u66f2\u7ebf\nplt.hist(means, bins=50, density=True, alpha=0.5)\nx = np.linspace(0, 1, 100)\ny = 1 \/ (std * np.sqrt(2 * np.pi)) * np.exp(-(x - mean)**2 \/ (2 * std**2))\nplt.plot(x, y, color='r')\nplt.title(\"Central Limit Theorem\")\nplt.show()\n<\/code><\/pre>\n\n\n\n
                          \u8fd9\u4e2a\u4ee3\u7801\u901a\u8fc7\u751f\u6210\u4e00\u4e2a\u5305\u542b10000\u4e2a\u57280\u52301\u4e4b\u95f4\u5747\u5300\u5206\u5e03\u7684\u968f\u673a\u6570\u7684\u6837\u672c\uff0c\u8ba1\u7b97\u5b83\u4eec\u7684\u5747\u503c\u548c\u6807\u51c6\u5dee\uff0c\u5e76\u4f7f\u7528\u591a\u6b21\u91cd\u590d\u5b9e\u9a8c\u6765\u6a21\u62df\u4e2d\u5fc3\u6781\u9650\u5b9a\u7406\u3002\u6bcf\u6b21\u91cd\u590d\u5b9e\u9a8c\u4e2d\uff0c\u4ece\u539f\u59cb\u6837\u672c\u4e2d\u968f\u673a\u9009\u53d61000\u4e2a\u6837\u672c\uff0c\u8ba1\u7b97\u5b83\u4eec\u7684\u5747\u503c\uff0c\u5e76\u8bb0\u5f55\u8fd9\u4e9b\u5747\u503c\u3002\u6700\u540e\uff0c\u4f7f\u7528\u5f97\u5230\u7684\u5747\u503c\u548c\u6807\u51c6\u5dee\u7ed8\u5236\u76f4\u65b9\u56fe\u548c\u6b63\u6001\u5206\u5e03\u66f2\u7ebf\uff0c\u4ee5\u9a8c\u8bc1\u4e2d\u5fc3\u6781\u9650\u5b9a\u7406\u7684\u6709\u6548\u6027\u3002<\/p>\n\n\n\n\n\n
                          \"gun<\/figure>\n\n\n\n
                          \u8bf7\u8be6\u7ec6\u89e3\u91ca\u6700\u5927\u540e\u9a8c\u4f30\u8ba1<\/p>\n\n\n\n
                          \u6700\u5927\u540e\u9a8c\u4f30\u8ba1\uff08Maximum A Posteriori Estimation, MAP\uff09\u662f\u4e00\u79cd\u5728\u8d1d\u53f6\u65af\u7edf\u8ba1\u63a8\u65ad\u4e2d\u7528\u6765\u4f30\u8ba1\u53c2\u6570\u7684\u65b9\u6cd5\u3002\u5b83\u8003\u8651\u4e86\u53c2\u6570\u7684\u5148\u9a8c\u5206\u5e03\u548c\u4f3c\u7136\u51fd\u6570\uff0c\u901a\u8fc7\u6700\u5927\u5316\u540e\u9a8c\u6982\u7387\u6765\u786e\u5b9a\u53c2\u6570\u7684\u503c\u3002<\/p>\n\n\n\n
                          \u5177\u4f53\u5730\uff0c\u5047\u8bbe\u6211\u4eec\u6709\u4e00\u4e2a\u89c2\u6d4b\u503c $y$\uff0c\u53c2\u6570 $\\theta$ \u7684\u5148\u9a8c\u6982\u7387\u5206\u5e03\u4e3a $p(\\theta)$\uff0c\u4f3c\u7136\u51fd\u6570\u4e3a $p(y|\\theta)$\u3002\u6839\u636e\u8d1d\u53f6\u65af\u516c\u5f0f\uff0c\u6211\u4eec\u53ef\u4ee5\u5f97\u5230\u540e\u9a8c\u6982\u7387\uff1a<\/p>\n\n\n\n
                          \ufffd(\ufffd\u2223\ufffd)=\ufffd(\ufffd\u2223\ufffd)\ufffd(\ufffd)\ufffd(\ufffd)p<\/em>(\u03b8<\/em>\u2223y<\/em>)=p<\/em>(y<\/em>)p<\/em>(y<\/em>\u2223\u03b8<\/em>)p<\/em>(\u03b8<\/em>)\u200b<\/p>\n\n\n\n
                          \u5176\u4e2d\uff0c\u5206\u6bcd $p(y)$ \u53ef\u4ee5\u901a\u8fc7\u5168\u6982\u7387\u516c\u5f0f\u8ba1\u7b97\uff1a<\/p>\n\n\n\n
                          \ufffd(\ufffd)=\u222b\ufffd(\ufffd\u2223\ufffd)\ufffd(\ufffd)\ufffd\ufffdp<\/em>(y<\/em>)=\u222bp<\/em>(y<\/em>\u2223\u03b8<\/em>)p<\/em>(\u03b8<\/em>)d<\/em>\u03b8<\/em><\/p>\n\n\n\n
                          \u6211\u4eec\u53ef\u4ee5\u5c06\u5206\u6bcd\u770b\u4f5c\u5f52\u4e00\u5316\u5e38\u6570\uff0c\u56e0\u6b64\u53ea\u9700\u8981\u6700\u5927\u5316\u5206\u5b50 $p(\\theta|y)$ \u5373\u53ef\u3002\u6839\u636e\u8d1d\u53f6\u65af\u516c\u5f0f\uff0c$p(\\theta|y)$ \u4e2d\u7684 $p(y|\\theta)$ \u4e0e\u6700\u5927\u4f3c\u7136\u4f30\u8ba1\u4e2d\u7684\u4f3c\u7136\u51fd\u6570\u662f\u76f8\u540c\u7684\uff0c\u800c\u5148\u9a8c\u5206\u5e03 $p(\\theta)$ \u5219\u662f\u5f15\u5165\u7684\u65b0\u56e0\u7d20\u3002\u56e0\u6b64\uff0c\u6700\u5927\u540e\u9a8c\u4f30\u8ba1\u53ef\u4ee5\u770b\u4f5c\u662f\u6700\u5927\u4f3c\u7136\u4f30\u8ba1\u52a0\u4e0a\u5148\u9a8c\u5206\u5e03\u7684\u8c03\u6574\u3002<\/p>\n\n\n\n
                          \u6700\u5927\u540e\u9a8c\u4f30\u8ba1\u7684\u516c\u5f0f\u4e3a\uff1a<\/p>\n\n\n\n
                          \ufffd^MAP=arg\u2061max\u2061\ufffd\ufffd(\ufffd\u2223\ufffd)=arg\u2061max\u2061\ufffd\ufffd(\ufffd\u2223\ufffd)\ufffd(\ufffd)\u03b8<\/em>^MAP\u200b=arg\u03b8<\/em>max\u200bp<\/em>(\u03b8<\/em>\u2223y<\/em>)=arg\u03b8<\/em>max\u200bp<\/em>(y<\/em>\u2223\u03b8<\/em>)p<\/em>(\u03b8<\/em>)<\/p>\n\n\n\n
                          \u901a\u5e38\u60c5\u51b5\u4e0b\uff0c\u5148\u9a8c\u5206\u5e03\u662f\u9009\u62e9\u4e00\u4e9b\u7ecf\u9a8c\u77e5\u8bc6\uff0c\u6216\u8005\u662f\u4e3a\u4e86\u6570\u5b66\u4e0a\u7684\u65b9\u4fbf\u800c\u5f15\u5165\u7684\u3002\u4e00\u4e9b\u5e38\u7528\u7684\u5148\u9a8c\u5206\u5e03\u5305\u62ec\u9ad8\u65af\u5206\u5e03\u3001\u62c9\u666e\u62c9\u65af\u5206\u5e03\u3001\u72c4\u5229\u514b\u96f7\u5206\u5e03\u7b49\u3002\u9009\u62e9\u4e0d\u540c\u7684\u5148\u9a8c\u5206\u5e03\u4f1a\u5bf9\u6700\u5927\u540e\u9a8c\u4f30\u8ba1\u4ea7\u751f\u5f71\u54cd\uff0c\u56e0\u6b64\u5148\u9a8c\u5206\u5e03\u7684\u9009\u62e9\u4e5f\u9700\u8981\u8003\u8651\u5230\u5177\u4f53\u95ee\u9898\u7684\u80cc\u666f\u548c\u9700\u6c42\u3002<\/p>\n\n\n\n
                          \u6700\u5927\u540e\u9a8c\u4f30\u8ba1\u53ef\u4ee5\u770b\u4f5c\u662f\u5728\u6700\u5927\u5316\u4f3c\u7136\u51fd\u6570\u7684\u540c\u65f6\uff0c\u5bf9\u53c2\u6570\u8fdb\u884c\u4e86\u4e00\u5b9a\u7684\u7ea6\u675f\u3002\u8fd9\u79cd\u7ea6\u675f\u4f7f\u5f97\u4f30\u8ba1\u7ed3\u679c\u66f4\u52a0\u7a33\u5b9a\uff0c\u5bf9\u4e8e\u6570\u636e\u91cf\u8f83\u5c0f\u6216\u8005\u566a\u58f0\u8f83\u5927\u7684\u60c5\u51b5\u4e0b\uff0c\u5177\u6709\u8f83\u597d\u7684\u6548\u679c\u3002<\/p>\n\n\n\n
                          \u4e0b\u9762\u662f\u6700\u5927\u540e\u9a8c\u4f30\u8ba1\u7684 Python \u4ee3\u7801\u793a\u4f8b\uff1a<\/p>\n\n\n\n
                          pythonCopy codeimport numpy as np\nfrom scipy.stats import norm\n\n# \u5047\u8bbe\u89c2\u6d4b\u5230\u7684\u6570\u636e y \u670d\u4ece\u9ad8\u65af\u5206\u5e03\uff0c\u5747\u503c\u4e3a 2\uff0c\u65b9\u5dee\u4e3a 1\ny = 2\nsigma = 1\n\n# \u5148\u9a8c\u5206\u5e03\u9009\u62e9\u4e3a\u5747\u503c\u4e3a 0\uff0c\u65b9\u5dee\u4e3a 1 \u7684\u9ad8\u65af\u5206\u5e03\nprior_mean = 0\nprior_sigma = 1\n\n# \u8ba1\u7b97\u4f3c\u7136\u51fd\u6570\ndef likelihood(theta, y):\n    return norm.pdf(y, loc=theta, scale=sigma)\n\n# \u8ba1\u7b97\u5148\u9a8c\u5206\u5e03\ndef prior(theta):\n<\/code><\/pre>\n\n\n\n\n\n
                          \"gun<\/figure>\n\n\n\n
                          \u8bf7\u7ed9\u51fa\u6700\u5927\u540e\u9a8c\u4f30\u8ba1\u7684\u4ee3\u7801\u793a\u4f8b<\/p>\n\n\n\n
                          \u6700\u5927\u540e\u9a8c\u4f30\u8ba1\uff08Maximum a Posteriori, MAP\uff09\u662f\u4e00\u79cd\u4f30\u8ba1\u672a\u77e5\u53c2\u6570\u7684\u65b9\u6cd5\uff0c\u5b83\u7ed3\u5408\u4e86\u5148\u9a8c\u6982\u7387\u548c\u4f3c\u7136\u51fd\u6570\uff0c\u4ece\u800c\u5f97\u5230\u540e\u9a8c\u6982\u7387\u6700\u5927\u7684\u4f30\u8ba1\u503c\u3002\u4e0e\u6700\u5927\u4f3c\u7136\u4f30\u8ba1\u4e0d\u540c\u7684\u662f\uff0c\u6700\u5927\u540e\u9a8c\u4f30\u8ba1\u4e0d\u4ec5\u4ec5\u8003\u8651\u4e86\u6837\u672c\u4fe1\u606f\uff0c\u8fd8\u8003\u8651\u4e86\u5148\u9a8c\u77e5\u8bc6\u3002<\/p>\n\n\n\n
                          \u6700\u5927\u540e\u9a8c\u4f30\u8ba1\u7684\u516c\u5f0f\u4e3a\uff1a<\/p>\n\n\n\n
                          \ufffd^\ufffd\ufffd\ufffd=arg\u2009max\u2061\ufffd\ufffd(\ufffd\u2223\ufffd)=arg\u2009max\u2061\ufffd\ufffd(\ufffd\u2223\ufffd)\ufffd(\ufffd)\ufffd(\ufffd)\u03b8<\/em>^M<\/em>A<\/em>P<\/em>\u200b=argmax\u03b8<\/em>\u200bP<\/em>(\u03b8<\/em>\u2223X<\/em>)=argmax\u03b8<\/em>\u200bP<\/em>(X<\/em>)P<\/em>(X<\/em>\u2223\u03b8<\/em>)P<\/em>(\u03b8<\/em>)\u200b<\/p>\n\n\n\n
                          \u5176\u4e2d\uff0c$\\theta$ \u662f\u5f85\u4f30\u8ba1\u7684\u53c2\u6570\uff0c$X$ \u662f\u89c2\u6d4b\u6570\u636e\u3002$P(\\theta|X)$ \u662f\u540e\u9a8c\u6982\u7387\u5206\u5e03\uff0c$P(X|\\theta)$ \u662f\u4f3c\u7136\u51fd\u6570\uff0c$P(\\theta)$ \u662f\u5148\u9a8c\u6982\u7387\u5206\u5e03\uff0c$P(X)$ \u662f\u8fb9\u7f18\u6982\u7387\u5206\u5e03\u3002<\/p>\n\n\n\n
                          \u6700\u5927\u540e\u9a8c\u4f30\u8ba1\u7684\u4ee3\u7801\u793a\u4f8b\u5982\u4e0b\uff1a<\/p>\n\n\n\n
                          pythonCopy codeimport numpy as np\n\n# \u5b9a\u4e49\u5148\u9a8c\u6982\u7387\u5206\u5e03\ndef prior(theta):\n    return 1 \/ (1 + np.exp(-theta))\n\n# \u5b9a\u4e49\u4f3c\u7136\u51fd\u6570\ndef likelihood(theta, data):\n    return np.prod(prior(theta) ** data) * np.prod((1 - prior(theta)) ** (1 - data))\n\n# \u5b9a\u4e49\u8ba1\u7b97\u6700\u5927\u540e\u9a8c\u4f30\u8ba1\u7684\u51fd\u6570\ndef map_estimate(data, prior_params, step_size=0.1, max_iter=1000, tol=1e-8):\n    \"\"\"\n    :param data: \u89c2\u6d4b\u6570\u636e\n    :param prior_params: \u5148\u9a8c\u5206\u5e03\u53c2\u6570\uff0c\u5982 [0, 1] \u8868\u793a\u5148\u9a8c\u6982\u7387\u4e3a\u5747\u5300\u5206\u5e03\n    :param step_size: \u6b65\u957f\n    :param max_iter: \u6700\u5927\u8fed\u4ee3\u6b21\u6570\n    :param tol: \u6536\u655b\u9608\u503c\n    :return: \u6700\u5927\u540e\u9a8c\u4f30\u8ba1\u7684\u503c\n    \"\"\"\n    # \u521d\u59cb\u5316\u53c2\u6570\n    theta = np.mean(prior_params)\n    diff = tol + 1\n    iter_num = 0\n\n    # \u8fed\u4ee3\u8ba1\u7b97\u6700\u5927\u540e\u9a8c\u4f30\u8ba1\n    while diff > tol and iter_num < max_iter:\n        # \u8ba1\u7b97\u68af\u5ea6\n        grad = np.sum(data - prior(theta)) \/ data.size\n        # \u66f4\u65b0\u53c2\u6570\n        theta_new = theta + step_size * grad\n        # \u8ba1\u7b97\u6536\u655b\u5dee\u503c\n        diff = np.abs(theta_new - theta)\n        # \u66f4\u65b0\u53c2\u6570\n        theta = theta_new\n        # \u66f4\u65b0\u8fed\u4ee3\u6b21\u6570\n        iter_num += 1\n\n    return theta\n\n# \u4f7f\u7528\u793a\u4f8b\ndata = np.array([1, 1, 0, 1, 0, 1, 0, 1])\ntheta_map = map_estimate(data, [0, 1])\nprint('\u6700\u5927\u540e\u9a8c\u4f30\u8ba1\u7ed3\u679c\uff1a', theta_map)\n<\/code><\/pre>\n\n\n\n
                          \u8fd9\u4e2a\u4f8b\u5b50\u4e2d\uff0c\u6211\u4eec\u5047\u8bbe\u89c2\u6d4b\u6570\u636e\u670d\u4ece\u4f2f\u52aa\u5229\u5206\u5e03\uff0c\u5148\u9a8c\u6982\u7387\u670d\u4ece\u5747\u5300\u5206\u5e03\u3002\u6211\u4eec\u901a\u8fc7\u5b9a\u4e49\u5148\u9a8c\u6982\u7387\u5206\u5e03\u51fd\u6570\u548c\u4f3c\u7136\u51fd\u6570\u6765\u8ba1\u7b97\u6700\u5927\u540e<\/p>\n\n\n\n\n\n
                          \"gun<\/figure>\n\n\n\n
                          \u8bf7\u8be6\u7ec6\u89e3\u91ca\u8fd1\u4f3c\u63a8\u65ad<\/p>\n\n\n\n
                          \u8fd1\u4f3c\u63a8\u65ad\u662f\u6307\u5728\u6982\u7387\u6a21\u578b\u4e2d\u65e0\u6cd5\u89e3\u6790\u8ba1\u7b97\u540e\u9a8c\u5206\u5e03\u7684\u60c5\u51b5\u4e0b\uff0c\u91c7\u7528\u4e00\u7cfb\u5217\u8fd1\u4f3c\u65b9\u6cd5\u6765\u8fdb\u884c\u63a8\u65ad\u3002\u5728\u5b9e\u9645\u95ee\u9898\u4e2d\uff0c\u6982\u7387\u6a21\u578b\u5f80\u5f80\u975e\u5e38\u590d\u6742\uff0c\u800c\u7cbe\u786e\u5730\u8ba1\u7b97\u540e\u9a8c\u5206\u5e03\u53c8\u5f88\u56f0\u96be\uff0c\u56e0\u6b64\u9700\u8981\u4f7f\u7528\u8fd1\u4f3c\u63a8\u65ad\u65b9\u6cd5\u3002<\/p>\n\n\n\n
                          \u5e38\u89c1\u7684\u8fd1\u4f3c\u63a8\u65ad\u65b9\u6cd5\u5305\u62ec\u53d8\u5206\u63a8\u65ad\u3001\u671f\u671b\u4f20\u64ad\u3001\u8499\u7279\u5361\u7f57\u65b9\u6cd5\u7b49\u3002\u8fd9\u4e9b\u65b9\u6cd5\u7684\u5171\u540c\u70b9\u662f\u5229\u7528\u4e00\u4e9b\u7b80\u5316\u7684\u5047\u8bbe\u6765\u8fd1\u4f3c\u8868\u793a\u540e\u9a8c\u5206\u5e03\uff0c\u4ece\u800c\u7b80\u5316\u63a8\u65ad\u95ee\u9898\u3002<\/p>\n\n\n\n
                          \u5176\u4e2d\uff0c\u53d8\u5206\u63a8\u65ad\u662f\u4e00\u79cd\u57fa\u4e8e\u4f18\u5316\u7684\u65b9\u6cd5\uff0c\u5b83\u901a\u8fc7\u6700\u5c0f\u5316\u539f\u59cb\u540e\u9a8c\u5206\u5e03\u4e0e\u4e00\u4e2a\u6613\u4e8e\u8ba1\u7b97\u7684\u5206\u5e03\u4e4b\u95f4\u7684\u5dee\u5f02\u6765\u8fdb\u884c\u8fd1\u4f3c\u63a8\u65ad\u3002\u671f\u671b\u4f20\u64ad\u662f\u4e00\u79cd\u57fa\u4e8e\u56fe\u6a21\u578b\u7684\u65b9\u6cd5\uff0c\u5b83\u5c06\u95ee\u9898\u8f6c\u5316\u4e3a\u8ba1\u7b97\u5404\u4e2a\u8282\u70b9\u7684\u671f\u671b\u503c\uff0c\u4ece\u800c\u8fd1\u4f3c\u8868\u793a\u540e\u9a8c\u5206\u5e03\u3002\u8499\u7279\u5361\u7f57\u65b9\u6cd5\u5219\u662f\u901a\u8fc7\u968f\u673a\u91c7\u6837\u6765\u8fd1\u4f3c\u8ba1\u7b97\u540e\u9a8c\u5206\u5e03\u3002<\/p>\n\n\n\n
                          \u8fd1\u4f3c\u63a8\u65ad\u662f\u6982\u7387\u56fe\u6a21\u578b\u4e2d\u975e\u5e38\u91cd\u8981\u7684\u4e00\u73af\uff0c\u5b83\u4e0d\u4ec5\u53ef\u4ee5\u7528\u4e8e\u63a8\u65ad\uff0c\u8fd8\u53ef\u4ee5\u7528\u4e8e\u6982\u7387\u6a21\u578b\u7684\u5b66\u4e60\u548c\u4f18\u5316\u3002<\/p>\n\n\n\n\n\n
                          \"gun<\/figure>\n\n\n\n
                          \u8bf7\u7ed9\u51fa\u8fd1\u4f3c\u63a8\u65ad\u7684\u4ee3\u7801\u793a\u4f8b<\/p>\n\n\n\n
                          \u8fd1\u4f3c\u63a8\u65ad\u662f\u4e00\u79cd\u901a\u8fc7\u91c7\u6837\u3001\u4f18\u5316\u6216\u5176\u4ed6\u65b9\u6cd5\u8fd1\u4f3c\u8ba1\u7b97\u65e0\u6cd5\u89e3\u6790\u6c42\u89e3\u7684\u6982\u7387\u5206\u5e03\u7684\u65b9\u6cd5\u3002\u4e0b\u9762\u662f\u4e00\u4e9b\u5e38\u89c1\u7684\u8fd1\u4f3c\u63a8\u65ad\u7b97\u6cd5\u7684\u7b80\u5355\u4ee3\u7801\u793a\u4f8b\uff1a<\/p>\n\n\n\n
                            \n
                          1. \u8499\u7279\u5361\u6d1b\u65b9\u6cd5<\/li>\n<\/ol>\n\n\n\n
                            \u8499\u7279\u5361\u6d1b\u65b9\u6cd5\u662f\u4e00\u79cd\u901a\u8fc7\u968f\u673a\u91c7\u6837\u6765\u4f30\u8ba1\u6982\u7387\u5206\u5e03\u7684\u65b9\u6cd5\u3002\u4e0b\u9762\u662f\u4e00\u4e2a\u7b80\u5355\u7684\u793a\u4f8b\uff0c\u6f14\u793a\u5982\u4f55\u4f7f\u7528\u8499\u7279\u5361\u6d1b\u65b9\u6cd5\u6765\u4f30\u8ba1\u51fd\u6570 $f(x)=\\frac{1}{\\sqrt{2\\pi}}e^{-x^2\/2}$ \u5728\u533a\u95f4 $[0,1]$ \u4e0a\u7684\u79ef\u5206\uff1a<\/p>\n\n\n\n
                            pythonCopy codeimport random\n\ndef f(x):\n    return 1 \/ (2**0.5 * 3.1415**0.5) * math.exp(-x**2 \/ 2)\n\nN = 1000000\nsum = 0\nfor i in range(N):\n    x = random.uniform(0, 1)\n    sum += f(x)\n\nresult = sum \/ N\nprint(result)\n<\/code><\/pre>\n\n\n\n
                              \n
                            1. \u53d8\u5206\u63a8\u65ad<\/li>\n<\/ol>\n\n\n\n
                              \u53d8\u5206\u63a8\u65ad\u662f\u4e00\u79cd\u901a\u8fc7\u4f18\u5316\u4e00\u4e2a\u7b80\u5316\u7684\u8fd1\u4f3c\u5206\u5e03\u6765\u8fd1\u4f3c\u771f\u5b9e\u7684\u540e\u9a8c\u5206\u5e03\u7684\u65b9\u6cd5\u3002\u4e0b\u9762\u662f\u4e00\u4e2a\u7b80\u5355\u7684\u793a\u4f8b\uff0c\u6f14\u793a\u5982\u4f55\u4f7f\u7528\u53d8\u5206\u63a8\u65ad\u6765\u4f30\u8ba1\u4e00\u4e9b\u7b80\u5355\u7684\u9ad8\u65af\u6df7\u5408\u6a21\u578b\u7684\u53c2\u6570\uff1a<\/p>\n\n\n\n
                              pythonCopy codeimport numpy as np\nfrom scipy.stats import norm\n\ndef var_inference(data, K, max_iter=100):\n    # Initialize means, variances and mixture proportions\n    N, D = data.shape\n    mu = np.random.randn(K, D)\n    sigma = np.ones((K, D))\n    pi = np.ones(K) \/ K\n    \n    # Iterate until convergence\n    for i in range(max_iter):\n        # Compute the approximate posterior\n        q = np.zeros((N, K))\n        for k in range(K):\n            q[:,k] = pi[k] * norm(mu[k], sigma[k]).pdf(data).prod(axis=1)\n        q \/= q.sum(axis=1, keepdims=True)\n        \n        # Update the mixture proportions\n        pi = q.mean(axis=0)\n        \n        # Update the means and variances\n        for k in range(K):\n            mu[k] = (q[:,k,None] * data).sum(axis=0) \/ q[:,k].sum()\n            sigma[k] = ((q[:,k,None] * (data - mu[k])**2).sum(axis=0)\n                       \/ q[:,k].sum())\n    \n    return mu, sigma, pi\n<\/code><\/pre>\n\n\n\n
                              \u8fd9\u4e2a\u51fd\u6570\u91c7\u7528\u4e00\u4e9b\u89c2\u5bdf\u5230\u7684\u6570\u636e\uff0c\u5047\u8bbe\u5b83\u4eec\u6765\u81ea\u4e00\u4e2a\u9ad8\u65af\u6df7\u5408\u6a21\u578b\uff0c\u5e76\u901a\u8fc7\u53d8\u5206\u63a8\u65ad\u6765\u4f30\u8ba1\u6bcf\u4e2a\u7ec4\u4ef6\u7684\u5747\u503c\u548c\u65b9\u5dee\u4ee5\u53ca\u6df7\u5408\u6bd4\u4f8b\u3002<\/p>\n\n\n\n
                                \n
                              1. \u9a6c\u5c14\u53ef\u592b\u94fe\u8499\u7279\u5361\u7f57\u6cd5\uff08MCMC\uff09<\/li>\n<\/ol>\n\n\n\n
                                MCMC \u662f\u4e00\u79cd\u901a\u8fc7\u6784\u9020\u4e00\u4e2a\u9a6c\u5c14\u53ef\u592b\u94fe\u6765\u6a21\u62df\u5206\u5e03\u7684\u65b9\u6cd5\u3002\u4e0b\u9762\u662f\u4e00\u4e2a\u7b80\u5355\u7684\u793a\u4f8b\uff0c\u6f14\u793a\u5982\u4f55\u4f7f\u7528 Metropolis-Hastings \u7b97\u6cd5\u4ece\u4e8c\u5143\u9ad8\u65af\u5206\u5e03\u4e2d\u91c7\u6837\uff1a<\/p>\n\n\n\n
                                pythonCopy codeimport numpy as np\nfrom scipy.stats import multivariate_normal\n\ndef met\n<\/code><\/pre>\n\n\n\n\n\n
                                \"gun<\/figure>\n\n\n\n
                                \u8bf7\u8be6\u7ec6\u89e3\u91ca\u9192\u7720\u7b97\u6cd5<\/p>\n\n\n\n
                                \u9192\u7720\u7b97\u6cd5\uff08Wake-Sleep Algorithm\uff09\u662f\u4e00\u79cd\u7528\u4e8e\u5b66\u4e60\u6982\u7387\u751f\u6210\u6a21\u578b\u53c2\u6570\u7684\u7b97\u6cd5\uff0c\u6700\u521d\u7531Hinton\u7b49\u4eba\u63d0\u51fa\u3002\u5b83\u7684\u6838\u5fc3\u601d\u60f3\u662f\u901a\u8fc7\u8fed\u4ee3\u7684\u65b9\u5f0f\uff0c\u4ece\u672a\u6807\u8bb0\u7684\u6570\u636e\u4e2d\u5b66\u4e60\u6982\u7387\u751f\u6210\u6a21\u578b\u7684\u53c2\u6570\u3002\u7b97\u6cd5\u4e3b\u8981\u5206\u4e3a\u4e24\u4e2a\u90e8\u5206\uff1a\u9192\uff08Wake\uff09\u548c\u7761\u7720\uff08Sleep\uff09\u3002<\/p>\n\n\n\n
                                \u5728\u9192\u7720\u7b97\u6cd5\u7684\u201c\u9192\u201d\u90e8\u5206\uff0c\u9996\u5148\u5229\u7528\u5df2\u77e5\u7684\u53c2\u6570\u4ece\u6570\u636e\u4e2d\u62bd\u53d6\u6837\u672c\uff0c\u7136\u540e\u8ba1\u7b97\u5bf9\u6570\u6982\u7387\u5e76\u8ba1\u7b97\u68af\u5ea6\u3002\u8fd9\u4e00\u8fc7\u7a0b\u7c7b\u4f3c\u4e8e\u7528\u6700\u5927\u4f3c\u7136\u4f30\u8ba1\u6cd5\uff08MLE\uff09\u8fdb\u884c\u8bad\u7ec3\u3002\u5728\u201c\u7761\u7720\u201d\u90e8\u5206\uff0c\u6211\u4eec\u4f7f\u7528\u53cd\u5411\u63a8\u7406\u6765\u751f\u6210\u6570\u636e\u6837\u672c\uff0c\u8ba1\u7b97\u6bcf\u4e2a\u9690\u53d8\u91cf\u7684\u540e\u9a8c\u6982\u7387\uff0c\u5e76\u5229\u7528\u8fd9\u4e9b\u540e\u9a8c\u6982\u7387\u6765\u4f30\u8ba1\u53c2\u6570\u3002<\/p>\n\n\n\n
                                \u603b\u7684\u6765\u8bf4\uff0c\u9192\u7720\u7b97\u6cd5\u7684\u76ee\u7684\u662f\u6700\u5c0f\u5316\u6a21\u578b\u751f\u6210\u6837\u672c\u7684\u635f\u5931\u51fd\u6570\uff0c\u8be5\u51fd\u6570\u5305\u62ec\u4e24\u4e2a\u90e8\u5206\uff1a\u4e00\u4e2a\u662f\u7531\u9192\u90e8\u5206\u8d21\u732e\u7684\u8bef\u5dee\uff0c\u53e6\u4e00\u4e2a\u662f\u7531\u7761\u7720\u90e8\u5206\u8d21\u732e\u7684\u8bef\u5dee\u3002\u901a\u8fc7\u8fed\u4ee3\u8fd9\u4e2a\u8fc7\u7a0b\uff0c\u6211\u4eec\u53ef\u4ee5\u9010\u6e10\u4f18\u5316\u6a21\u578b\u7684\u53c2\u6570\u3002<\/p>\n\n\n\n
                                \u4ee5\u4e0b\u662f\u9192\u7720\u7b97\u6cd5\u7684\u4f2a\u4ee3\u7801\uff1a<\/p>\n\n\n\n
                                perlCopy coderepeat until convergence:\n    # Wake phase\n    for each training example x:\n        h ~ P(h|x, theta)  # Sample from the conditional distribution P(h|x, theta)\n        gradient = dlogP(x,h)\/dtheta  # Compute gradient of log-likelihood with respect to theta\n        update theta using the gradient\n        \n    # Sleep phase\n    for each sample h:\n        x ~ P(x|h, theta)  # Sample from the conditional distribution P(x|h, theta)\n        q(h|x) = P(h|x, theta)  # Compute the posterior distribution over h given x and theta\n        gradient = dlogP(x,h)\/dtheta  # Compute gradient of log-likelihood with respect to theta\n        update theta using the gradient estimated using q(h|x)\n<\/code><\/pre>\n\n\n\n
                                \u5176\u4e2d\uff0cx<\/code> \u8868\u793a\u8f93\u5165\u6570\u636e\uff0ch<\/code> \u8868\u793a\u6f5c\u5728\u53d8\u91cf\uff0ctheta<\/code> \u8868\u793a\u6a21\u578b\u7684\u53c2\u6570\uff0cP(h|x, theta)<\/code> \u8868\u793a\u7ed9\u5b9a\u8f93\u5165 x<\/code> \u548c\u53c2\u6570 theta<\/code> \u65f6\uff0ch<\/code> \u7684\u540e\u9a8c\u5206\u5e03\uff0cP(x|h, theta)<\/code> \u8868\u793a\u7ed9\u5b9a h<\/code> \u548c\u53c2\u6570 theta<\/code> \u65f6\uff0cx<\/code> \u7684\u6761\u4ef6\u5206\u5e03\uff0cq(h|x)<\/code> \u8868\u793a\u7ed9\u5b9a x<\/code> \u548c\u53c2\u6570 theta<\/code> \u65f6\uff0ch<\/code> \u7684\u8fd1\u4f3c\u540e\u9a8c\u5206\u5e03\u3002<\/p>\n\n\n\n
                                \u5728\u4ee3\u7801\u5b9e\u73b0\u65f6\uff0c\u53ef\u4ee5\u4f7f\u7528\u6982\u7387\u7f16\u7a0b\u5de5\u5177\uff08\u5982Pyro\u3001Stan\u7b49\uff09\u6765\u7b80\u5316\u7b97\u6cd5\u7684\u5b9e\u73b0\u8fc7\u7a0b\u3002<\/p>\n\n\n\n\n\n
                                \"gun<\/figure>\n\n\n\n
                                \u8bf7\u7ed9\u51fa\u9192\u7720\u7b97\u6cd5\u7684\u4ee3\u7801\u793a\u4f8b<\/p>\n\n\n\n
                                \u5f88\u62b1\u6b49\uff0c\u7531\u4e8e\u9192\u76ee\u7b97\u6cd5\u4e0d\u662f\u4e00\u4e2a\u7279\u5b9a\u7684\u7b97\u6cd5\uff0c\u800c\u662f\u4e00\u4e2a\u901a\u7528\u7684\u601d\u60f3\uff0c\u56e0\u6b64\u65e0\u6cd5\u7ed9\u51fa\u5177\u4f53\u7684\u4ee3\u7801\u793a\u4f8b\u3002\u9192\u76ee\u7b97\u6cd5\u7684\u5b9e\u73b0\u65b9\u5f0f\u548c\u5177\u4f53\u7684\u5e94\u7528\u573a\u666f\u6709\u5173\uff0c\u53ef\u4ee5\u4f7f\u7528\u4e0d\u540c\u7684\u6982\u7387\u6a21\u578b\u548c\u63a8\u65ad\u7b97\u6cd5\u6765\u5b9e\u73b0\u3002\u4ee5\u4e0b\u662f\u4e00\u4e2a\u901a\u7528\u7684\u9192\u76ee\u7b97\u6cd5\u4f2a\u4ee3\u7801\uff1a<\/p>\n\n\n\n
                                makefileCopy code# Initialize the model parameters\ntheta = initial_parameters()\n\n# Initialize the \"sleeping\" variables\nz = None\n\n# Initialize the number of iterations\nnum_iterations = 0\n\nwhile not converged:\n    # Choose a \"sleeping\" variable\n    z = sample_sleeping_variable(theta)\n    \n    # Compute the \"awakening\" distribution\n    p = compute_awakening_distribution(z, theta)\n    \n    # Choose a \"wakeful\" value for the \"sleeping\" variable\n    z_new = sample_wakeful_value(p)\n    \n    # Update the model parameters\n    theta = update_parameters(z_new, z, theta)\n    \n    # Increment the number of iterations\n    num_iterations += 1\n    \n    # Check for convergence\n    converged = check_convergence(theta, num_iterations)\n<\/code><\/pre>\n\n\n\n
                                \u5728\u8fd9\u4e2a\u4f2a\u4ee3\u7801\u4e2d\uff0ctheta<\/code>\u662f\u6a21\u578b\u53c2\u6570\u7684\u5411\u91cf\uff0cz<\/code>\u662f\u201c\u7761\u7720\u201d\u53d8\u91cf\uff0c\u5b83\u53ef\u4ee5\u5728\u6bcf\u6b21\u8fed\u4ee3\u4e2d\u88ab\u66f4\u65b0\uff0cnum_iterations<\/code>\u662f\u8fed\u4ee3\u6b21\u6570\uff0cconverged<\/code>\u662f\u4e00\u4e2a\u5e03\u5c14\u53d8\u91cf\uff0c\u8868\u793a\u662f\u5426\u8fbe\u5230\u4e86\u6536\u655b\u6761\u4ef6\u3002\u5728\u6bcf\u6b21\u8fed\u4ee3\u4e2d\uff0c\u6211\u4eec\u9996\u5148\u9009\u62e9\u4e00\u4e2a\u201c\u7761\u7720\u201d\u53d8\u91cfz<\/code>\uff0c\u7136\u540e\u8ba1\u7b97\u201c\u89c9\u9192\u201d\u5206\u5e03p<\/code>\uff0c\u5e76\u6839\u636e\u5b83\u9009\u62e9\u4e00\u4e2a\u201c\u89c9\u9192\u201d\u503cz_new<\/code>\u3002\u7136\u540e\uff0c\u6211\u4eec\u4f7f\u7528z_new<\/code>\u548cz<\/code>\u66f4\u65b0\u6a21\u578b\u53c2\u6570theta<\/code>\uff0c\u5e76\u589e\u52a0num_iterations<\/code>\u3002\u6700\u540e\uff0c\u6211\u4eec\u68c0\u67e5\u6536\u655b\u6761\u4ef6\uff0c\u5982\u679c\u8fbe\u5230\u4e86\u5c31\u505c\u6b62\u8fed\u4ee3\uff0c\u5426\u5219\u7ee7\u7eed\u4e0b\u4e00\u6b21\u8fed\u4ee3\u3002<\/p>\n\n\n\n\n\n
                                \"gun<\/figure>\n\n\n\n
                                \u8bf7\u8be6\u7ec6\u89e3\u91ca\u73bb\u5c14\u5179\u66fc\u673a<\/p>\n\n\n\n
                                \u73bb\u5c14\u5179\u66fc\u673a\uff08Boltzmann machine\uff09\u662f\u4e00\u79cd\u6982\u7387\u751f\u6210\u6a21\u578b\uff0c\u5b83\u53ef\u4ee5\u88ab\u770b\u4f5c\u662f\u4e00\u79cd\u5e26\u9690\u5c42\u7684\u53d7\u9650\u73bb\u5c14\u5179\u66fc\u673a\u3002\u5b83\u7531\u53ef\u89c1\u5c42\u548c\u9690\u5c42\u7ec4\u6210\uff0c\u53ef\u89c1\u5c42\u548c\u9690\u5c42\u4e4b\u95f4\u7684\u795e\u7ecf\u5143\u4e4b\u95f4\u5b58\u5728\u6743\u91cd\u3002\u73bb\u5c14\u5179\u66fc\u673a\u7684\u76ee\u6807\u662f\u5b66\u4e60\u51fa\u8fd9\u4e9b\u6743\u91cd\uff0c\u4ee5\u4fbf\u6a21\u578b\u53ef\u4ee5\u751f\u6210\u65b0\u7684\u6837\u672c\u6216\u8bc6\u522b\u7ed9\u5b9a\u7684\u6837\u672c\u3002<\/p>\n\n\n\n
                                \u5728\u73bb\u5c14\u5179\u66fc\u673a\u4e2d\uff0c\u6bcf\u4e2a\u795e\u7ecf\u5143\u90fd\u662f\u4e00\u4e2a\u4e8c\u5143\u53d8\u91cf\uff080\u62161\uff09\u3002\u4e00\u4e2a\u795e\u7ecf\u5143\u7684\u72b6\u6001\u53d6\u51b3\u4e8e\u5176\u4ed6\u795e\u7ecf\u5143\u7684\u72b6\u6001\u548c\u5b83\u4e0e\u5176\u4ed6\u795e\u7ecf\u5143\u4e4b\u95f4\u7684\u6743\u91cd\u3002\u7ed9\u5b9a\u795e\u7ecf\u5143\u7684\u72b6\u6001\uff0c\u8be5\u795e\u7ecf\u5143\u88ab\u6fc0\u6d3b\u7684\u6982\u7387\u53ef\u4ee5\u901a\u8fc7\u4e00\u4e2asigmoid\u51fd\u6570\u6765\u8ba1\u7b97\uff0csigmoid\u51fd\u6570\u4e2d\u7684\u53c2\u6570\u662f\u8be5\u795e\u7ecf\u5143\u7684\u8f93\u5165\u548c\u6743\u91cd\u4e4b\u548c\u3002<\/p>\n\n\n\n
                                \u73bb\u5c14\u5179\u66fc\u673a\u7684\u8bad\u7ec3\u8fc7\u7a0b\u662f\u901a\u8fc7\u6700\u5927\u5316\u8bad\u7ec3\u6570\u636e\u7684\u4f3c\u7136\u51fd\u6570\u6765\u5b8c\u6210\u7684\u3002\u5728\u8bad\u7ec3\u4e2d\uff0c\u901a\u8fc7\u968f\u673a\u62bd\u53d6\u6837\u672c\u6765\u66f4\u65b0\u6743\u91cd\u3002\u73bb\u5c14\u5179\u66fc\u673a\u901a\u5e38\u4f7f\u7528Gibbs\u91c7\u6837\u6765\u751f\u6210\u65b0\u7684\u6837\u672c\u3002<\/p>\n\n\n\n
                                \u73bb\u5c14\u5179\u66fc\u673a\u53ef\u4ee5\u7528\u4e8e\u8bb8\u591a\u4efb\u52a1\uff0c\u5305\u62ec\u56fe\u50cf\u548c\u97f3\u9891\u7684\u751f\u6210\u3001\u5206\u7c7b\u548c\u964d\u7ef4\u7b49\u3002<\/p>\n\n\n\n
                                \u4e0b\u9762\u662f\u4e00\u4e2a\u7b80\u5355\u7684\u73bb\u5c14\u5179\u66fc\u673a\u7684Python\u793a\u4f8b\uff1a<\/p>\n\n\n\n
                                pythonCopy codeimport numpy as np\n\nclass BoltzmannMachine:\n    def __init__(self, num_visible, num_hidden):\n        self.num_visible = num_visible\n        self.num_hidden = num_hidden\n        self.weights = np.random.randn(num_visible, num_hidden)\n        self.visible_bias = np.zeros(num_visible)\n        self.hidden_bias = np.zeros(num_hidden)\n        \n    def sigmoid(self, x):\n        return 1.0 \/ (1.0 + np.exp(-x))\n        \n    def sample_hidden(self, visible):\n        weighted_sum = np.dot(visible, self.weights) + self.hidden_bias\n        hidden_prob = self.sigmoid(weighted_sum)\n        return (hidden_prob > np.random.rand(self.num_hidden)).astype(int)\n    \n    def sample_visible(self, hidden):\n        weighted_sum = np.dot(hidden, self.weights.T) + self.visible_bias\n        visible_prob = self.sigmoid(weighted_sum)\n        return (visible_prob > np.random.rand(self.num_visible)).astype(int)\n    \n    def gibbs_sampling(self, visible, num_samples):\n        for i in range(num_samples):\n            hidden = self.sample_hidden(visible)\n            visible = self.sample_visible(hidden)\n        return visible\n    \n    def train(self, data, learning_rate=0.1, num_epochs=100):\n        num_examples = data.shape[0]\n        for epoch in range(num_epochs):\n            for i in range(num_examples):\n                visible = data[i]\n                hidden_prob = self.sigmoid(np.dot(visible, self.weights) + self.hidden_bias)\n                hidden_state = (hidden_prob > np.random.rand(self.num_hidden)).astype(int)\n                positive_grad = np.outer(visible, hidden_prob)\n                negative_grad = np.outer(visible, self.sigmoid(np.dot(hidden_state, self.weights.T) + self.visible_bias))\n                self.weights += learning_rate * (positive_grad - negative_grad)\n                self.hidden_bias += learning\n<\/code><\/pre>\n\n\n\n\n\n
                                \"gun<\/figure>\n\n\n\n
                                \u8bf7\u8be6\u7ec6\u89e3\u91ca\u4fe1\u5ff5\u7f51\u7edc<\/p>\n\n\n\n
                                \u4fe1\u5ff5\u7f51\u7edc\uff08Belief Network\uff09\u662f\u4e00\u79cd\u6982\u7387\u56fe\u6a21\u578b\uff0c\u4e5f\u88ab\u79f0\u4e3a\u8d1d\u53f6\u65af\u7f51\u7edc\u6216\u8005\u8d1d\u53f6\u65af\u4fe1\u5ff5\u7f51\u7edc\u3002\u5b83\u662f\u57fa\u4e8e\u6709\u5411\u65e0\u73af\u56fe\uff08DAG\uff09\u7684\u4e00\u79cd\u6982\u7387\u56fe\u6a21\u578b\uff0c\u7528\u4e8e\u8868\u793a\u53d8\u91cf\u4e4b\u95f4\u7684\u6982\u7387\u5173\u7cfb\uff0c\u5176\u4e2d\u6bcf\u4e2a\u8282\u70b9\u8868\u793a\u4e00\u4e2a\u53d8\u91cf\uff0c\u6bcf\u4e2a\u8fb9\u8868\u793a\u4e24\u4e2a\u53d8\u91cf\u4e4b\u95f4\u7684\u6982\u7387\u5173\u7cfb\u3002\u4fe1\u5ff5\u7f51\u7edc\u53ef\u4ee5\u7528\u4e8e\u63a8\u65ad\u548c\u6982\u7387\u63a8\u65ad\uff0c\u4f8b\u5982\u5206\u7c7b\u3001\u56de\u5f52\u3001\u805a\u7c7b\u3001\u5f02\u5e38\u68c0\u6d4b\u7b49\u3002<\/p>\n\n\n\n
                                \u5728\u4fe1\u5ff5\u7f51\u7edc\u4e2d\uff0c\u8282\u70b9\u5206\u4e3a\u4e24\u7c7b\uff1a\u89c2\u5bdf\u8282\u70b9\u548c\u9690\u53d8\u91cf\u8282\u70b9\u3002\u89c2\u5bdf\u8282\u70b9\u5bf9\u5e94\u4e8e\u5df2\u77e5\u7684\u53d8\u91cf\uff0c\u9690\u53d8\u91cf\u8282\u70b9\u5bf9\u5e94\u4e8e\u672a\u77e5\u7684\u53d8\u91cf\u3002\u6bcf\u4e2a\u8282\u70b9\u90fd\u6709\u4e00\u4e2a\u6761\u4ef6\u6982\u7387\u5206\u5e03\uff0c\u7528\u4e8e\u63cf\u8ff0\u8be5\u8282\u70b9\u7684\u53d6\u503c\u4e0e\u5176\u7236\u8282\u70b9\u7684\u53d6\u503c\u4e4b\u95f4\u7684\u5173\u7cfb\u3002\u901a\u8fc7\u8fd9\u4e9b\u6761\u4ef6\u6982\u7387\u5206\u5e03\u548c\u8d1d\u53f6\u65af\u516c\u5f0f\uff0c\u53ef\u4ee5\u8ba1\u7b97\u7ed9\u5b9a\u89c2\u5bdf\u8282\u70b9\u7684\u6761\u4ef6\u4e0b\uff0c\u9690\u53d8\u91cf\u8282\u70b9\u7684\u540e\u9a8c\u6982\u7387\u5206\u5e03\uff0c\u4ece\u800c\u8fdb\u884c\u63a8\u65ad\u3002<\/p>\n\n\n\n
                                \u4fe1\u5ff5\u7f51\u7edc\u53ef\u4ee5\u901a\u8fc7\u8d1d\u53f6\u65af\u7f51\u7ed3\u6784\u7684\u5b66\u4e60\u6765\u81ea\u52a8\u63a8\u5bfc\u6761\u4ef6\u6982\u7387\u5206\u5e03\uff0c\u5e38\u7528\u7684\u5b66\u4e60\u7b97\u6cd5\u5305\u62ec\u6700\u5927\u4f3c\u7136\u4f30\u8ba1\u548c\u8d1d\u53f6\u65af\u7f51\u7edc\u7ed3\u6784\u5b66\u4e60\u3002<\/p>\n\n\n\n
                                \u4ee5\u4e0b\u662f\u4e00\u4e2a\u7b80\u5355\u7684\u4fe1\u5ff5\u7f51\u7edc\u7684\u4f8b\u5b50\uff0c\u5176\u4e2d\u89c2\u5bdf\u8282\u70b9\u4e3a\u7070\u8272\u8282\u70b9\uff0c\u9690\u53d8\u91cf\u8282\u70b9\u4e3a\u767d\u8272\u8282\u70b9\uff1a<\/p>\n\n\n\n
                                \"belief_network_example\"\/<\/figure>\n\n\n\n
                                \u5728\u8fd9\u4e2a\u4f8b\u5b50\u4e2d\uff0c\u6211\u4eec\u53ef\u4ee5\u770b\u5230\uff0c\u89c2\u5bdf\u8282\u70b9\u201c\u53d1\u70ed\u201d\u548c\u201c\u54b3\u55fd\u201d\u662f\u76f4\u63a5\u76f8\u8fde\u7684\uff0c\u56e0\u4e3a\u53d1\u70ed\u548c\u54b3\u55fd\u53ef\u80fd\u662f\u7531\u540c\u4e00\u79cd\u75be\u75c5\u5f15\u8d77\u7684\u3002\u800c\u9690\u53d8\u91cf\u8282\u70b9\u201c\u611f\u5192\u201d\u548c\u201c\u6d41\u611f\u201d\u5219\u662f\u7531\u53d1\u70ed\u548c\u54b3\u55fd\u8fd9\u4e24\u4e2a\u89c2\u5bdf\u8282\u70b9\u5171\u540c\u5f71\u54cd\u7684\uff0c\u56e0\u6b64\u5b83\u4eec\u4e4b\u95f4\u4e5f\u5b58\u5728\u6982\u7387\u5173\u7cfb\u3002<\/p>\n\n\n\n
                                \u5728\u8fdb\u884c\u63a8\u65ad\u65f6\uff0c\u6211\u4eec\u53ef\u4ee5\u8f93\u5165\u4e00\u4e9b\u89c2\u5bdf\u8282\u70b9\u7684\u53d6\u503c\uff0c\u7136\u540e\u4f7f\u7528\u4fe1\u5ff5\u7f51\u7edc\u7684\u6761\u4ef6\u6982\u7387\u5206\u5e03\u8ba1\u7b97\u9690\u53d8\u91cf\u8282\u70b9\u7684\u540e\u9a8c\u6982\u7387\u5206\u5e03\u3002\u4f8b\u5982\uff0c\u5982\u679c\u6211\u4eec\u77e5\u9053\u4e00\u4e2a\u4eba\u53d1\u70ed\u548c\u54b3\u55fd\u4e86\uff0c\u90a3\u4e48\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u4fe1\u5ff5\u7f51\u7edc\u8ba1\u7b97\u51fa\u4ed6\u611f\u67d3\u4e86\u611f\u5192\u6216\u8005\u6d41\u611f\u7684\u540e\u9a8c\u6982\u7387\u5206\u5e03\uff0c\u4ece\u800c\u8fdb\u884c\u5206\u7c7b\u6216\u8005\u5176\u4ed6\u7684\u63a8\u65ad\u4efb\u52a1\u3002<\/p>\n\n\n\n\n\n
                                \"gun<\/figure>\n\n\n\n
                                \u8bf7\u7ed9\u51fa\u91ca\u4fe1\u5ff5\u7f51\u7edc\u7684\u4ee3\u7801\u793a\u4f8b<\/p>\n\n\n\n
                                \u4fe1\u5ff5\u7f51\u7edc\u662f\u4e00\u79cd\u7528\u4e8e\u65e0\u5411\u56fe\u6a21\u578b\u7684\u6982\u7387\u56fe\u6a21\u578b\uff0c\u5b83\u7531\u591a\u4e2a\u4e8c\u5143\u8282\u70b9\u7ec4\u6210\uff0c\u6bcf\u4e2a\u8282\u70b9\u53ef\u4ee5\u8868\u793a\u4e00\u4e2a\u89c2\u6d4b\u53d8\u91cf\u6216\u4e00\u4e2a\u6f5c\u5728\u53d8\u91cf\uff0c\u5b83\u4eec\u4e4b\u95f4\u76f8\u4e92\u4f5c\u7528\uff0c\u6784\u6210\u4e00\u4e2a\u8054\u5408\u5206\u5e03\u3002\u5728\u4fe1\u5ff5\u7f51\u7edc\u4e2d\uff0c\u6f5c\u5728\u53d8\u91cf\u662f\u672a\u77e5\u7684\uff0c\u800c\u89c2\u6d4b\u53d8\u91cf\u662f\u5df2\u77e5\u7684\u3002\u4fe1\u5ff5\u7f51\u7edc\u53ef\u4ee5\u7528\u4e8e\u5206\u7c7b\u3001\u805a\u7c7b\u3001\u8bed\u97f3\u8bc6\u522b\u7b49\u591a\u4e2a\u9886\u57df\u3002<\/p>\n\n\n\n
                                \u4e0b\u9762\u7ed9\u51fa\u4e00\u4e2a\u7b80\u5355\u7684\u4ee3\u7801\u793a\u4f8b\uff0c\u5c55\u793a\u5982\u4f55\u4f7f\u7528Python\u7684PyMC3\u5e93\u5b9e\u73b0\u4e00\u4e2a\u7b80\u5355\u7684\u4fe1\u5ff5\u7f51\u7edc\u3002\u672c\u4f8b\u4e2d\u4f7f\u7528\u4fe1\u5ff5\u7f51\u7edc\u5b9e\u73b0\u4e8c\u5206\u7c7b\u4efb\u52a1\uff0c\u6570\u636e\u96c6\u662f\u9e22\u5c3e\u82b1\u6570\u636e\u96c6\u3002\u4fe1\u5ff5\u7f51\u7edc\u7684\u7ed3\u6784\u5982\u4e0b\uff1a<\/p>\n\n\n\n
                                \"Belief<\/figure>\n\n\n\n
                                \u5176\u4e2d\uff0c$X_1$\u8868\u793a\u82b1\u843c\u957f\u5ea6\uff0c$X_2$\u8868\u793a\u82b1\u843c\u5bbd\u5ea6\uff0c$X_3$\u8868\u793a\u82b1\u74e3\u957f\u5ea6\uff0c$X_4$\u8868\u793a\u82b1\u74e3\u5bbd\u5ea6\uff0c$Y$\u8868\u793a\u9e22\u5c3e\u82b1\u7684\u79cd\u7c7b\u3002<\/p>\n\n\n\n
                                \u4ee3\u7801\u5b9e\u73b0\u5982\u4e0b\uff1a<\/p>\n\n\n\n
                                pythonCopy codeimport pymc3 as pm\nimport numpy as np\nimport pandas as pd\nfrom sklearn.datasets import load_iris\n\n# load iris dataset\niris = load_iris()\ndata = pd.DataFrame(data= np.c_[iris['data'], iris['target']],\n                     columns= iris['feature_names'] + ['target'])\n\n# define model\nwith pm.Model() as model:\n    # priors\n    alpha = pm.Normal('alpha', mu=0, sd=10)\n    beta1 = pm.Normal('beta1', mu=0, sd=10)\n    beta2 = pm.Normal('beta2', mu=0, sd=10)\n    beta3 = pm.Normal('beta3', mu=0, sd=10)\n    beta4 = pm.Normal('beta4', mu=0, sd=10)\n    mu = alpha + beta1 * data['sepal length (cm)'] + beta2 * data['sepal width (cm)'] + \\\n         beta3 * data['petal length (cm)'] + beta4 * data['petal width (cm)']\n    # sigmoid function\n    theta = pm.invlogit(mu)\n    # likelihood\n    y_obs = pm.Bernoulli('y_obs', p=theta, observed=data['target'])\n    \n# inference\nwith model:\n    trace = pm.sample(10000, tune=5000)\n<\/code><\/pre>\n\n\n\n
                                \u5728\u4ee3\u7801\u4e2d\uff0c\u9996\u5148\u5bfc\u5165\u4e86\u5fc5\u8981\u7684\u5e93\uff0c\u5e76\u8f7d\u5165\u9e22\u5c3e\u82b1\u6570\u636e\u96c6\u3002\u7136\u540e\u4f7f\u7528PyMC3\u5b9a\u4e49\u4e86\u4e00\u4e2a\u4fe1\u5ff5\u7f51\u7edc\u6a21\u578b\uff0c\u8be5\u6a21\u578b\u4f7f\u7528\u9ad8\u65af\u5206\u5e03\u4f5c\u4e3a\u5148\u9a8c\u5206\u5e03\uff0c\u4f7f\u7528\u903b\u8f91\u65af\u8482\u51fd\u6570\u4f5c\u4e3a\u6fc0\u6d3b\u51fd\u6570\u3002\u6700\u540e\uff0c\u4f7f\u7528\u9a6c\u5c14\u53ef\u592b\u8499\u7279\u5361\u7f57\u65b9\u6cd5\uff08MCMC\uff09\u91c7\u6837\u5f97\u5230\u540e\u9a8c\u5206\u5e03\uff0c\u83b7\u5f97\u53c2\u6570\u7684\u6700\u4f18\u503c\u3002<\/p>\n\n\n\n\n\n
                                \"gun<\/figure>\n\n\n\n
                                \u8bf7\u8be6\u7ec6\u89e3\u91ca\u5bf9\u6297\u7f51\u7edc<\/p>\n\n\n\n
                                \u5bf9\u6297\u7f51\u7edc\uff08Generative Adversarial Networks\uff0cGANs\uff09\u662f\u4e00\u79cd\u6df1\u5ea6\u5b66\u4e60\u7684\u65e0\u76d1\u7763\u5b66\u4e60\u7b97\u6cd5\uff0c\u901a\u8fc7\u4e24\u4e2a\u795e\u7ecf\u7f51\u7edc\u4e4b\u95f4\u7684\u5bf9\u6297\u6765\u5b66\u4e60\u6570\u636e\u5206\u5e03\uff0c\u751f\u6210\u65b0\u7684\u3001\u4e0e\u771f\u5b9e\u6570\u636e\u7c7b\u4f3c\u7684\u6570\u636e\u3002<\/p>\n\n\n\n
                                GANs\u7531\u4e00\u4e2a\u751f\u6210\u5668\u7f51\u7edc\u548c\u4e00\u4e2a\u5224\u522b\u5668\u7f51\u7edc\u7ec4\u6210\u3002\u751f\u6210\u5668\u7f51\u7edc\u5c06\u566a\u58f0\u5411\u91cf\u4f5c\u4e3a\u8f93\u5165\uff0c\u8f93\u51fa\u751f\u6210\u7684\u6570\u636e\uff0c\u800c\u5224\u522b\u5668\u7f51\u7edc\u5219\u5c06\u751f\u6210\u7684\u6570\u636e\u6216\u771f\u5b9e\u6570\u636e\u4f5c\u4e3a\u8f93\u5165\uff0c\u8f93\u51fa\u4e00\u4e2a\u6982\u7387\u503c\uff0c\u8868\u793a\u8f93\u5165\u6570\u636e\u662f\u771f\u5b9e\u6570\u636e\u7684\u6982\u7387\u3002\u8fd9\u4e24\u4e2a\u7f51\u7edc\u901a\u8fc7\u53cd\u590d\u8fed\u4ee3\uff0c\u4f7f\u751f\u6210\u5668\u7f51\u7edc\u4e0d\u65ad\u751f\u6210\u66f4\u63a5\u8fd1\u771f\u5b9e\u6570\u636e\u7684\u6837\u672c\uff0c\u540c\u65f6\u5224\u522b\u5668\u7f51\u7edc\u4e0d\u65ad\u63d0\u9ad8\u5224\u65ad\u8f93\u5165\u6570\u636e\u771f\u5b9e\u6027\u7684\u51c6\u786e\u7387\u3002<\/p>\n\n\n\n
                                GANs\u7684\u635f\u5931\u51fd\u6570\u53ef\u4ee5\u5199\u6210\uff1a<\/p>\n\n\n\n
                                min\u2061\ufffdmax\u2061\ufffd\ufffd(\ufffd,\ufffd)=\ufffd\ufffd\u223c\ufffd\ufffd\ufffd\ufffd\ufffd(\ufffd)[log\u2061\ufffd(\ufffd)]+\ufffd\ufffd\u223c\ufffd\ufffd(\ufffd)[log\u2061(1\u2212\ufffd(\ufffd(\ufffd)))]G<\/em>min\u200bD<\/em>max\u200bV<\/em>(D<\/em>,G<\/em>)=Ex<\/em>\u223cp<\/em>d<\/em>a<\/em>t<\/em>a<\/em>\u200b(x<\/em>)\u200b[logD<\/em>(x<\/em>)]+Ez<\/em>\u223cp<\/em>z<\/em>\u200b(z<\/em>)\u200b[log(1\u2212D<\/em>(G<\/em>(z<\/em>)))]<\/p>\n\n\n\n
                                \u5176\u4e2d\uff0c$D(x)$\u8868\u793a\u5224\u522b\u5668\u7f51\u7edc\u5bf9\u771f\u5b9e\u6570\u636e$x$\u7684\u6982\u7387\u8f93\u51fa\uff0c$G(z)$\u8868\u793a\u751f\u6210\u5668\u7f51\u7edc\u5bf9\u566a\u58f0$z$\u7684\u751f\u6210\u7ed3\u679c\uff0c$p_{data}(x)$\u548c$p_z(z)$\u5206\u522b\u8868\u793a\u771f\u5b9e\u6570\u636e\u548c\u566a\u58f0\u7684\u5206\u5e03\u3002<\/p>\n\n\n\n
                                \u5728\u8bad\u7ec3\u8fc7\u7a0b\u4e2d\uff0c\u9996\u5148\u901a\u8fc7\u968f\u673a\u566a\u58f0\u751f\u6210\u4e00\u6279\u5047\u6570\u636e\uff0c\u7136\u540e\u5c06\u771f\u5b9e\u6570\u636e\u548c\u751f\u6210\u7684\u5047\u6570\u636e\u653e\u5165\u5224\u522b\u5668\u7f51\u7edc\u4e2d\uff0c\u8ba1\u7b97\u4e24\u8005\u7684\u635f\u5931\u51fd\u6570\uff0c\u5e76\u53cd\u5411\u4f20\u64ad\u66f4\u65b0\u5224\u522b\u5668\u7f51\u7edc\u7684\u53c2\u6570\u3002\u63a5\u7740\u5c06\u968f\u673a\u566a\u58f0\u653e\u5165\u751f\u6210\u5668\u7f51\u7edc\u4e2d\u751f\u6210\u65b0\u7684\u5047\u6570\u636e\uff0c\u5e76\u5c06\u5176\u653e\u5165\u5224\u522b\u5668\u7f51\u7edc\u4e2d\u8ba1\u7b97\u635f\u5931\u51fd\u6570\uff0c\u53cd\u5411\u4f20\u64ad\u66f4\u65b0\u751f\u6210\u5668\u7f51\u7edc\u7684\u53c2\u6570\u3002<\/p>\n\n\n\n
                                \u4ee5\u4e0b\u662f\u4f7f\u7528PyTorch\u5b9e\u73b0GANs\u7684\u4ee3\u7801\u793a\u4f8b\uff1a<\/p>\n\n\n\n\n\n
                                \"gun<\/figure>\n\n\n\n
                                \u8bf7\u7ed9\u51fa\u5bf9\u6297\u7f51\u7edc\u5927\u4ee3\u7801\u793a\u4f8b<\/p>\n\n\n\n
                                \u5bf9\u6297\u7f51\u7edc\uff08Generative Adversarial Networks\uff0c\u7b80\u79f0GAN\uff09\u662f\u4e00\u79cd\u6df1\u5ea6\u5b66\u4e60\u6a21\u578b\uff0c\u5b83\u7531\u4e24\u4e2a\u795e\u7ecf\u7f51\u7edc\u7ec4\u6210\uff1a\u751f\u6210\u5668\uff08Generator\uff09\u548c\u5224\u522b\u5668\uff08Discriminator\uff09\u3002\u751f\u6210\u5668\u8bd5\u56fe\u751f\u6210\u80fd\u591f\u9a97\u8fc7\u5224\u522b\u5668\u7684\u5047\u6570\u636e\uff0c\u800c\u5224\u522b\u5668\u5219\u8bd5\u56fe\u5224\u65ad\u6570\u636e\u662f\u771f\u5b9e\u6570\u636e\u8fd8\u662f\u751f\u6210\u5668\u751f\u6210\u7684\u5047\u6570\u636e\u3002\u901a\u8fc7\u5bf9\u6297\u7684\u8fc7\u7a0b\uff0c\u751f\u6210\u5668\u4e0d\u65ad\u5b66\u4e60\u751f\u6210\u66f4\u771f\u5b9e\u7684\u6570\u636e\uff0c\u800c\u5224\u522b\u5668\u5219\u4e0d\u65ad\u5b66\u4e60\u66f4\u597d\u5730\u5224\u522b\u771f\u5b9e\u6570\u636e\u548c\u751f\u6210\u7684\u5047\u6570\u636e\u3002<\/p>\n\n\n\n
                                \u4ee5\u4e0b\u662f\u4e00\u4e2a\u7b80\u5355\u7684PyTorch\u5b9e\u73b0\uff1a<\/p>\n\n\n\n
                                pythonCopy codeimport torch\nimport torch.nn as nn\nimport torch.optim as optim\nimport torchvision.datasets as dset\nimport torchvision.transforms as transforms\nfrom torch.utils.data import DataLoader\nfrom torchvision.utils import save_image\n\n# \u5b9a\u4e49\u751f\u6210\u5668\u7f51\u7edc\nclass Generator(nn.Module):\n    def __init__(self, input_dim, output_dim):\n        super(Generator, self).__init__()\n        self.fc = nn.Sequential(\n            nn.Linear(input_dim, 256),\n            nn.BatchNorm1d(256),\n            nn.ReLU(),\n            nn.Linear(256, 512),\n            nn.BatchNorm1d(512),\n            nn.ReLU(),\n            nn.Linear(512, 1024),\n            nn.BatchNorm1d(1024),\n            nn.ReLU(),\n            nn.Linear(1024, output_dim),\n            nn.Tanh()\n        )\n\n    def forward(self, x):\n        x = self.fc(x)\n        return x\n\n# \u5b9a\u4e49\u5224\u522b\u5668\u7f51\u7edc\nclass Discriminator(nn.Module):\n    def __init__(self, input_dim):\n        super(Discriminator, self).__init__()\n        self.fc = nn.Sequential(\n            nn.Linear(input_dim, 512),\n            nn.LeakyReLU(0.2),\n            nn.Linear(512, 256),\n            nn.LeakyReLU(0.2),\n            nn.Linear(256, 1),\n            nn.Sigmoid()\n        )\n\n    def forward(self, x):\n        x = self.fc(x)\n        return x\n\n# \u5b9a\u4e49\u8d85\u53c2\u6570\nbatch_size = 128\nlearning_rate = 0.0002\nnum_epochs = 100\ninput_dim = 100\noutput_dim = 784\ndevice = torch.device('cuda' if torch.cuda.is_available() else 'cpu')\n\n# \u52a0\u8f7dMNIST\u6570\u636e\u96c6\ntransform = transforms.Compose([\n    transforms.ToTensor(),\n    transforms.Normalize(mean=[0.5], std=[0.5])\n])\ntrain_dataset = dset.MNIST(root='.\/data', train=True, transform=transform, download=True)\ntrain_loader = DataLoader(train_dataset, batch_size=batch_size, shuffle=True)\n\n# \u521d\u59cb\u5316\u7f51\u7edc\nG = Generator(input_dim, output_dim).to(device)\nD = Discriminator(output_dim).to(device)\n\n# \u5b9a\u4e49\u4f18\u5316\u5668\u548c\u635f\u5931\u51fd\u6570\nG_optimizer = optim.Adam(G.parameters(), lr=learning_rate)\nD_optimizer = optim.Adam(D.parameters(), lr=learning_rate)\ncriterion = nn.BCELoss()\n\n# \u8bad\u7ec3\u7f51\u7edc\nfor epoch in range(num_epochs):\n    for i, (real_images, _) in enumerate(train_loader):\n        real_images = real_images.to(device)\n        real_labels = torch.ones(real_images.size(0), 1).to(device)\n        fake_labels = torch.zeros(real_images.size(0), 1).to(device)\n\n        # \u8bad\u7ec3\u5224\u522b\u5668\n        D_real_outputs = D(real_images.view(real_images.size(0), -1))\n       \n<\/code><\/pre>\n\n\n\n\n\n
                                \"gun<\/figure>\n\n\n\n
                                \u8bf7\u8be6\u7ec6\u89e3\u91ca\u81ea\u7f16\u7801\u5668<\/p>\n\n\n\n
                                \u81ea\u7f16\u7801\u5668\uff08Autoencoder\uff09\u662f\u4e00\u79cd\u7528\u4e8e\u6570\u636e\u538b\u7f29\u548c\u964d\u7ef4\u7684\u795e\u7ecf\u7f51\u7edc\u6a21\u578b\u3002\u5b83\u5305\u542b\u4e00\u4e2a\u7f16\u7801\u5668\u548c\u4e00\u4e2a\u89e3\u7801\u5668\uff0c\u53ef\u4ee5\u5c06\u9ad8\u7ef4\u6570\u636e\u8f6c\u6362\u4e3a\u4f4e\u7ef4\u7f16\u7801\u5e76\u91cd\u5efa\u539f\u59cb\u6570\u636e\u3002\u81ea\u7f16\u7801\u5668\u901a\u5e38\u7528\u4e8e\u65e0\u76d1\u7763\u5b66\u4e60\uff0c\u53ef\u4ee5\u5728\u4e0d\u9700\u8981\u6807\u7b7e\u7684\u60c5\u51b5\u4e0b\u5bf9\u6570\u636e\u8fdb\u884c\u5efa\u6a21\u548c\u5b66\u4e60\u3002<\/p>\n\n\n\n
                                \u81ea\u7f16\u7801\u5668\u7684\u6838\u5fc3\u601d\u60f3\u662f\u901a\u8fc7\u7f16\u7801\u5668\u5c06\u539f\u59cb\u6570\u636e\u8f6c\u6362\u4e3a\u4f4e\u7ef4\u7f16\u7801\uff0c\u5e76\u901a\u8fc7\u89e3\u7801\u5668\u5c06\u7f16\u7801\u8f6c\u6362\u56de\u539f\u59cb\u6570\u636e\u3002\u7f16\u7801\u5668\u548c\u89e3\u7801\u5668\u901a\u5e38\u90fd\u662f\u795e\u7ecf\u7f51\u7edc\uff0c\u5176\u4e2d\u7f16\u7801\u5668\u8d1f\u8d23\u5c06\u539f\u59cb\u6570\u636e\u538b\u7f29\u5230\u4e00\u4e2a\u4f4e\u7ef4\u7a7a\u95f4\u4e2d\uff0c\u800c\u89e3\u7801\u5668\u5219\u8d1f\u8d23\u5c06\u8fd9\u4e2a\u4f4e\u7ef4\u7f16\u7801\u8f6c\u6362\u56de\u539f\u59cb\u6570\u636e\u3002\u81ea\u7f16\u7801\u5668\u7684\u8bad\u7ec3\u76ee\u6807\u662f\u6700\u5c0f\u5316\u539f\u59cb\u6570\u636e\u4e0e\u91cd\u6784\u6570\u636e\u4e4b\u95f4\u7684\u5dee\u5f02\uff0c\u901a\u5e38\u4f7f\u7528\u5747\u65b9\u8bef\u5dee\uff08MSE\uff09\u4f5c\u4e3a\u635f\u5931\u51fd\u6570\u3002<\/p>\n\n\n\n
                                \u81ea\u7f16\u7801\u5668\u7684\u79cd\u7c7b\u5f88\u591a\uff0c\u6700\u7b80\u5355\u7684\u662f\u5355\u5c42\u7684\u5168\u8fde\u63a5\u81ea\u7f16\u7801\u5668\u3002\u4e0b\u9762\u662f\u4e00\u4e2a\u57fa\u4e8ePyTorch\u5b9e\u73b0\u7684\u5355\u5c42\u5168\u8fde\u63a5\u81ea\u7f16\u7801\u5668\u7684\u4ee3\u7801\u793a\u4f8b\uff1a<\/p>\n\n\n\n
                                pythonCopy codeimport torch\nimport torch.nn as nn\nimport torch.optim as optim\n\nclass Autoencoder(nn.Module):\n    def __init__(self, input_dim, encoding_dim):\n        super(Autoencoder, self).__init__()\n        self.encoder = nn.Linear(input_dim, encoding_dim)\n        self.decoder = nn.Linear(encoding_dim, input_dim)\n\n    def forward(self, x):\n        encoded = self.encoder(x)\n        decoded = self.decoder(encoded)\n        return decoded\n\n# \u5b9a\u4e49\u8bad\u7ec3\u6570\u636e\ntrain_data = torch.randn(1000, 20)\n\n# \u521d\u59cb\u5316\u81ea\u7f16\u7801\u5668\nautoencoder = Autoencoder(input_dim=20, encoding_dim=10)\n\n# \u5b9a\u4e49\u4f18\u5316\u5668\u548c\u635f\u5931\u51fd\u6570\noptimizer = optim.Adam(autoencoder.parameters(), lr=0.01)\ncriterion = nn.MSELoss()\n\n# \u8bad\u7ec3\u81ea\u7f16\u7801\u5668\nfor epoch in range(100):\n    # \u524d\u5411\u4f20\u64ad\n    output = autoencoder(train_data)\n    loss = criterion(output, train_data)\n\n    # \u53cd\u5411\u4f20\u64ad\u548c\u4f18\u5316\n    optimizer.zero_grad()\n    loss.backward()\n    optimizer.step()\n\n    # \u6253\u5370\u635f\u5931\n    print('Epoch [{}\/{}], Loss: {:.4f}'.format(epoch+1, 100, loss.item()))\n<\/code><\/pre>\n\n\n\n
                                \u8fd9\u4e2a\u793a\u4f8b\u4e2d\uff0c\u6211\u4eec\u9996\u5148\u5b9a\u4e49\u4e86\u4e00\u4e2aAutoencoder<\/code>\u7c7b\uff0c\u5b83\u5305\u542b\u4e00\u4e2a\u7f16\u7801\u5668\u548c\u4e00\u4e2a\u89e3\u7801\u5668\u3002\u7f16\u7801\u5668\u662f\u4e00\u4e2ann.Linear<\/code>\u5c42\uff0c\u5c06\u8f93\u5165\u768420\u7ef4\u6570\u636e\u538b\u7f29\u523010\u7ef4\u3002\u89e3\u7801\u5668\u4e5f\u662f\u4e00\u4e2ann.Linear<\/code>\u5c42\uff0c\u5c0610\u7ef4\u7f16\u7801\u8f6c\u6362\u56de20\u7ef4\u6570\u636e\u3002\u6211\u4eec\u4f7f\u7528\u5747\u65b9\u8bef\u5dee\u4f5c\u4e3a\u635f\u5931\u51fd\u6570\uff0c\u4f18\u5316\u5668\u9009\u62e9Adam\u4f18\u5316\u5668\u3002\u5728\u6bcf\u4e2aepoch\u4e2d\uff0c\u6211\u4eec\u5bf9\u8bad\u7ec3\u6570\u636e\u8fdb\u884c\u524d\u5411\u4f20\u64ad\u3001\u8ba1\u7b97\u635f\u5931\u3001\u53cd\u5411\u4f20\u64ad\u548c\u4f18\u5316\uff0c\u5e76\u6253\u5370\u635f\u5931\u3002<\/p>\n\n\n\n\n\n
                                \"gun<\/figure>\n\n\n\n
                                \u8bf7\u8be6\u7ec6\u89e3\u91ca\u751f\u6210\u968f\u673a\u7f51\u7edc<\/p>\n\n\n\n
                                \u751f\u6210\u968f\u673a\u7f51\u7edc\uff08Generative Random Network\uff09\u662f\u4e00\u7c7b\u7528\u4e8e\u751f\u6210\u6570\u636e\u7684\u795e\u7ecf\u7f51\u7edc\u6a21\u578b\uff0c\u5176\u76ee\u6807\u662f\u4ece\u4e00\u7ec4\u968f\u673a\u566a\u58f0\u4e2d\u751f\u6210\u5177\u6709\u7c7b\u4f3c\u4e8e\u8bad\u7ec3\u6570\u636e\u7684\u65b0\u6570\u636e\u3002\u4e0e\u4f20\u7edf\u7684\u76d1\u7763\u5b66\u4e60\u6a21\u578b\u4e0d\u540c\uff0c\u751f\u6210\u968f\u673a\u7f51\u7edc\u7684\u76ee\u6807\u662f\u5b66\u4e60\u6570\u636e\u5206\u5e03\u672c\u8eab\u800c\u4e0d\u662f\u5b66\u4e60\u8f93\u5165\u548c\u8f93\u51fa\u4e4b\u95f4\u7684\u6620\u5c04\u5173\u7cfb\u3002<\/p>\n\n\n\n
                                \u751f\u6210\u968f\u673a\u7f51\u7edc\u901a\u5e38\u7531\u4e24\u4e2a\u90e8\u5206\u7ec4\u6210\uff1a\u751f\u6210\u5668\u548c\u5224\u522b\u5668\u3002\u751f\u6210\u5668\u662f\u4e00\u4e2a\u795e\u7ecf\u7f51\u7edc\uff0c\u5b83\u63a5\u53d7\u4e00\u4e2a\u968f\u673a\u5411\u91cf\u4f5c\u4e3a\u8f93\u5165\uff0c\u5e76\u5c1d\u8bd5\u751f\u6210\u7c7b\u4f3c\u4e8e\u8bad\u7ec3\u6570\u636e\u7684\u65b0\u6570\u636e\u3002\u5224\u522b\u5668\u662f\u53e6\u4e00\u4e2a\u795e\u7ecf\u7f51\u7edc\uff0c\u5b83\u63a5\u53d7\u751f\u6210\u5668\u751f\u6210\u7684\u6570\u636e\u548c\u8bad\u7ec3\u6570\u636e\u4f5c\u4e3a\u8f93\u5165\uff0c\u5e76\u5c1d\u8bd5\u533a\u5206\u54ea\u4e9b\u6570\u636e\u662f\u7531\u751f\u6210\u5668\u751f\u6210\u7684\uff0c\u54ea\u4e9b\u6570\u636e\u662f\u771f\u5b9e\u7684\u8bad\u7ec3\u6570\u636e\u3002<\/p>\n\n\n\n
                                \u751f\u6210\u5668\u548c\u5224\u522b\u5668\u662f\u901a\u8fc7\u5bf9\u6297\u8bad\u7ec3\u65b9\u5f0f\u8fdb\u884c\u8bad\u7ec3\u7684\uff0c\u5373\u751f\u6210\u5668\u8bd5\u56fe\u6b3a\u9a97\u5224\u522b\u5668\uff0c\u4f7f\u5176\u65e0\u6cd5\u533a\u5206\u751f\u6210\u5668\u751f\u6210\u7684\u6570\u636e\u548c\u771f\u5b9e\u6570\u636e\uff0c\u800c\u5224\u522b\u5668\u5219\u5c3d\u53ef\u80fd\u51c6\u786e\u5730\u533a\u5206\u8fd9\u4e24\u79cd\u6570\u636e\u3002\u5728\u8bad\u7ec3\u8fc7\u7a0b\u4e2d\uff0c\u751f\u6210\u5668\u4f1a\u4e0d\u65ad\u5c1d\u8bd5\u751f\u6210\u66f4\u63a5\u8fd1\u4e8e\u771f\u5b9e\u6570\u636e\u7684\u65b0\u6570\u636e\uff0c\u800c\u5224\u522b\u5668\u5219\u4e0d\u65ad\u5b66\u4e60\u5982\u4f55\u66f4\u597d\u5730\u533a\u5206\u751f\u6210\u7684\u6570\u636e\u548c\u771f\u5b9e\u6570\u636e\u3002<\/p>\n\n\n\n
                                \u751f\u6210\u968f\u673a\u7f51\u7edc\u662f\u4e00\u7c7b\u975e\u5e38\u5f3a\u5927\u7684\u6a21\u578b\uff0c\u53ef\u4ee5\u7528\u4e8e\u751f\u6210\u5404\u79cd\u7c7b\u578b\u7684\u6570\u636e\uff0c\u5305\u62ec\u56fe\u50cf\u3001\u8bed\u97f3\u3001\u6587\u672c\u7b49\u3002\u8fd1\u5e74\u6765\uff0c\u751f\u6210\u968f\u673a\u7f51\u7edc\u5728\u8ba1\u7b97\u673a\u89c6\u89c9\u3001\u81ea\u7136\u8bed\u8a00\u5904\u7406\u7b49\u9886\u57df\u53d6\u5f97\u4e86\u8bb8\u591a\u91cd\u8981\u7684\u6210\u679c\uff0c\u4f8b\u5982GAN\u3001VAE\u7b49\u6a21\u578b\u3002<\/p>\n\n\n\n
                                \u4e0b\u9762\u662f\u4e00\u4e2a\u7b80\u5355\u7684\u751f\u6210\u968f\u673a\u7f51\u7edc\u7684\u4ee3\u7801\u793a\u4f8b\uff0c\u4f7f\u7528PyTorch\u5b9e\u73b0\uff1a<\/p>\n\n\n\n
                                pythonCopy codeimport torch\nimport torch.nn as nn\nimport torch.optim as optim\n\n# \u5b9a\u4e49\u751f\u6210\u5668\u6a21\u578b\nclass Generator(nn.Module):\n    def __init__(self, input_size, output_size):\n        super(Generator, self).__init__()\n        self.fc1 = nn.Linear(input_size, 128)\n        self.fc2 = nn.Linear(128, 256)\n        self.fc3 = nn.Linear(256, output_size)\n\n    def forward(self, x):\n        x = torch.relu(self.fc1(x))\n        x = torch.relu(self.fc2(x))\n        x = torch.tanh(self.fc3(x))\n        return x\n\n# \u5b9a\u4e49\u5224\u522b\u5668\u6a21\u578b\nclass Discriminator(nn.Module):\n    def __init__(self, input_size):\n        super(Discriminator, self).__init__()\n        self.fc1 = nn.Linear(input_size, 256)\n        self.fc2 = nn.Linear(256, 128)\n        self.fc3 = nn.Linear(128, 1)\n\n    def forward(self, x):\n        x = torch.relu(self.fc1(x))\n        x = torch.relu(self.fc2(x))\n        x = torch.sigmoid(self.fc3(x))\n        return x\n\n# \u5b9a\u4e49\u8bad\u7ec3\u6570\u636e\u96c6\ndataset = torch.randn(1000, 10)\n\n# \u5b9a\u4e49\u751f\u6210\u5668\u548c\u5224\u522b\u5668\u6a21\u578b\ngenerator =\n<\/code><\/pre>\n\n\n\n\n\n
                                \"gun<\/figure>\n\n\n\n
                                2 \/ 2<\/p>\n\n\n\n
                                \u8bf7\u8be6\u7ec6\u89e3\u91ca\u7279\u5f81\u5206\u89e3<\/p>\n\n\n\n
                                \u7279\u5f81\u5206\u89e3\uff08eigendecomposition\uff09\u662f\u7ebf\u6027\u4ee3\u6570\u4e2d\u7684\u4e00\u79cd\u91cd\u8981\u6280\u672f\uff0c\u7528\u4e8e\u5c06\u4e00\u4e2a\u77e9\u9635\u5206\u89e3\u6210\u4e00\u7ec4\u7279\u5f81\u5411\u91cf\u548c\u7279\u5f81\u503c\u3002\u5728\u673a\u5668\u5b66\u4e60\u4e2d\uff0c\u7279\u5f81\u5206\u89e3\u5e7f\u6cdb\u5e94\u7528\u4e8e\u964d\u7ef4\u3001\u805a\u7c7b\u3001\u56fe\u50cf\u5904\u7406\u3001\u4fe1\u53f7\u5904\u7406\u7b49\u9886\u57df\u3002<\/p>\n\n\n\n
                                \u7ed9\u5b9a\u4e00\u4e2a\u65b9\u9635A\uff0c\u5176\u7279\u5f81\u5411\u91cf\uff08eigenvectors\uff09v\u548c\u7279\u5f81\u503c\uff08eigenvalues\uff09\u03bb\u6ee1\u8db3\u4e0b\u5217\u65b9\u7a0b\uff1a<\/p>\n\n\n\n
                                $A v = \\lambda v$<\/p>\n\n\n\n
                                \u5176\u4e2d\uff0cv\u4e3aA\u7684\u4e00\u4e2a\u975e\u96f6\u5411\u91cf\uff0c\u03bb\u4e3aA\u5bf9\u5e94\u7684\u6807\u91cf\u3002\u53ef\u4ee5\u770b\u51fa\uff0c\u7279\u5f81\u5411\u91cfv\u5728\u77e9\u9635A\u4f5c\u7528\u4e0b\u53ea\u4f1a\u88ab\u62c9\u4f38\uff08\u5373\u7f29\u653e\uff09\u800c\u4e0d\u4f1a\u6539\u53d8\u65b9\u5411\u3002<\/p>\n\n\n\n
                                \u77e9\u9635\u7684\u7279\u5f81\u5206\u89e3\u662f\u5c06\u77e9\u9635\u5206\u89e3\u4e3a\u5982\u4e0b\u5f62\u5f0f\u7684\u4e58\u79ef\uff1a<\/p>\n\n\n\n
                                $A = Q\\Lambda Q^{-1}$<\/p>\n\n\n\n
                                \u5176\u4e2d\uff0cQ\u662f\u7279\u5f81\u5411\u91cf\u77e9\u9635\uff0c$\\Lambda$\u662f\u7279\u5f81\u503c\u77e9\u9635\uff0c\u5b83\u662f\u4e00\u4e2a\u5bf9\u89d2\u77e9\u9635\uff0c\u5bf9\u89d2\u7ebf\u4e0a\u7684\u5143\u7d20\u662fA\u7684\u7279\u5f81\u503c\u3002<\/p>\n\n\n\n
                                \u7279\u5f81\u5206\u89e3\u6709\u5f88\u591a\u5e94\u7528\uff0c\u5176\u4e2d\u4e00\u79cd\u662f\u901a\u8fc7\u7279\u5f81\u5206\u89e3\u5c06\u9ad8\u7ef4\u6570\u636e\u964d\u5230\u4f4e\u7ef4\u7a7a\u95f4\u3002\u5047\u8bbe\u6709\u4e00\u4e2a$m\\times n$\u7684\u77e9\u9635X\uff0c\u5176\u4e2dm\u662f\u6837\u672c\u6570\u91cf\uff0cn\u662f\u7279\u5f81\u6570\u91cf\u3002\u5bf9X\u7684\u534f\u65b9\u5dee\u77e9\u9635\u8fdb\u884c\u7279\u5f81\u5206\u89e3\uff0c\u53ef\u4ee5\u5f97\u5230\u7279\u5f81\u5411\u91cf\u77e9\u9635Q\u548c\u7279\u5f81\u503c\u77e9\u9635$\\Lambda$\u3002\u5047\u8bbe\u6211\u4eec\u8981\u5c06\u6570\u636e\u964d\u5230k\u7ef4\uff0c\u5219\u53ef\u4ee5\u9009\u53d6\u524dk\u4e2a\u7279\u5f81\u5411\u91cf\u6784\u6210\u77e9\u9635P\uff0c\u5bf9\u539f\u59cb\u6570\u636e\u77e9\u9635X\u8fdb\u884c\u53d8\u6362\u5f97\u5230\uff1a<\/p>\n\n\n\n
                                $Y = XP$<\/p>\n\n\n\n
                                \u5176\u4e2dY\u662f$m\\times k$\u7684\u4f4e\u7ef4\u77e9\u9635\u3002<\/p>\n\n\n\n
                                \u7279\u5f81\u5206\u89e3\u53ef\u4ee5\u7528numpy\u5e93\u4e2d\u7684linalg.eig\u51fd\u6570\u8fdb\u884c\u8ba1\u7b97\u3002\u4ee5\u4e0b\u662f\u4e00\u4e2a\u793a\u4f8b\u4ee3\u7801\uff1a<\/p>\n\n\n\n
                                pythonCopy codeimport numpy as np\n\n# \u6784\u9020\u4e00\u4e2a\u968f\u673a\u65b9\u9635\nA = np.random.randn(4, 4)\n\n# \u8ba1\u7b97\u7279\u5f81\u503c\u548c\u7279\u5f81\u5411\u91cf\neigenvalues, eigenvectors = np.linalg.eig(A)\n\n# \u8f93\u51fa\u7ed3\u679c\nprint(\"\u7279\u5f81\u503c\uff1a\", eigenvalues)\nprint(\"\u7279\u5f81\u5411\u91cf\uff1a\", eigenvectors)\n<\/code><\/pre>\n\n\n\n
                                \u8f93\u51fa\uff1a<\/p>\n\n\n\n
                                cssCopy code\u7279\u5f81\u503c\uff1a [ 1.8968701  -0.69828757  0.26170379 -0.17312339]\n\u7279\u5f81\u5411\u91cf\uff1a [[-0.22224377 -0.78475713 -0.30187412  0.49040844]\n             [-0.53964232  0.45716123 -0.41770218 -0.57201119]\n             [-0.61705617  0.26380099  0.85354129 -0.00299044]\n             [-0.52358056 -0.28419712 -0.06005597  0.69985807]]\n<\/code><\/pre>\n\n\n\n
                                \u8fd9\u4e2a<\/p>\n\n\n\n\n\n
                                \"gun<\/figure>\n\n\n\n
                                \u8bf7\u8be6\u7ec6\u89e3\u91ca\u9010\u5c42\u9884\u8bad\u7ec3<\/p>\n\n\n\n
                                \u9010\u5c42\u9884\u8bad\u7ec3\uff08pre-training\uff09\u662f\u4e00\u79cd\u7528\u4e8e\u8bad\u7ec3\u6df1\u5ea6\u795e\u7ecf\u7f51\u7edc\u7684\u6280\u672f\uff0c\u901a\u5e38\u5728\u4f7f\u7528\u5927\u91cf\u6807\u8bb0\u6570\u636e\u8bad\u7ec3\u6df1\u5ea6\u795e\u7ecf\u7f51\u7edc\u4e4b\u524d\uff0c\u5148\u4f7f\u7528\u672a\u6807\u8bb0\u6570\u636e\u8fdb\u884c\u9884\u8bad\u7ec3\uff0c\u4ee5\u63d0\u9ad8\u6a21\u578b\u7684\u6027\u80fd\u3002\u8be5\u65b9\u6cd5\u901a\u5e38\u7528\u4e8e\u5177\u6709\u8bb8\u591a\u5c42\u7684\u795e\u7ecf\u7f51\u7edc\uff0c\u56e0\u4e3a\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u6807\u8bb0\u6570\u636e\u53ef\u80fd\u4e0d\u8db3\u4ee5\u6709\u6548\u5730\u8bad\u7ec3\u6240\u6709\u5c42\u3002<\/p>\n\n\n\n
                                \u9010\u5c42\u9884\u8bad\u7ec3\u7684\u601d\u60f3\u662f\u4ece\u6700\u5e95\u5c42\u5f00\u59cb\u8bad\u7ec3\u6a21\u578b\uff0c\u7136\u540e\u9010\u5c42\u5411\u4e0a\u8bad\u7ec3\u3002\u5728\u8bad\u7ec3\u6bcf\u4e00\u5c42\u65f6\uff0c\u8be5\u5c42\u88ab\u89c6\u4e3a\u4e00\u4e2a\u81ea\u7f16\u7801\u5668\uff0c\u5176\u8f93\u5165\u548c\u8f93\u51fa\u662f\u76f8\u540c\u7684\u3002\u8be5\u5c42\u7684\u6743\u91cd\u88ab\u5b66\u4e60\u4ee5\u6700\u5c0f\u5316\u8f93\u5165\u548c\u8f93\u51fa\u4e4b\u95f4\u7684\u91cd\u6784\u8bef\u5dee\u3002\u5728\u5b8c\u6210\u6bcf\u4e2a\u81ea\u7f16\u7801\u5668\u7684\u8bad\u7ec3\u540e\uff0c\u8be5\u5c42\u7684\u6743\u91cd\u88ab\u7528\u4f5c\u4e0b\u4e00\u5c42\u7684\u521d\u59cb\u6743\u91cd\uff0c\u5e76\u91cd\u590d\u8be5\u8fc7\u7a0b\uff0c\u76f4\u5230\u8bad\u7ec3\u6240\u6709\u5c42\u4e3a\u6b62\u3002<\/p>\n\n\n\n
                                \u9010\u5c42\u9884\u8bad\u7ec3\u7684\u4f18\u70b9\u662f\u5b83\u53ef\u4ee5\u63d0\u9ad8\u6a21\u578b\u7684\u6cdb\u5316\u80fd\u529b\uff0c\u7279\u522b\u662f\u5728\u4f7f\u7528\u5c11\u91cf\u6807\u8bb0\u6570\u636e\u7684\u60c5\u51b5\u4e0b\u3002\u6b64\u5916\uff0c\u5b83\u8fd8\u53ef\u4ee5\u52a0\u5feb\u8bad\u7ec3\u901f\u5ea6\uff0c\u56e0\u4e3a\u5728\u6bcf\u4e2a\u81ea\u7f16\u7801\u5668\u7684\u8bad\u7ec3\u671f\u95f4\uff0c\u53ea\u9700\u8981\u8003\u8651\u8be5\u5c42\u7684\u8f93\u5165\u548c\u8f93\u51fa\uff0c\u800c\u4e0d\u9700\u8981\u8003\u8651\u6574\u4e2a\u6a21\u578b\u7684\u8f93\u5165\u548c\u8f93\u51fa\u3002<\/p>\n\n\n\n
                                \u4e0b\u9762\u662f\u4e00\u4e2a\u7b80\u5355\u7684\u9010\u5c42\u9884\u8bad\u7ec3\u7684\u4ee3\u7801\u793a\u4f8b\uff0c\u7528\u4e8e\u9884\u8bad\u7ec3\u4e00\u4e2a\u591a\u5c42\u611f\u77e5\u673a\u6a21\u578b\uff1a<\/p>\n\n\n\n
                                pythonCopy codeimport numpy as np\nimport tensorflow as tf\n\n# \u5b9a\u4e49\u6bcf\u4e00\u5c42\u7684\u81ea\u7f16\u7801\u5668\u6a21\u578b\nclass Autoencoder(tf.keras.Model):\n    def __init__(self, hidden_dim):\n        super(Autoencoder, self).__init__()\n        self.hidden_dim = hidden_dim\n        self.encoder = tf.keras.layers.Dense(hidden_dim, activation='relu')\n        self.decoder = tf.keras.layers.Dense(784, activation='sigmoid')\n\n    def call(self, inputs):\n        encoded = self.encoder(inputs)\n        decoded = self.decoder(encoded)\n        return decoded\n\n# \u5b9a\u4e49\u591a\u5c42\u611f\u77e5\u673a\u6a21\u578b\nclass MLP(tf.keras.Model):\n    def __init__(self, hidden_dims):\n        super(MLP, self).__init__()\n        self.hidden_dims = hidden_dims\n        self.autoencoders = [Autoencoder(hidden_dim) for hidden_dim in hidden_dims]\n        self.num_layers = len(hidden_dims)\n\n    def call(self, inputs):\n        x = inputs\n        for i in range(self.num_layers):\n            x = self.autoencoders[i](x)\n        return x\n\n# \u52a0\u8f7d\u6570\u636e\u96c6\n(x_train, y_train), (x_test, y_test) = tf.keras.datasets.mnist.load_data()\nx_train = np.reshape(x_train, (60000, 784))\nx_test = np.reshape(x_test, (10000, 784))\nx_train = x_train.astype('float32') \/ 255\nx_test = x_test.astype('float32') \/ 255\n\n# \u5b9a\u4e49\u6a21\u578b\nmlp = MLP([512, 256, 128])\n\n# \u9010\u5c42\u9884\n<\/code><\/pre>\n\n\n\n\n\n
                                \"gun<\/figure>\n\n\n\n
                                \u8bf7\u8be6\u7ec6\u89e3\u91ca\u91cd\u8981\u6027\u91c7\u6837<\/p>\n\n\n\n
                                \u91cd\u8981\u6027\u91c7\u6837\uff08Importance Sampling\uff09\u662f\u4e00\u79cd\u7528\u4e8e\u8ba1\u7b97\u6982\u7387\u5206\u5e03\u7684\u7edf\u8ba1\u65b9\u6cd5\uff0c\u5b83\u53ef\u4ee5\u5229\u7528\u4e00\u4e2a\u7b80\u5355\u7684\u6982\u7387\u5206\u5e03\u6765\u4f30\u8ba1\u4e00\u4e2a\u96be\u4ee5\u8ba1\u7b97\u7684\u590d\u6742\u6982\u7387\u5206\u5e03\u7684\u671f\u671b\u503c\u6216\u79ef\u5206\u3002<\/p>\n\n\n\n
                                \u5177\u4f53\u6765\u8bf4\uff0c\u5047\u8bbe\u6211\u4eec\u8981\u4f30\u8ba1\u4e00\u4e2a\u590d\u6742\u6982\u7387\u5206\u5e03 $p(x)$ \u7684\u67d0\u4e2a\u51fd\u6570 $f(x)$ \u7684\u671f\u671b\u503c $\\mathbb{E}_{x\\sim p}[f(x)]$\uff0c\u4f46\u662f $p(x)$ \u96be\u4ee5\u76f4\u63a5\u91c7\u6837\u6216\u8ba1\u7b97\uff0c\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u4e00\u4e2a\u7b80\u5355\u6982\u7387\u5206\u5e03 $q(x)$ \u6765\u8fd1\u4f3c $p(x)$\uff0c\u5e76\u901a\u8fc7\u5bf9 $q(x)$ \u8fdb\u884c\u91c7\u6837\uff0c\u6765\u8ba1\u7b97 $f(x)$ \u7684\u671f\u671b\u503c\u7684\u4f30\u8ba1\u503c\uff1a<\/p>\n\n\n\n
                                \ufffd\ufffd\u223c\ufffd[\ufffd(\ufffd)]=\u222b\ufffd(\ufffd)\ufffd(\ufffd)\ufffd\ufffd=\u222b\ufffd(\ufffd)\ufffd(\ufffd)\ufffd(\ufffd)\ufffd(\ufffd)\ufffd\ufffd\u22481\ufffd\u2211\ufffd=1\ufffd\ufffd\ufffd\ufffd(\ufffd\ufffd)Ex<\/em>\u223cp<\/em>\u200b[f<\/em>(x<\/em>)]=\u222bf<\/em>(x<\/em>)p<\/em>(x<\/em>)d<\/em>x<\/em>=\u222bf<\/em>(x<\/em>)q<\/em>(x<\/em>)p<\/em>(x<\/em>)\u200bq<\/em>(x<\/em>)d<\/em>x<\/em>\u2248N<\/em>1\u200bi<\/em>=1\u2211N<\/em>\u200bw<\/em>i<\/em>\u200bf<\/em>(x<\/em>i<\/em>\u200b)<\/p>\n\n\n\n
                                \u5176\u4e2d\uff0c$x_i$ \u662f\u4ece\u7b80\u5355\u6982\u7387\u5206\u5e03 $q(x)$ \u4e2d\u91c7\u6837\u7684\u6837\u672c\uff0c$w_i=p(x_i)\/q(x_i)$ \u662f\u4e00\u4e2a\u6743\u91cd\uff0c\u53ef\u4ee5\u770b\u4f5c\u662f $q(x)$ \u5bf9\u4e8e\u4f30\u8ba1 $\\mathbb{E}_{x\\sim p}[f(x)]$ \u7684\u8d21\u732e\u5ea6\u3002<\/p>\n\n\n\n
                                \u91cd\u8981\u6027\u91c7\u6837\u7684\u4f18\u70b9\u5728\u4e8e\uff0c\u5373\u4f7f\u590d\u6742\u6982\u7387\u5206\u5e03 $p(x)$ \u96be\u4ee5\u76f4\u63a5\u91c7\u6837\u6216\u8ba1\u7b97\uff0c\u6211\u4eec\u4ecd\u53ef\u4ee5\u901a\u8fc7\u5f15\u5165\u4e00\u4e2a\u7b80\u5355\u6982\u7387\u5206\u5e03 $q(x)$ \u6765\u8fd1\u4f3c $p(x)$\uff0c\u5e76\u5f97\u5230\u4e00\u4e2a\u6709\u6548\u7684\u4f30\u8ba1\u503c\u3002\u4f46\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u5f53 $q(x)$ \u548c $p(x)$ \u5728\u5206\u5e03\u5f62\u72b6\u4e0a\u5dee\u5f02\u8f83\u5927\u65f6\uff0c\u91cd\u8981\u6027\u91c7\u6837\u7684\u6548\u679c\u4f1a\u53d8\u5f97\u4e0d\u7a33\u5b9a\uff0c\u751a\u81f3\u4f1a\u5931\u6548\u3002<\/p>\n\n\n\n
                                \u4e0b\u9762\u662f\u4e00\u4e2a\u7b80\u5355\u7684 Python \u4ee3\u7801\u793a\u4f8b\uff0c\u7528\u4e8e\u5b9e\u73b0\u91cd\u8981\u6027\u91c7\u6837\u7684\u4f30\u8ba1\u8fc7\u7a0b\uff1a<\/p>\n\n\n\n
                                pythonCopy codeimport numpy as np\n\ndef importance_sampling(p, q, f, n_samples):\n    \"\"\"\n    \u91cd\u8981\u6027\u91c7\u6837\n    :param p: \u590d\u6742\u5206\u5e03\u7684\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\n    :param q: \u7b80\u5355\u5206\u5e03\u7684\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\n    :param f: \u9700\u8981\u8ba1\u7b97\u671f\u671b\u7684\u51fd\u6570\n    :param n_samples: \u91c7\u6837\u6570\u76ee\n    :return: \u671f\u671b\u7684\u4f30\u8ba1\u503c\n    \"\"\"\n    samples = q.rvs(n_samples)  # \u4ece\u7b80\u5355\u5206\u5e03 q \u4e2d\u91c7\u6837\n    weights = p.pdf(samples) \/ q.pdf(samples)  # \u8ba1\u7b97\u6743\u91cd\n    return np.mean(weights * f(samples))  # \u8ba1\u7b97\u671f\u671b\u7684\u4f30\u8ba1\u503c\n<\/code><\/pre>\n\n\n\n
                                \u5176\u4e2d\uff0cp<\/code> \u548c q<\/code> \u662f scipy.stats<\/code> \u4e2d\u7684\u6982\u7387\u5206\u5e03\u5bf9\u8c61\uff0cf<\/code> \u662f\u9700\u8981\u8ba1\u7b97\u671f\u671b\u7684\u51fd\u6570\uff0cn_samples<\/code> \u662f\u91c7\u6837\u6570\u76ee\u3002\u8be5\u51fd\u6570\u8fd4\u56de\u671f\u671b\u7684\u4f30\u8ba1\u503c\u3002<\/p>\n\n\n\n\n\n
                                \"gun<\/figure>\n\n\n\n
                                \u8bf7\u8be6\u7ec6\u89e3\u91ca\u8868\u793a\u5b66\u4e60<\/p>\n\n\n\n
                                \u8868\u793a\u5b66\u4e60\uff08representation learning\uff09\u662f\u6307\u901a\u8fc7\u5bf9\u539f\u59cb\u6570\u636e\u8fdb\u884c\u5b66\u4e60\uff0c\u81ea\u52a8\u5730\u53d1\u6398\u6570\u636e\u7684\u5185\u5728\u7ed3\u6784\uff0c\u5e76\u5c06\u6570\u636e\u6620\u5c04\u5230\u4e00\u4e2a\u66f4\u52a0\u62bd\u8c61\u548c\u6709\u610f\u4e49\u7684\u8868\u793a\u7a7a\u95f4\u4e2d\u3002\u5728\u8fd9\u4e2a\u65b0\u7684\u8868\u793a\u7a7a\u95f4\u4e2d\uff0c\u6570\u636e\u5177\u6709\u66f4\u597d\u7684\u53ef\u89e3\u91ca\u6027\u3001\u66f4\u9ad8\u7684\u62bd\u8c61\u5c42\u6b21\u3001\u66f4\u597d\u7684\u5206\u7c7b\u80fd\u529b\u7b49\u4f18\u70b9\u3002\u8868\u793a\u5b66\u4e60\u662f\u673a\u5668\u5b66\u4e60\u9886\u57df\u4e2d\u7684\u4e00\u4e2a\u91cd\u8981\u7814\u7a76\u65b9\u5411\uff0c\u5e7f\u6cdb\u5e94\u7528\u4e8e\u8ba1\u7b97\u673a\u89c6\u89c9\u3001\u81ea\u7136\u8bed\u8a00\u5904\u7406\u3001\u63a8\u8350\u7cfb\u7edf\u7b49\u9886\u57df\u3002<\/p>\n\n\n\n
                                \u8868\u793a\u5b66\u4e60\u53ef\u4ee5\u5206\u4e3a\u65e0\u76d1\u7763\u8868\u793a\u5b66\u4e60\u548c\u6709\u76d1\u7763\u8868\u793a\u5b66\u4e60\u3002\u65e0\u76d1\u7763\u8868\u793a\u5b66\u4e60\u7684\u76ee\u6807\u662f\u5b66\u4e60\u4e00\u4e2a\u597d\u7684\u8868\u793a\u7a7a\u95f4\uff0c\u4e0d\u9700\u8981\u6807\u6ce8\u6570\u636e\uff1b\u6709\u76d1\u7763\u8868\u793a\u5b66\u4e60\u7684\u76ee\u6807\u662f\u5b66\u4e60\u4e00\u4e2a\u597d\u7684\u8868\u793a\u7a7a\u95f4\uff0c\u540c\u65f6\u5229\u7528\u6807\u6ce8\u6570\u636e\u8fdb\u884c\u8bad\u7ec3\u3002<\/p>\n\n\n\n
                                \u5e38\u89c1\u7684\u8868\u793a\u5b66\u4e60\u65b9\u6cd5\u5305\u62ec\u81ea\u7f16\u7801\u5668\u3001\u6df1\u5ea6\u4fe1\u5ff5\u7f51\u7edc\u3001\u81ea\u7ec4\u7ec7\u6620\u5c04\u3001\u5377\u79ef\u795e\u7ecf\u7f51\u7edc\u7b49\u3002\u8fd9\u4e9b\u65b9\u6cd5\u5728\u6df1\u5ea6\u5b66\u4e60\u9886\u57df\u5f97\u5230\u4e86\u5e7f\u6cdb\u5e94\u7528\uff0c\u53d6\u5f97\u4e86\u5f88\u597d\u7684\u6548\u679c\u3002<\/p>\n\n\n\n
                                \u8868\u793a\u5b66\u4e60\u7684\u4e3b\u8981\u4f18\u70b9\u662f\u80fd\u591f\u63d0\u53d6\u6570\u636e\u7684\u62bd\u8c61\u7279\u5f81\uff0c\u964d\u4f4e\u6570\u636e\u7ef4\u5ea6\uff0c\u63d0\u9ad8\u6570\u636e\u5904\u7406\u6548\u7387\uff0c\u540c\u65f6\u80fd\u591f\u63d0\u9ad8\u6570\u636e\u7684\u5206\u7c7b\u51c6\u786e\u6027\u3001\u964d\u4f4e\u8fc7\u62df\u5408\u7b49\u95ee\u9898\u3002<\/p>\n\n\n\n\n\n
                                \"gun<\/figure>\n\n\n\n
                                \u8bf7\u7ed9\u51fa\u8868\u793a\u5b66\u4e60\u7684\u4ee3\u7801\u793a\u4f8b<\/p>\n\n\n\n
                                \u8868\u793a\u5b66\u4e60\u662f\u4e00\u7c7b\u673a\u5668\u5b66\u4e60\u65b9\u6cd5\uff0c\u65e8\u5728\u4ece\u539f\u59cb\u6570\u636e\u4e2d\u5b66\u4e60\u8868\u793a\u6216\u7279\u5f81\uff0c\u4ee5\u4fbf\u66f4\u597d\u5730\u63cf\u8ff0\u6570\u636e\u548c\u89e3\u51b3\u76f8\u5173\u4efb\u52a1\u3002\u8868\u793a\u5b66\u4e60\u901a\u5e38\u53ef\u4ee5\u5206\u4e3a\u76d1\u7763\u5b66\u4e60\u548c\u65e0\u76d1\u7763\u5b66\u4e60\u4e24\u79cd\u7c7b\u578b\u3002\u4ee5\u4e0b\u662f\u4e00\u4e2a\u65e0\u76d1\u7763\u5b66\u4e60\u4e2d\u57fa\u4e8e\u81ea\u7f16\u7801\u5668\u7684\u8868\u793a\u5b66\u4e60\u7684\u4ee3\u7801\u793a\u4f8b\u3002<\/p>\n\n\n\n
                                pythonCopy codeimport tensorflow as tf\n\n# \u5b9a\u4e49\u81ea\u7f16\u7801\u5668\u6a21\u578b\nclass Autoencoder(tf.keras.Model):\n    def __init__(self, input_dim, hidden_dim):\n        super(Autoencoder, self).__init__()\n        self.encoder = tf.keras.layers.Dense(units=hidden_dim, activation='relu')\n        self.decoder = tf.keras.layers.Dense(units=input_dim, activation='sigmoid')\n\n    def call(self, input_features):\n        encoded = self.encoder(input_features)\n        decoded = self.decoder(encoded)\n        return decoded\n\n# \u52a0\u8f7d\u6570\u636e\n(x_train, y_train), _ = tf.keras.datasets.mnist.load_data()\nx_train = x_train.astype('float32') \/ 255.\nx_train = x_train.reshape((x_train.shape[0], x_train.shape[1] * x_train.shape[2]))\n\n# \u5b9a\u4e49\u81ea\u7f16\u7801\u5668\u6a21\u578b\nautoencoder = Autoencoder(input_dim=x_train.shape[1], hidden_dim=64)\n\n# \u7f16\u8bd1\u6a21\u578b\nautoencoder.compile(optimizer='adam', loss='binary_crossentropy')\n\n# \u8bad\u7ec3\u6a21\u578b\nautoencoder.fit(x_train, x_train, epochs=50, batch_size=256, shuffle=True, validation_split=0.2)\n\n# \u83b7\u53d6\u7f16\u7801\u5668\u6a21\u578b\nencoder = autoencoder.encoder\n\n# \u4f7f\u7528\u7f16\u7801\u5668\u6a21\u578b\u5c06\u6570\u636e\u8f6c\u6362\u4e3a\u7279\u5f81\u8868\u793a\nfeatures = encoder(x_train)\n<\/code><\/pre>\n\n\n\n
                                \u5728\u8fd9\u4e2a\u793a\u4f8b\u4e2d\uff0c\u6211\u4eec\u4f7f\u7528 MNIST \u6570\u636e\u96c6\u8bad\u7ec3\u4e86\u4e00\u4e2a\u57fa\u4e8e\u81ea\u7f16\u7801\u5668\u7684\u8868\u793a\u5b66\u4e60\u6a21\u578b\u3002\u6211\u4eec\u9996\u5148\u5b9a\u4e49\u4e86\u4e00\u4e2a\u81ea\u7f16\u7801\u5668\u6a21\u578b\uff0c\u5305\u62ec\u4e00\u4e2a\u7f16\u7801\u5668\u548c\u4e00\u4e2a\u89e3\u7801\u5668\uff0c\u7136\u540e\u4f7f\u7528\u8bad\u7ec3\u6570\u636e\u5bf9\u5176\u8fdb\u884c\u8bad\u7ec3\u3002\u5728\u8bad\u7ec3\u8fc7\u7a0b\u4e2d\uff0c\u6211\u4eec\u4f7f\u7528\u4e86 Adam \u4f18\u5316\u5668\u548c\u4e8c\u5143\u4ea4\u53c9\u71b5\u635f\u5931\u51fd\u6570\u3002\u8bad\u7ec3\u5b8c\u6210\u540e\uff0c\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u8c03\u7528\u7f16\u7801\u5668\u90e8\u5206\u6765\u83b7\u53d6\u7279\u5f81\u8868\u793a\u3002<\/p>\n\n\n\n\n\n
                                \"gun<\/figure>\n\n\n\n
                                \u8bf7\u8be6\u7ec6\u89e3\u91ca\u7a00\u758f\u7f16\u7801<\/p>\n\n\n\n
                                \u7a00\u758f\u7f16\u7801\u662f\u4e00\u79cd\u65e0\u76d1\u7763\u5b66\u4e60\u7b97\u6cd5\uff0c\u7528\u4e8e\u4ece\u8f93\u5165\u6570\u636e\u4e2d\u63d0\u53d6\u6f5c\u5728\u7684\u7279\u5f81\u3002\u5176\u57fa\u672c\u601d\u60f3\u662f\u5728\u8f93\u5165\u6570\u636e\u4e2d\u5bfb\u627e\u7a00\u758f\u7684\u8868\u793a\u65b9\u5f0f\uff0c\u5373\u53ea\u6709\u5f88\u5c11\u7684\u795e\u7ecf\u5143\u88ab\u6fc0\u6d3b\uff0c\u5176\u4ed6\u795e\u7ecf\u5143\u5904\u4e8e\u672a\u6fc0\u6d3b\u72b6\u6001\u3002\u8fd9\u79cd\u8868\u793a\u65b9\u5f0f\u53ef\u4ee5\u6709\u6548\u5730\u51cf\u5c11\u8f93\u5165\u6570\u636e\u7684\u7ef4\u5ea6\uff0c\u5e76\u4e14\u53ef\u4ee5\u4f7f\u6a21\u578b\u66f4\u5177\u6709\u9c81\u68d2\u6027\u548c\u6cdb\u5316\u80fd\u529b\u3002<\/p>\n\n\n\n
                                \u7a00\u758f\u7f16\u7801\u7684\u76ee\u6807\u51fd\u6570\u53ef\u4ee5\u5f62\u5f0f\u5316\u4e3a\uff1a<\/p>\n\n\n\n
                                $J(\\theta) = \\frac{1}{2n}\\sum_{i=1}^{n}\\lVert x_i – \\hat{x}i \\rVert^2 + \\lambda \\sum<\/em>{j=1}^{m} \\lVert w_j \\rVert_1$<\/p>\n\n\n\n
                                \u5176\u4e2d\uff0c$n$\u8868\u793a\u8bad\u7ec3\u6837\u672c\u7684\u6570\u91cf\uff0c$m$\u8868\u793a\u7f16\u7801\u5668\u7684\u795e\u7ecf\u5143\u6570\u91cf\uff0c$x_i$\u8868\u793a\u7b2c$i$\u4e2a\u8f93\u5165\u6837\u672c\uff0c$\\hat{x}_i$\u8868\u793a\u8be5\u6837\u672c\u7684\u91cd\u6784\uff0c$w_j$\u8868\u793a\u7f16\u7801\u5668\u7684\u7b2c$j$\u4e2a\u795e\u7ecf\u5143\u7684\u6743\u91cd\u5411\u91cf\uff0c$\\lambda$\u662f\u6b63\u5219\u5316\u53c2\u6570\u3002\u76ee\u6807\u51fd\u6570\u7684\u7b2c\u4e00\u9879\u8868\u793a\u91cd\u6784\u8bef\u5dee\uff0c\u7b2c\u4e8c\u9879\u8868\u793a\u7a00\u758f\u6027\u7ea6\u675f\u3002<\/p>\n\n\n\n
                                \u7a00\u758f\u7f16\u7801\u7684\u8bad\u7ec3\u8fc7\u7a0b\u53ef\u4ee5\u91c7\u7528\u53cd\u5411\u4f20\u64ad\u7b97\u6cd5\u548c\u68af\u5ea6\u4e0b\u964d\u65b9\u6cd5\u3002\u8bad\u7ec3\u65f6\uff0c\u8f93\u5165\u6837\u672c\u901a\u8fc7\u7f16\u7801\u5668\u5f97\u5230\u4e00\u7ec4\u7a00\u758f\u7684\u8868\u793a\uff0c\u518d\u901a\u8fc7\u89e3\u7801\u5668\u5f97\u5230\u91cd\u6784\u3002\u6839\u636e\u91cd\u6784\u8bef\u5dee\u548c\u7a00\u758f\u6027\u7ea6\u675f\uff0c\u901a\u8fc7\u68af\u5ea6\u4e0b\u964d\u6cd5\u6765\u66f4\u65b0\u7f16\u7801\u5668\u548c\u89e3\u7801\u5668\u7684\u6743\u91cd\u3002<\/p>\n\n\n\n
                                \u4ee5\u4e0b\u662f\u4f7f\u7528Python\u5b9e\u73b0\u7a00\u758f\u7f16\u7801\u7684\u793a\u4f8b\u4ee3\u7801\uff1a<\/p>\n\n\n\n
                                pythonCopy codeimport numpy as np\nimport matplotlib.pyplot as plt\n\nclass SparseAutoencoder(object):\n    \n    def __init__(self, input_size, hidden_size, sparsity_param, beta):\n        self.input_size = input_size\n        self.hidden_size = hidden_size\n        self.sparsity_param = sparsity_param\n        self.beta = beta\n        \n        # \u968f\u673a\u521d\u59cb\u5316\u6743\u91cd\u77e9\u9635\n        self.W1 = np.random.randn(hidden_size, input_size)\n        self.b1 = np.zeros((hidden_size, 1))\n        self.W2 = np.random.randn(input_size, hidden_size)\n        self.b2 = np.zeros((input_size, 1))\n        \n    def sigmoid(self, x):\n        return 1 \/ (1 + np.exp(-x))\n    \n    def forward(self, x):\n        z2 = np.dot(self.W1, x) + self.b1\n        a2 = self.sigmoid(z2)\n        z3 = np.dot(self.W2, a2) + self.b2\n        a3 = self.sigmoid(z3)\n        return z2, a2, z3, a3\n    \n    def sparse(self, rho, rho_hat):\n        return rho * np.log(rho \/ rho_hat) + (1 - rho) * np.log((1 - rho) \/ (1 - rho_hat))\n    \n    def cost(self, x):\n        m = x.shape[1]\n        z2, a2, z3, a3 = self.forward(x\n<\/code><\/pre>\n\n\n\n\n\n
                                \"gun<\/figure>\n\n\n\n
                                \u8bf7\u8be6\u7ec6\u89e3\u91ca\u56e0\u5b50\u5206\u6790<\/p>\n\n\n\n
                                \u56e0\u5b50\u5206\u6790\u662f\u4e00\u79cd\u7edf\u8ba1\u65b9\u6cd5\uff0c\u7528\u4e8e\u4ece\u89c2\u6d4b\u5230\u7684\u53d8\u91cf\u4e2d\u8bc6\u522b\u6f5c\u5728\u7684\u56e0\u5b50\u3002\u5728\u56e0\u5b50\u5206\u6790\u4e2d\uff0c\u6211\u4eec\u5047\u8bbe\u89c2\u6d4b\u5230\u7684\u53d8\u91cf\u662f\u7531\u6f5c\u5728\u56e0\u5b50\u548c\u968f\u673a\u8bef\u5dee\u5171\u540c\u51b3\u5b9a\u7684\u3002\u8fd9\u4e9b\u6f5c\u5728\u56e0\u5b50\u65e0\u6cd5\u76f4\u63a5\u89c2\u6d4b\u5230\uff0c\u4f46\u5b83\u4eec\u5bf9\u89c2\u6d4b\u5230\u7684\u53d8\u91cf\u4ea7\u751f\u5f71\u54cd\u3002<\/p>\n\n\n\n
                                \u56e0\u5b50\u5206\u6790\u7684\u76ee\u6807\u662f\u63a8\u65ad\u51fa\u6f5c\u5728\u56e0\u5b50\u7684\u6570\u91cf\u548c\u6bcf\u4e2a\u89c2\u6d4b\u53d8\u91cf\u4e0e\u6bcf\u4e2a\u56e0\u5b50\u4e4b\u95f4\u7684\u5173\u7cfb\u3002\u901a\u8fc7\u5c06\u8fd9\u4e9b\u56e0\u5b50\u7528\u4e8e\u6570\u636e\u7684\u964d\u7ef4\uff0c\u53ef\u4ee5\u5e2e\u52a9\u6211\u4eec\u66f4\u597d\u5730\u7406\u89e3\u6570\u636e\u4e2d\u7684\u6a21\u5f0f\u548c\u7ed3\u6784\u3002<\/p>\n\n\n\n
                                \u5177\u4f53\u6765\u8bf4\uff0c\u5047\u8bbe\u6211\u4eec\u6709p\u4e2a\u89c2\u6d4b\u53d8\u91cf\u548cq\u4e2a\u6f5c\u5728\u56e0\u5b50\uff0c\u53ef\u4ee5\u5c06\u56e0\u5b50\u5206\u6790\u6a21\u578b\u8868\u793a\u4e3a\uff1a<\/p>\n\n\n\n
                                X = LF + E<\/p>\n\n\n\n
                                \u5176\u4e2d\uff0cX\u662f\u4e00\u4e2ap\u7ef4\u5411\u91cf\uff0c\u8868\u793a\u89c2\u6d4b\u5230\u7684\u53d8\u91cf\u3002L\u662f\u4e00\u4e2ap\u00d7q\u7684\u56e0\u5b50\u8f7d\u8377\u77e9\u9635\uff0c\u8868\u793a\u6bcf\u4e2a\u89c2\u6d4b\u53d8\u91cf\u4e0e\u6bcf\u4e2a\u56e0\u5b50\u4e4b\u95f4\u7684\u5173\u7cfb\u3002F\u662f\u4e00\u4e2aq\u7ef4\u5411\u91cf\uff0c\u8868\u793a\u6f5c\u5728\u56e0\u5b50\u3002E\u662f\u4e00\u4e2ap\u7ef4\u5411\u91cf\uff0c\u8868\u793a\u968f\u673a\u8bef\u5dee\u3002<\/p>\n\n\n\n
                                \u4e3a\u4e86\u627e\u5230\u6700\u4f73\u7684\u56e0\u5b50\u8f7d\u8377\u77e9\u9635L\u548c\u6f5c\u5728\u56e0\u5b50F\uff0c\u9700\u8981\u6700\u5927\u5316\u4f3c\u7136\u51fd\u6570\u3002\u8fd9\u53ef\u4ee5\u901a\u8fc7\u4f7f\u7528EM\u7b97\u6cd5\u6216\u6700\u5c0f\u5316\u91cd\u6784\u8bef\u5dee\u7b49\u65b9\u6cd5\u6765\u5b8c\u6210\u3002<\/p>\n\n\n\n
                                \u4ee5\u4e0b\u662f\u4e00\u4e2a\u7b80\u5355\u7684\u56e0\u5b50\u5206\u6790\u7684Python\u4ee3\u7801\u793a\u4f8b\uff1a<\/p>\n\n\n\n
                                pythonCopy codeimport numpy as np\nfrom sklearn.decomposition import FactorAnalysis\n\n# \u751f\u6210\u968f\u673a\u6570\u636e\nnp.random.seed(0)\nX = np.random.rand(100, 5)\n\n# \u4f7f\u7528sklearn\u4e2d\u7684\u56e0\u5b50\u5206\u6790\u51fd\u6570\nfa = FactorAnalysis(n_components=2)\nfa.fit(X)\n\n# \u6253\u5370\u56e0\u5b50\u8f7d\u8377\u77e9\u9635\nprint(fa.components_)\n\n# \u8f6c\u6362\u6570\u636e\u5230\u6f5c\u5728\u56e0\u5b50\u7a7a\u95f4\nF = fa.transform(X)\nprint(F)\n<\/code><\/pre>\n\n\n\n
                                \u5728\u8fd9\u4e2a\u793a\u4f8b\u4e2d\uff0c\u6211\u4eec\u751f\u6210\u4e86\u4e00\u4e2a100\u00d75\u7684\u968f\u673a\u6570\u636e\u77e9\u9635X\uff0c\u5e76\u4f7f\u7528sklearn\u4e2d\u7684FactorAnalysis\u51fd\u6570\u8fdb\u884c\u56e0\u5b50\u5206\u6790\u3002\u6211\u4eec\u5c06\u6f5c\u5728\u56e0\u5b50\u6570\u91cf\u8bbe\u7f6e\u4e3a2\uff0c\u5e76\u6253\u5370\u51fa\u56e0\u5b50\u8f7d\u8377\u77e9\u9635\u3002\u6700\u540e\uff0c\u6211\u4eec\u5c06\u6570\u636e\u8f6c\u6362\u5230\u6f5c\u5728\u56e0\u5b50\u7a7a\u95f4\uff0c\u5e76\u6253\u5370\u7ed3\u679c\u3002<\/p>\n\n\n\n\n\n
                                \"gun<\/figure>\n\n\n\n
                                \u8bf7\u8be6\u7ec6\u89e3\u91ca\u5f20\u91cf<\/p>\n\n\n\n
                                \u5728\u6570\u5b66\u4e2d\uff0c\u5f20\u91cf\u662f\u4e00\u79cd\u591a\u7ef4\u6570\u7ec4\uff0c\u53ef\u4ee5\u8868\u793a\u5411\u91cf\u3001\u77e9\u9635\u548c\u5176\u4ed6\u66f4\u9ad8\u7ef4\u7684\u6570\u636e\u7ed3\u6784\u3002\u4e0e\u6807\u91cf\uff08\u96f6\u9636\u5f20\u91cf\uff09\u548c\u5411\u91cf\uff08\u4e00\u9636\u5f20\u91cf\uff09\u4e0d\u540c\uff0c\u5f20\u91cf\u5177\u6709\u591a\u4e2a\u6307\u6807\uff0c\u4e5f\u79f0\u4e3a\u9636\u6570\u6216\u7ef4\u6570\u3002\u4f8b\u5982\uff0c\u4e00\u4e2a\u4e8c\u9636\u5f20\u91cf\u53ef\u4ee5\u8868\u793a\u4e00\u4e2a\u77e9\u9635\uff0c\u6709\u4e24\u4e2a\u6307\u6807\uff0c\u5982\u884c\u548c\u5217\u3002\u4e00\u4e2a\u4e09\u9636\u5f20\u91cf\u5219\u5177\u6709\u4e09\u4e2a\u6307\u6807\uff0c\u53ef\u4ee5\u8868\u793a\u4e00\u4e2a\u7acb\u4f53\u6570\u7ec4\u6216\u7acb\u65b9\u4f53\uff0c\u6bcf\u4e2a\u5143\u7d20\u5177\u6709\u4e09\u4e2a\u5750\u6807\uff0c\u5982\u6df1\u5ea6\u3001\u884c\u548c\u5217\u3002<\/p>\n\n\n\n
                                \u5728\u673a\u5668\u5b66\u4e60\u4e2d\uff0c\u5f20\u91cf\u662f\u975e\u5e38\u91cd\u8981\u7684\u6570\u636e\u7c7b\u578b\uff0c\u901a\u5e38\u7528\u4e8e\u8868\u793a\u8f93\u5165\u6570\u636e\u3001\u4e2d\u95f4\u5c42\u7279\u5f81\u548c\u8f93\u51fa\u6570\u636e\u3002\u4f8b\u5982\uff0c\u56fe\u50cf\u6570\u636e\u901a\u5e38\u8868\u793a\u4e3a\u56db\u9636\u5f20\u91cf\uff0c\u5176\u56db\u4e2a\u6307\u6807\u5206\u522b\u662f\u6837\u672c\u6570\u3001\u901a\u9053\u6570\u3001\u9ad8\u5ea6\u548c\u5bbd\u5ea6\u3002\u8bed\u97f3\u6570\u636e\u901a\u5e38\u8868\u793a\u4e3a\u4e09\u9636\u5f20\u91cf\uff0c\u5176\u4e09\u4e2a\u6307\u6807\u5206\u522b\u662f\u6837\u672c\u6570\u3001\u65f6\u95f4\u6b65\u6570\u548c\u97f3\u9891\u91c7\u6837\u6570\u3002<\/p>\n\n\n\n
                                \u5728\u5b9e\u73b0\u6df1\u5ea6\u5b66\u4e60\u7b97\u6cd5\u65f6\uff0c\u6211\u4eec\u901a\u5e38\u4f7f\u7528\u5f20\u91cf\u4f5c\u4e3a\u6570\u636e\u7ed3\u6784\u6765\u5b58\u50a8\u548c\u64cd\u4f5c\u8f93\u5165\u3001\u8f93\u51fa\u3001\u6743\u91cd\u548c\u68af\u5ea6\u7b49\u6570\u636e\u3002\u5f20\u91cf\u7684\u7279\u6027\u4f7f\u5f97\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u5411\u91cf\u5316\u7684\u65b9\u5f0f\u9ad8\u6548\u5730\u6267\u884c\u77e9\u9635\u4e58\u6cd5\u3001\u5377\u79ef\u548c\u6c60\u5316\u7b49\u8ba1\u7b97\uff0c\u4ece\u800c\u52a0\u901f\u6df1\u5ea6\u5b66\u4e60\u7b97\u6cd5\u7684\u8bad\u7ec3\u548c\u63a8\u7406\u8fc7\u7a0b\u3002<\/p>\n\n\n\n
                                \u4ee5\u4e0b\u662f\u4e00\u4e2a\u793a\u4f8b\u4ee3\u7801\uff0c\u4f7f\u7528Python\u548cNumPy\u5e93\u521b\u5efa\u4e00\u4e2a\u4e8c\u9636\u5f20\u91cf\uff0c\u5e76\u8ba1\u7b97\u5b83\u7684\u77e9\u9635\u4e58\u79ef\uff1a<\/p>\n\n\n\n
                                pythonCopy codeimport numpy as np\n\n# \u521b\u5efa\u4e00\u4e2a2x3\u7684\u77e9\u9635\u4f5c\u4e3a\u5f20\u91cf\ntensor = np.array([[1, 2, 3], [4, 5, 6]])\n\n# \u521b\u5efa\u4e00\u4e2a3x2\u7684\u77e9\u9635\u4f5c\u4e3a\u6743\u91cd\nweights = np.array([[2, 3], [4, 5], [6, 7]])\n\n# \u8ba1\u7b97\u5f20\u91cf\u4e0e\u6743\u91cd\u7684\u77e9\u9635\u4e58\u79ef\nresult = np.dot(tensor, weights)\n\nprint(result)\n<\/code><\/pre>\n\n\n\n
                                \u8f93\u51fa\uff1a<\/p>\n\n\n\n
                                luaCopy code[[28 34]\n [64 79]]\n<\/code><\/pre>\n\n\n\n
                                \u5728\u8fd9\u4e2a\u4f8b\u5b50\u4e2d\uff0c\u6211\u4eec\u521b\u5efa\u4e86\u4e00\u4e2a\u4e8c\u9636\u5f20\u91cftensor<\/code>\u548c\u4e00\u4e2a\u4e8c\u9636\u6743\u91cdweights<\/code>\uff0c\u5e76\u8ba1\u7b97\u5b83\u4eec\u7684\u77e9\u9635\u4e58\u79ef\u3002\u7531\u4e8e\u5f20\u91cf\u548c\u6743\u91cd\u7684\u7ef4\u5ea6\u5339\u914d\uff0c\u77e9\u9635\u4e58\u79ef\u53ef\u4ee5\u6210\u529f\u8ba1\u7b97\uff0c\u5e76\u8fd4\u56de\u4e00\u4e2a\u65b0\u7684\u4e8c\u9636\u5f20\u91cf\u4f5c\u4e3a\u7ed3\u679c\u3002<\/p>\n\n\n\n\n\n
                                \"gun<\/figure>\n\n\n\n
                                \u8bf7\u8be6\u7ec6\u89e3\u91ca\u7ebf\u6027\u76f8\u5173<\/p>\n\n\n\n
                                \u7ebf\u6027\u76f8\u5173\u6307\u7684\u662f\u5b58\u5728\u4e00\u4e9b\u975e\u96f6\u7684\u7cfb\u6570\uff0c\u4f7f\u5f97\u4e00\u4e2a\u5411\u91cf\u53ef\u4ee5\u8868\u793a\u4e3a\u53e6\u4e00\u4e2a\u6216\u591a\u4e2a\u5411\u91cf\u7684\u7ebf\u6027\u7ec4\u5408\u3002\u5982\u679c\u4e00\u4e2a\u5411\u91cf\u53ef\u4ee5\u8868\u793a\u4e3a\u53e6\u4e00\u4e2a\u6216\u591a\u4e2a\u5411\u91cf\u7684\u7ebf\u6027\u7ec4\u5408\uff0c\u90a3\u4e48\u8fd9\u4e9b\u5411\u91cf\u5c31\u88ab\u79f0\u4e3a\u7ebf\u6027\u76f8\u5173\u7684\u3002<\/p>\n\n\n\n
                                \u4f8b\u5982\uff0c\u5047\u8bbe\u6709\u4e24\u4e2a\u5411\u91cfv1\u548cv2\uff0c\u5982\u679c\u5b58\u5728\u4e00\u4e9b\u4e0d\u5168\u4e3a\u96f6\u7684\u5e38\u6570c1\u548cc2\uff0c\u4f7f\u5f97c1v1 + c2<\/em>v2 = 0\uff0c\u90a3\u4e48v1\u548cv2\u5c31\u662f\u7ebf\u6027\u76f8\u5173\u7684\u3002\u5982\u679c\u4e0d\u5b58\u5728\u8fd9\u6837\u7684\u5e38\u6570\uff0c\u90a3\u4e48v1\u548cv2\u5c31\u662f\u7ebf\u6027\u65e0\u5173\u7684\u3002<\/p>\n\n\n\n
                                \u7ebf\u6027\u76f8\u5173\u7684\u5411\u91cf\u5728\u67d0\u4e9b\u60c5\u51b5\u4e0b\u53ef\u80fd\u4e0d\u592a\u6709\u7528\uff0c\u56e0\u4e3a\u5b83\u4eec\u4e0d\u80fd\u63d0\u4f9b\u66f4\u591a\u7684\u4fe1\u606f\uff0c\u800c\u53ea\u662f\u63d0\u4f9b\u4e86\u5197\u4f59\u7684\u4fe1\u606f\u3002\u5728\u4e00\u4e9b\u7ebf\u6027\u4ee3\u6570\u7684\u5e94\u7528\u4e2d\uff0c\u6211\u4eec\u5e0c\u671b\u901a\u8fc7\u9009\u62e9\u7ebf\u6027\u65e0\u5173\u7684\u5411\u91cf\uff0c\u6765\u6784\u5efa\u4e00\u4e2a\u66f4\u52a0\u7d27\u51d1\u548c\u6709\u6548\u7684\u8868\u793a\u65b9\u6cd5\u3002<\/p>\n\n\n\n
                                \u7ebf\u6027\u76f8\u5173\u4e0e\u7ebf\u6027\u65e0\u5173\u7684\u6982\u5ff5\u4e5f\u53ef\u4ee5\u63a8\u5e7f\u5230\u77e9\u9635\u548c\u5f20\u91cf\u7684\u60c5\u51b5\u3002\u5728\u77e9\u9635\u4e2d\uff0c\u5982\u679c\u77e9\u9635\u7684\u5217\u5411\u91cf\u7ebf\u6027\u76f8\u5173\uff0c\u90a3\u4e48\u8fd9\u4e2a\u77e9\u9635\u5c31\u662f\u7ebf\u6027\u76f8\u5173\u7684\u3002\u5728\u5f20\u91cf\u4e2d\uff0c\u5982\u679c\u67d0\u4e2a\u5f20\u91cf\u53ef\u4ee5\u8868\u793a\u4e3a\u5176\u4ed6\u5f20\u91cf\u7684\u7ebf\u6027\u7ec4\u5408\uff0c\u90a3\u4e48\u8fd9\u4e9b\u5f20\u91cf\u5c31\u662f\u7ebf\u6027\u76f8\u5173\u7684\u3002<\/p>\n\n\n\n\n\n
                                \"gun<\/figure>\n\n\n\n
                                \u8bf7\u7ed9\u51fa\u7ebf\u6027\u76f8\u5173\u7684\u793a\u4f8b<\/p>\n\n\n\n
                                \u5047\u8bbe\u6709\u4e24\u4e2a\u5411\u91cf $u = \\begin{bmatrix} 1 \\ 2 \\ 3 \\end{bmatrix}$ \u548c $v = \\begin{bmatrix} 2 \\ 4 \\ 6 \\end{bmatrix}$\u3002\u5b83\u4eec\u662f\u7ebf\u6027\u76f8\u5173\u7684\uff0c\u56e0\u4e3a $v$ \u53ef\u4ee5\u901a\u8fc7 $u$ \u7684\u500d\u6570\u6765\u8868\u793a\uff1a$v = 2u$\u3002<\/p>\n\n\n\n
                                \u7ebf\u6027\u76f8\u5173\u610f\u5473\u7740\u5b58\u5728\u4e00\u4e2a\u5411\u91cf\u53ef\u4ee5\u8868\u793a\u4e3a\u53e6\u4e00\u4e2a\u5411\u91cf\u7684\u7ebf\u6027\u7ec4\u5408\u3002\u5982\u679c\u5411\u91cf $u$ \u548c $v$ \u7ebf\u6027\u65e0\u5173\uff0c\u90a3\u4e48\u5b83\u4eec\u4e0d\u80fd\u8868\u793a\u4e3a\u5f7c\u6b64\u7684\u7ebf\u6027\u7ec4\u5408\u3002<\/p>\n\n\n\n\n\n
                                \"gun<\/figure>\n\n\n\n
                                \u8bf7\u7ed9\u51fa\u7ebf\u6027\u76f8\u5173\u7684\u793a\u4f8b\u4ee3\u7801<\/p>\n\n\n\n
                                \u4e0b\u9762\u662f\u4e00\u4e2a\u7b80\u5355\u7684\u793a\u4f8b\u4ee3\u7801\uff0c\u6f14\u793a\u5982\u4f55\u68c0\u67e5\u5411\u91cf\u4e4b\u95f4\u7684\u7ebf\u6027\u76f8\u5173\u6027\uff1a<\/p>\n\n\n\n
                                pythonCopy codeimport numpy as np\n\n# \u751f\u6210\u4e24\u4e2a\u7ebf\u6027\u76f8\u5173\u7684\u5411\u91cf\nx = np.array([1, 2, 3])\ny = np.array([2, 4, 6])\n\n# \u8ba1\u7b97\u5411\u91cf\u4e4b\u95f4\u7684\u76f8\u5173\u7cfb\u6570\ncorr = np.corrcoef(x, y)[0, 1]\n\nif corr == 1.0:\n    print(\"\u5411\u91cf\u5b8c\u5168\u7ebf\u6027\u76f8\u5173\")\nelif corr == -1.0:\n    print(\"\u5411\u91cf\u5b8c\u5168\u7ebf\u6027\u65e0\u5173\")\nelse:\n    print(\"\u5411\u91cf\u7ebf\u6027\u76f8\u5173\uff0c\u4f46\u4e0d\u5b8c\u5168\u7ebf\u6027\u76f8\u5173\")\n<\/code><\/pre>\n\n\n\n
                                \u5728\u4e0a\u9762\u7684\u793a\u4f8b\u4e2d\uff0c\u6211\u4eec\u751f\u6210\u4e86\u4e24\u4e2a\u7ebf\u6027\u76f8\u5173\u7684\u5411\u91cf\uff0c\u7136\u540e\u4f7f\u7528numpy<\/code>\u4e2d\u7684corrcoef<\/code>\u51fd\u6570\u8ba1\u7b97\u5b83\u4eec\u4e4b\u95f4\u7684\u76f8\u5173\u7cfb\u6570\u3002\u5982\u679c\u76f8\u5173\u7cfb\u6570\u4e3a1\uff0c\u5219\u5411\u91cf\u5b8c\u5168\u7ebf\u6027\u76f8\u5173\uff1b\u5982\u679c\u76f8\u5173\u7cfb\u6570\u4e3a-1\uff0c\u5219\u5411\u91cf\u5b8c\u5168\u7ebf\u6027\u65e0\u5173\uff1b\u5426\u5219\u5411\u91cf\u7ebf\u6027\u76f8\u5173\uff0c\u4f46\u4e0d\u5b8c\u5168\u7ebf\u6027\u76f8\u5173\u3002<\/p>\n\n\n\n\n\n
                                \"gun<\/figure>\n\n\n\n
                                \u8bf7\u8be6\u7ec6\u89e3\u91ca\u65b9\u5dee<\/p>\n\n\n\n
                                \u65b9\u5dee\u662f\u63cf\u8ff0\u968f\u673a\u53d8\u91cf\u53d6\u503c\u7684\u79bb\u6563\u7a0b\u5ea6\u7684\u4e00\u4e2a\u7edf\u8ba1\u91cf\uff0c\u7528\u4e8e\u8861\u91cf\u6570\u636e\u96c6\u5408\u4e2d\u6240\u6709\u6570\u636e\u4e0e\u5176\u5e73\u5747\u503c\u7684\u5dee\u5f02\u7a0b\u5ea6\u3002\u65b9\u5dee\u8d8a\u5927\uff0c\u8868\u793a\u6570\u636e\u7684\u79bb\u6563\u7a0b\u5ea6\u8d8a\u5927\uff0c\u6570\u636e\u5206\u5e03\u8d8a\u5206\u6563\u3002<\/p>\n\n\n\n
                                \u65b9\u5dee\u7684\u516c\u5f0f\u5982\u4e0b\uff1a<\/p>\n\n\n\n
                                Var\u2061(\ufffd)=1\ufffd\u2211\ufffd=1\ufffd(\ufffd\ufffd\u2212\ufffd\u02c9)2Var(X<\/em>)=n<\/em>1\u200bi<\/em>=1\u2211n<\/em>\u200b(x<\/em>i<\/em>\u200b\u2212x<\/em>\u02c9)2<\/p>\n\n\n\n
                                \u5176\u4e2d\uff0c$X$ \u4e3a\u968f\u673a\u53d8\u91cf\uff0c$x_i$ \u4e3a $X$ \u7684\u7b2c $i$ \u6b21\u89c2\u6d4b\u503c\uff0c$\\bar{x}$ \u4e3a $X$ \u7684\u5747\u503c\uff0c$n$ \u4e3a\u6837\u672c\u5927\u5c0f\u3002<\/p>\n\n\n\n
                                \u65b9\u5dee\u7684\u8ba1\u7b97\u8fc7\u7a0b\u53ef\u4ee5\u5206\u4e3a\u4ee5\u4e0b\u51e0\u6b65\uff1a<\/p>\n\n\n\n
                                  \n
                                1. \u8ba1\u7b97\u6570\u636e\u96c6\u7684\u5e73\u5747\u503c $\\bar{x}$\u3002<\/li>\n\n\n\n
                                2. \u5bf9\u4e8e\u6bcf\u4e2a\u89c2\u6d4b\u503c $x_i$\uff0c\u8ba1\u7b97 $(x_i-\\bar{x})^2$\u3002<\/li>\n\n\n\n
                                3. \u5c06\u6240\u6709\u8ba1\u7b97\u5f97\u5230\u7684 $(x_i-\\bar{x})^2$ \u6c42\u548c\u3002<\/li>\n\n\n\n
                                4. \u5c06\u6c42\u548c\u5f97\u5230\u7684\u503c\u9664\u4ee5\u6837\u672c\u5927\u5c0f $n$\uff0c\u5f97\u5230\u65b9\u5dee\u3002<\/li>\n<\/ol>\n\n\n\n
                                  \u5728\u5b9e\u9645\u5e94\u7528\u4e2d\uff0c\u65b9\u5dee\u901a\u5e38\u7528\u6765\u8861\u91cf\u6570\u636e\u7684\u79bb\u6563\u7a0b\u5ea6\uff0c\u5e76\u4e14\u8fd8\u53ef\u4ee5\u7528\u4e8e\u6784\u5efa\u5176\u4ed6\u7edf\u8ba1\u5b66\u6a21\u578b\uff0c\u5982\u65b9\u5dee\u5206\u6790\u3001\u56de\u5f52\u7b49\u3002<\/p>\n\n\n\n\n\n
                                  \"gun<\/figure>\n\n\n\n
                                  \u8bf7\u7ed9\u51fa\u65b9\u5dee\u7684\u793a\u4f8b\u4ee3\u7801<\/p>\n\n\n\n
                                  \u4e0b\u9762\u662f Python \u4e2d\u8ba1\u7b97\u4e00\u7ec4\u6570\u636e\u7684\u6837\u672c\u65b9\u5dee\u7684\u793a\u4f8b\u4ee3\u7801\uff1a<\/p>\n\n\n\n
                                  pythonCopy codeimport numpy as np\n\n# \u751f\u6210\u4e00\u7ec4\u6837\u672c\u6570\u636e\ndata = np.array([1, 2, 3, 4, 5])\n\n# \u8ba1\u7b97\u6837\u672c\u6570\u636e\u7684\u65b9\u5dee\nvariance = np.var(data, ddof=1)\n\nprint(\"\u6837\u672c\u65b9\u5dee\u4e3a\uff1a\", variance)\n<\/code><\/pre>\n\n\n\n
                                  \u8f93\u51fa\u7ed3\u679c\u4e3a\uff1a<\/p>\n\n\n\n
                                  Copy code\u6837\u672c\u65b9\u5dee\u4e3a\uff1a 2.5\n<\/code><\/pre>\n\n\n\n
                                  \u5176\u4e2d np.var()<\/code> \u51fd\u6570\u7528\u4e8e\u8ba1\u7b97\u65b9\u5dee\uff0cddof<\/code> \u53c2\u6570\u7528\u4e8e\u6307\u5b9a\u81ea\u7531\u5ea6\u7684\u4e2a\u6570\uff0c\u901a\u5e38\u53d6\u503c\u4e3a 1 \u8868\u793a\u4f7f\u7528\u6837\u672c\u65b9\u5dee\u3002<\/p>\n\n\n\n\n\n
                                  \"gun<\/figure>\n\n\n\n
                                  \u8bf7\u8be6\u7ec6\u89e3\u91ca\u635f\u5931\u51fd\u6570<\/p>\n\n\n\n
                                  \u635f\u5931\u51fd\u6570\uff08Loss Function\uff09\u662f\u7528\u4e8e\u5ea6\u91cf\u6a21\u578b\u9884\u6d4b\u7ed3\u679c\u4e0e\u5b9e\u9645\u6807\u7b7e\u4e4b\u95f4\u7684\u5dee\u5f02\u7684\u51fd\u6570\u3002\u5728\u8bad\u7ec3\u6a21\u578b\u65f6\uff0c\u6211\u4eec\u901a\u5e38\u4f1a\u5c06\u5176\u6700\u5c0f\u5316\uff0c\u4ee5\u4f7f\u6a21\u578b\u7684\u9884\u6d4b\u7ed3\u679c\u4e0e\u5b9e\u9645\u6807\u7b7e\u5c3d\u53ef\u80fd\u5730\u63a5\u8fd1\u3002<\/p>\n\n\n\n
                                  \u635f\u5931\u51fd\u6570\u7684\u5f62\u5f0f\u53d6\u51b3\u4e8e\u6240\u4f7f\u7528\u7684\u6a21\u578b\u548c\u4efb\u52a1\u3002\u4f8b\u5982\uff0c\u5bf9\u4e8e\u5206\u7c7b\u4efb\u52a1\uff0c\u5e38\u7528\u7684\u635f\u5931\u51fd\u6570\u6709\u4ea4\u53c9\u71b5\u635f\u5931\u51fd\u6570\uff08Cross-Entropy Loss\uff09\u3001Hinge Loss\u7b49\uff1b\u5bf9\u4e8e\u56de\u5f52\u4efb\u52a1\uff0c\u5e38\u7528\u7684\u635f\u5931\u51fd\u6570\u6709\u5747\u65b9\u8bef\u5dee\u635f\u5931\u51fd\u6570\uff08Mean Squared Error Loss\uff09\u3001\u5e73\u5747\u7edd\u5bf9\u8bef\u5dee\u635f\u5931\u51fd\u6570\uff08Mean Absolute Error Loss\uff09\u7b49\u3002<\/p>\n\n\n\n
                                  \u4ee5\u4e0b\u662f\u4e24\u4e2a\u5e38\u89c1\u7684\u635f\u5931\u51fd\u6570\u793a\u4f8b\uff1a<\/p>\n\n\n\n
                                    \n
                                  1. \u5747\u65b9\u8bef\u5dee\u635f\u5931\u51fd\u6570\uff1a\ufffd\ufffd\ufffd=1\ufffd\u2211\ufffd=1\ufffd(\ufffd\ufffd\u2212\ufffd\ufffd^)2MSE<\/em>=n<\/em>1\u200bi<\/em>=1\u2211n<\/em>\u200b(y<\/em>i<\/em>\u200b\u2212y<\/em>i<\/em>\u200b^\u200b)2\u5176\u4e2d\uff0c$y_i$\u662f\u5b9e\u9645\u6807\u7b7e\uff0c$\\hat{y_i}$\u662f\u6a21\u578b\u9884\u6d4b\u7684\u6807\u7b7e\uff0c$n$\u662f\u6837\u672c\u6570\u91cf\u3002<\/li>\n\n\n\n
                                  2. \u4ea4\u53c9\u71b5\u635f\u5931\u51fd\u6570\uff1a\ufffd\ufffd=\u22121\ufffd\u2211\ufffd=1\ufffd\u2211\ufffd=1\ufffd\ufffd\ufffd\ufffdlog\u2061(\ufffd\ufffd\ufffd^)CE<\/em>=\u2212n<\/em>1\u200bi<\/em>=1\u2211n<\/em>\u200bj<\/em>=1\u2211m<\/em>\u200by<\/em>ij<\/em>\u200blog(y<\/em>ij<\/em>\u200b^\u200b)\u5176\u4e2d\uff0c$y_{ij}$\u662f\u7b2c$i$\u4e2a\u6837\u672c\u7684\u5b9e\u9645\u6807\u7b7e\u7684\u7b2c$j$\u4e2a\u7c7b\u522b\u7684\u6982\u7387\u503c\uff0c$\\hat{y_{ij}}$\u662f\u6a21\u578b\u9884\u6d4b\u7684\u7b2c$i$\u4e2a\u6837\u672c\u7684\u7b2c$j$\u4e2a\u7c7b\u522b\u7684\u6982\u7387\u503c\uff0c$n$\u662f\u6837\u672c\u6570\u91cf\uff0c$m$\u662f\u7c7b\u522b\u6570\u91cf\u3002<\/li>\n<\/ol>\n\n\n\n
                                    \u5728\u5b9e\u9645\u4e2d\uff0c\u6211\u4eec\u9700\u8981\u6839\u636e\u5177\u4f53\u7684\u95ee\u9898\u9009\u62e9\u9002\u5f53\u7684\u635f\u5931\u51fd\u6570\uff0c\u4ee5\u4fbf\u80fd\u591f\u5728\u8bad\u7ec3\u8fc7\u7a0b\u4e2d\u5f97\u5230\u66f4\u597d\u7684\u7ed3\u679c\u3002<\/p>\n\n\n\n\n\n
                                    \"gun<\/figure>\n\n\n\n
                                    \u8bf7\u7ed9\u51fa\u4ea4\u53c9\u71b5\u635f\u5931\u51fd\u6570\u7684\u793a\u4f8b\u4ee3\u7801<\/p>\n\n\n\n
                                    \u4ea4\u53c9\u71b5\uff08Cross Entropy\uff09\u662f\u5e38\u7528\u7684\u635f\u5931\u51fd\u6570\u4e4b\u4e00\uff0c\u901a\u5e38\u7528\u4e8e\u5206\u7c7b\u95ee\u9898\u4e2d\u3002<\/p>\n\n\n\n
                                    \u5047\u8bbe\u6a21\u578b\u8f93\u51fa\u4e3a $y$\uff0c\u6807\u7b7e\u4e3a $t$\uff0c\u4ea4\u53c9\u71b5\u635f\u5931\u51fd\u6570\u4e3a\uff1a<\/p>\n\n\n\n
                                    \ufffd=\u2212\u2211\ufffd=1\ufffd\ufffd\ufffdlog\u2061\ufffd\ufffdL<\/em>=\u2212\u2211i<\/em>=1n<\/em>\u200bt<\/em>i<\/em>\u200blogy<\/em>i<\/em>\u200b<\/p>\n\n\n\n
                                    \u5176\u4e2d\uff0c$n$ \u662f\u5206\u7c7b\u6570\u76ee\uff0c$t_i$ \u662f\u7b2c $i$ \u4e2a\u6807\u7b7e\u7684\u53d6\u503c\uff080 \u6216 1\uff09\uff0c$y_i$ \u662f\u6a21\u578b\u8f93\u51fa\u7684\u7b2c $i$ \u4e2a\u5206\u7c7b\u7684\u6982\u7387\u3002<\/p>\n\n\n\n
                                    \u4e0b\u9762\u662f\u4e00\u4e2a\u4f7f\u7528 PyTorch \u5b9e\u73b0\u4ea4\u53c9\u71b5\u635f\u5931\u51fd\u6570\u7684\u793a\u4f8b\u4ee3\u7801\uff1a<\/p>\n\n\n\n
                                    pythonCopy codeimport torch.nn as nn\nimport torch\n\n# \u5047\u8bbe\u5206\u7c7b\u6570\u76ee\u4e3a 5\nn_class = 5\n# \u5047\u8bbe\u6a21\u578b\u8f93\u51fa\u4e3a 5 \u4e2a\u5206\u7c7b\u7684\u6982\u7387\uff0cbatch_size \u4e3a 10\ny = torch.randn(10, n_class)\n# \u5047\u8bbe\u6807\u7b7e\u4e3a\u968f\u673a\u7684\u5206\u7c7b\nt = torch.randint(0, n_class, size=(10,))\n\n# \u5b9a\u4e49\u4ea4\u53c9\u71b5\u635f\u5931\u51fd\u6570\nloss_fn = nn.CrossEntropyLoss()\n\n# \u8ba1\u7b97\u635f\u5931\nloss = loss_fn(y, t)\nprint(loss)\n<\/code><\/pre>\n\n\n\n
                                    \u4e0a\u8ff0\u4ee3\u7801\u4e2d\uff0c\u9996\u5148\u5b9a\u4e49\u4e86\u5206\u7c7b\u6570\u76ee\u548c\u6a21\u578b\u8f93\u51fa\u3002\u7136\u540e\u4f7f\u7528 PyTorch \u5185\u7f6e\u7684 CrossEntropyLoss<\/code> \u51fd\u6570\u5b9a\u4e49\u4e86\u4ea4\u53c9\u71b5\u635f\u5931\u51fd\u6570\uff0c\u6700\u540e\u8ba1\u7b97\u4e86\u635f\u5931\u3002<\/p>\n\n\n\n\n\n
                                    \"gun<\/figure>\n\n\n\n
                                    \u8bf7\u8be6\u7ec6\u89e3\u91ca\u7279\u5f81\u5206\u89e3<\/p>\n\n\n\n
                                    \u7279\u5f81\u5206\u89e3\uff08eigendecomposition\uff09\u662f\u4e00\u79cd\u5c06\u5bf9\u79f0\u77e9\u9635\u5206\u89e3\u6210\u7279\u5f81\u5411\u91cf\u548c\u7279\u5f81\u503c\u7684\u65b9\u6cd5\u3002\u5bf9\u79f0\u77e9\u9635\u5177\u6709\u5f88\u591a\u91cd\u8981\u7684\u6027\u8d28\uff0c\u5176\u4e2d\u4e4b\u4e00\u662f\u5b83\u7684\u7279\u5f81\u5411\u91cf\u662f\u6b63\u4ea4\u7684\uff0c\u8fd9\u4f7f\u5f97\u7279\u5f81\u5206\u89e3\u5728\u8bb8\u591a\u6570\u5b66\u548c\u7edf\u8ba1\u5b66\u5e94\u7528\u4e2d\u975e\u5e38\u6709\u7528\u3002<\/p>\n\n\n\n
                                    \u7279\u5f81\u5206\u89e3\u7684\u76ee\u6807\u662f\u5c06\u4e00\u4e2a\u5bf9\u79f0\u77e9\u9635 $A$ \u5206\u89e3\u6210\u7279\u5f81\u5411\u91cf\u548c\u7279\u5f81\u503c\u7684\u5f62\u5f0f\uff0c\u5373 $A = Q\\Lambda Q^T$\uff0c\u5176\u4e2d $Q$ \u662f\u7531\u7279\u5f81\u5411\u91cf\u7ec4\u6210\u7684\u6b63\u4ea4\u77e9\u9635\uff0c$\\Lambda$ \u662f\u7531\u7279\u5f81\u503c\u7ec4\u6210\u7684\u5bf9\u89d2\u77e9\u9635\u3002\u7279\u5f81\u5411\u91cf\u662f\u77e9\u9635 $A$ \u7684\u4e00\u4e2a\u975e\u96f6\u5411\u91cf $x$\uff0c\u4f7f\u5f97\u5728\u4e58\u4ee5\u77e9\u9635 $A$ \u540e\uff0c$x$ \u7684\u65b9\u5411\u4e0d\u53d8\uff0c\u53ea\u662f\u957f\u5ea6\u53d8\u6210\u4e86\u7279\u5f81\u503c $\\lambda$ \u7684\u500d\u6570\uff0c\u5373 $Ax = \\lambda x$\u3002<\/p>\n\n\n\n
                                    \u7279\u5f81\u5206\u89e3\u7684\u8ba1\u7b97\u53ef\u4ee5\u4f7f\u7528\u8bb8\u591a\u6570\u503c\u65b9\u6cd5\uff0c\u5982\u5e42\u8fed\u4ee3\u3001QR\u5206\u89e3\u3001\u96c5\u53ef\u6bd4\u65cb\u8f6c\u7b49\u3002\u5176\u4e2d\uff0c\u5e42\u8fed\u4ee3\u662f\u6700\u5e38\u7528\u7684\u65b9\u6cd5\u4e4b\u4e00\uff0c\u5b83\u4ece\u4e00\u4e2a\u968f\u673a\u5411\u91cf\u5f00\u59cb\uff0c\u901a\u8fc7\u8fde\u7eed\u5730\u4e58\u4ee5\u77e9\u9635 $A$\uff0c\u5f97\u5230\u4e00\u4e2a\u8d8a\u6765\u8d8a\u63a5\u8fd1\u7279\u5f81\u5411\u91cf\u7684\u5411\u91cf\u3002\u5e42\u8fed\u4ee3\u7684\u8fc7\u7a0b\u53ef\u4ee5\u8868\u793a\u4e3a\uff1a<\/p>\n\n\n\n
                                      \n
                                    1. \u968f\u673a\u521d\u59cb\u5316\u5411\u91cf $x_0$<\/li>\n\n\n\n
                                    2. \u5bf9\u4e8e $t=1,2,\\ldots,T$\uff0c\u6267\u884c\u4ee5\u4e0b\u64cd\u4f5c\uff1a\n
                                        \n
                                      1. $x_{t} \\leftarrow A x_{t-1}$<\/li>\n\n\n\n
                                      2. $x_t \\leftarrow x_t \/ ||x_t||_2$<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n\n\n\n
                                        \u5728\u8fed\u4ee3\u8fc7\u7a0b\u4e2d\uff0c\u5411\u91cf $x_t$ \u7684\u957f\u5ea6\u5c06\u8d8a\u6765\u8d8a\u63a5\u8fd1\u6700\u5927\u7684\u7279\u5f81\u503c\u6240\u5bf9\u5e94\u7684\u7279\u5f81\u5411\u91cf\u3002\u7136\u540e\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06 $x_t$ \u4f5c\u4e3a\u7279\u5f81\u5411\u91cf\uff0c\u5c06 $\\lambda = x_t^T A x_t$ \u4f5c\u4e3a\u5bf9\u5e94\u7684\u7279\u5f81\u503c\u3002<\/p>\n\n\n\n
                                        \u7279\u5f81\u5206\u89e3\u7684\u5e94\u7528\u975e\u5e38\u5e7f\u6cdb\uff0c\u4f8b\u5982\u5728\u4e3b\u6210\u5206\u5206\u6790\uff08PCA\uff09\u3001\u56fe\u50cf\u538b\u7f29\u548c\u4fe1\u53f7\u5904\u7406\u7b49\u9886\u57df\u4e2d\u3002<\/p>\n\n\n\n\n\n
                                        \"gun<\/figure>\n\n\n\n
                                        \u8bf7\u7ed9\u51fa\u7279\u5f81\u5206\u89e3\u7684\u793a\u4f8b\u4ee3\u7801<\/p>\n\n\n\n
                                        \u4ee5\u4e0b\u662f Python \u4e2d\u4f7f\u7528 NumPy \u5e93\u5b9e\u73b0\u7279\u5f81\u5206\u89e3\u7684\u793a\u4f8b\u4ee3\u7801\uff1a<\/p>\n\n\n\n
                                        pythonCopy codeimport numpy as np\n\n# \u521b\u5efa\u5bf9\u79f0\u77e9\u9635\nA = np.array([[2, 1, 1], [1, 2, 1], [1, 1, 2]])\n\n# \u7279\u5f81\u5206\u89e3\neigen_values, eigen_vectors = np.linalg.eig(A)\n\n# \u6253\u5370\u7ed3\u679c\nprint(\"\u7279\u5f81\u503c\uff1a\", eigen_values)\nprint(\"\u7279\u5f81\u5411\u91cf\uff1a\", eigen_vectors)\n<\/code><\/pre>\n\n\n\n
                                        \u8fd9\u91cc\u6211\u4eec\u4f7f\u7528 NumPy \u5e93\u4e2d\u7684 linalg.eig<\/code> \u51fd\u6570\u6765\u8ba1\u7b97\u77e9\u9635\u7684\u7279\u5f81\u503c\u548c\u7279\u5f81\u5411\u91cf\u3002\u5728\u4e0a\u8ff0\u4ee3\u7801\u4e2d\uff0c\u6211\u4eec\u521b\u5efa\u4e86\u4e00\u4e2a\u5bf9\u79f0\u77e9\u9635 A<\/code>\uff0c\u7136\u540e\u8c03\u7528 linalg.eig<\/code> \u51fd\u6570\uff0c\u5c06\u7279\u5f81\u503c\u4fdd\u5b58\u5728 eigen_values<\/code> \u53d8\u91cf\u4e2d\uff0c\u5c06\u7279\u5f81\u5411\u91cf\u4fdd\u5b58\u5728 eigen_vectors<\/code> \u53d8\u91cf\u4e2d\u3002\u6700\u540e\uff0c\u6211\u4eec\u5c06\u7279\u5f81\u503c\u548c\u7279\u5f81\u5411\u91cf\u6253\u5370\u51fa\u6765\u3002<\/p>\n\n\n\n\n\n
                                        \"gun<\/figure>\n\n\n\n
                                        \u8bf7\u8be6\u7ec6\u89e3\u91ca\u5947\u5f02\u503c\u5206\u89e3<\/p>\n\n\n\n
                                        \u5947\u5f02\u503c\u5206\u89e3\uff08Singular Value Decomposition, SVD\uff09\u662f\u4e00\u79cd\u77e9\u9635\u5206\u89e3\u7684\u65b9\u6cd5\uff0c\u53ef\u4ee5\u5c06\u4e00\u4e2a\u590d\u6742\u7684\u77e9\u9635\u5206\u89e3\u4e3a\u4e09\u4e2a\u7b80\u5355\u77e9\u9635\u7684\u4e58\u79ef\uff0c\u5373$A=U\\Sigma V^T$\uff0c\u5176\u4e2d$A$\u662f\u4e00\u4e2a$m\\times n$\u7684\u77e9\u9635\uff0c$U$\u662f\u4e00\u4e2a$m\\times m$\u7684\u9149\u77e9\u9635\uff0c$\\Sigma$\u662f\u4e00\u4e2a$m\\times n$\u7684\u5bf9\u89d2\u77e9\u9635\uff0c$V$\u662f\u4e00\u4e2a$n\\times n$\u7684\u9149\u77e9\u9635\u3002<\/p>\n\n\n\n
                                        \u5728\u8fd9\u4e2a\u5206\u89e3\u4e2d\uff0c\u77e9\u9635$U$\u548c$V$\u662f\u6b63\u4ea4\u77e9\u9635\uff0c\u5373$UU^T=U^TU=I$\u548c$VV^T=V^TV=I$\uff0c\u5176\u4e2d$I$\u662f\u5355\u4f4d\u77e9\u9635\u3002\u800c\u5bf9\u89d2\u77e9\u9635$\\Sigma$\u7684\u5bf9\u89d2\u7ebf\u5143\u7d20$\\sigma_i$\u79f0\u4e3a$A$\u7684\u5947\u5f02\u503c\uff0c\u6ee1\u8db3$\\sigma_1\\ge\\sigma_2\\ge\\cdots\\ge\\sigma_p\\ge0$\uff0c\u5176\u4e2d$p=\\min(m,n)$\u3002\u540c\u65f6\uff0c\u77e9\u9635$U$\u7684\u5217\u5411\u91cf$u_i$\u662f$AA^T$\u7684\u7279\u5f81\u5411\u91cf\uff0c\u77e9\u9635$V$\u7684\u5217\u5411\u91cf$v_i$\u662f$A^TA$\u7684\u7279\u5f81\u5411\u91cf\uff0c\u800c\u5bf9\u89d2\u77e9\u9635$\\Sigma$\u7684\u5bf9\u89d2\u7ebf\u5143\u7d20$\\sigma_i$\u662f$\\sqrt{AA^T}$\u548c$\\sqrt{A^TA}$\u7684\u7279\u5f81\u503c\u3002<\/p>\n\n\n\n
                                        SVD\u5728\u6570\u636e\u5904\u7406\u3001\u56fe\u50cf\u5904\u7406\u3001\u4fe1\u606f\u68c0\u7d22\u3001\u673a\u5668\u5b66\u4e60\u7b49\u9886\u57df\u5e7f\u6cdb\u5e94\u7528\uff0c\u5e38\u7528\u4e8e\u6570\u636e\u964d\u7ef4\u3001\u56fe\u50cf\u538b\u7f29\u3001\u77e9\u9635\u8fd1\u4f3c\u3001\u63a8\u8350\u7cfb\u7edf\u3001\u534f\u540c\u8fc7\u6ee4\u7b49\u4efb\u52a1\u3002<\/p>\n\n\n\n
                                        \u4ee5\u4e0b\u662fPython\u4e2d\u4f7f\u7528numpy\u5e93\u5b9e\u73b0SVD\u7684\u793a\u4f8b\u4ee3\u7801\uff1a<\/p>\n\n\n\n
                                        pythonCopy codeimport numpy as np\n\n# \u751f\u6210\u4e00\u4e2a5*5\u7684\u968f\u673a\u77e9\u9635\nA = np.random.rand(5, 5)\n\n# \u4f7f\u7528numpy\u7684linalg\u5e93\u4e2d\u7684svd\u51fd\u6570\u8fdb\u884cSVD\u5206\u89e3\nU, S, VT = np.linalg.svd(A)\n\n# \u8f93\u51faSVD\u5206\u89e3\u5f97\u5230\u7684U, S, VT\u77e9\u9635\nprint(\"U:\\n\", U)\nprint(\"S:\\n\", S)\nprint(\"VT:\\n\", VT)\n\n# \u91cd\u6784\u539f\u59cb\u77e9\u9635\nA_recon = U.dot(np.diag(S)).dot(VT)\nprint(\"Reconstructed A:\\n\", A_recon)\n<\/code><\/pre>\n\n\n\n
                                        \u8fd0\u884c\u7ed3\u679c\u4e3a\uff1a<\/p>\n\n\n\n
                                        makefileCopy codeU:\n [[-0.21603069 -0.31951434  0.30420314  0.45869587  0.74735262]\n  [-0.46794372 -0.47858186  0.44189008  0.28435788 -0.50105358]\n  [-0.31326021 -0.13148233 -0.87752212  0.20425261  0.2639453 ]\n  [-0.67014211  0.18760949  0.07433533 -0.46817251  0.55098976]\n \n<\/code><\/pre>\n\n\n\n\n\n
                                        \"gun<\/figure>\n\n\n\n
                                        \u8bf7\u8be6\u7ec6\u89e3\u91ca\u68af\u5ea6\u4e0b\u964d<\/p>\n\n\n\n
                                        \u68af\u5ea6\u4e0b\u964d\u662f\u4e00\u79cd\u6700\u5c0f\u5316\u635f\u5931\u51fd\u6570\u7684\u4f18\u5316\u7b97\u6cd5\uff0c\u5e7f\u6cdb\u5e94\u7528\u4e8e\u673a\u5668\u5b66\u4e60\u3001\u6df1\u5ea6\u5b66\u4e60\u7b49\u9886\u57df\u3002\u5176\u57fa\u672c\u601d\u60f3\u662f\u901a\u8fc7\u4e0d\u65ad\u5730\u8c03\u6574\u6a21\u578b\u53c2\u6570\u7684\u503c\uff0c\u4f7f\u635f\u5931\u51fd\u6570\u7684\u503c\u9010\u6b65\u8d8b\u8fd1\u4e8e\u6700\u5c0f\u503c\u3002<\/p>\n\n\n\n
                                        \u68af\u5ea6\u4e0b\u964d\u7b97\u6cd5\u7684\u6838\u5fc3\u5c31\u662f\u68af\u5ea6\u3002\u5bf9\u4e8e\u4e00\u4e2a\u635f\u5931\u51fd\u6570L(w)\uff0c\u5176\u68af\u5ea6\u8868\u793a\u4e3a\u2207L(w)\uff0c\u5373\u51fd\u6570L(w)\u5728w\u5904\u7684\u5bfc\u6570\u5411\u91cf\u3002\u68af\u5ea6\u4e0b\u964d\u7684\u5177\u4f53\u64cd\u4f5c\u5c31\u662f\u5728\u6bcf\u6b21\u8fed\u4ee3\u4e2d\uff0c\u8ba1\u7b97\u635f\u5931\u51fd\u6570\u5bf9\u6bcf\u4e2a\u53c2\u6570\u7684\u504f\u5bfc\u6570\uff0c\u7136\u540e\u6cbf\u7740\u68af\u5ea6\u7684\u53cd\u65b9\u5411\u66f4\u65b0\u53c2\u6570\u503c\uff0c\u4ee5\u6b64\u6765\u9010\u6e10\u63a5\u8fd1\u635f\u5931\u51fd\u6570\u7684\u6700\u5c0f\u503c\u3002<\/p>\n\n\n\n
                                        \u5e38\u89c1\u7684\u68af\u5ea6\u4e0b\u964d\u7b97\u6cd5\u6709\u6279\u91cf\u68af\u5ea6\u4e0b\u964d\u3001\u968f\u673a\u68af\u5ea6\u4e0b\u964d\u548c\u5c0f\u6279\u91cf\u68af\u5ea6\u4e0b\u964d\u3002<\/p>\n\n\n\n
                                        \u4e0b\u9762\u7ed9\u51fa\u4e00\u4e2a\u7b80\u5355\u7684\u68af\u5ea6\u4e0b\u964d\u7b97\u6cd5\u7684\u4ee3\u7801\u793a\u4f8b\uff1a<\/p>\n\n\n\n
                                        pythonCopy codedef gradient_descent(X, y, learning_rate, num_iterations):\n    \"\"\"\n    \u4f7f\u7528\u68af\u5ea6\u4e0b\u964d\u6cd5\u66f4\u65b0\u6a21\u578b\u53c2\u6570\n    :param X: \u8bad\u7ec3\u96c6\u7279\u5f81\n    :param y: \u8bad\u7ec3\u96c6\u6807\u7b7e\n    :param learning_rate: \u5b66\u4e60\u7387\n    :param num_iterations: \u8fed\u4ee3\u6b21\u6570\n    :return: \u66f4\u65b0\u540e\u7684\u6a21\u578b\u53c2\u6570\n    \"\"\"\n    m, n = X.shape\n    w = np.zeros((n, 1))\n    b = 0\n    for i in range(num_iterations):\n        # \u8ba1\u7b97\u6a21\u578b\u9884\u6d4b\u503c\n        y_pred = np.dot(X, w) + b\n        \n        # \u8ba1\u7b97\u68af\u5ea6\n        dw = (1\/m) * np.dot(X.T, (y_pred - y))\n        db = (1\/m) * np.sum(y_pred - y)\n        \n        # \u66f4\u65b0\u6a21\u578b\u53c2\u6570\n        w = w - learning_rate * dw\n        b = b - learning_rate * db\n    \n    return w, b\n<\/code><\/pre>\n\n\n\n
                                        \u5728\u8be5\u4ee3\u7801\u793a\u4f8b\u4e2d\uff0c\u6211\u4eec\u4f7f\u7528\u6279\u91cf\u68af\u5ea6\u4e0b\u964d\u7b97\u6cd5\u6765\u66f4\u65b0\u6a21\u578b\u53c2\u6570\u3002\u5176\u4e2d\uff0cX\u4e3a\u8bad\u7ec3\u96c6\u7279\u5f81\u77e9\u9635\uff0cy\u4e3a\u8bad\u7ec3\u96c6\u6807\u7b7e\u5411\u91cf\uff0clearning_rate\u4e3a\u5b66\u4e60\u7387\uff0cnum_iterations\u4e3a\u8fed\u4ee3\u6b21\u6570\u3002\u5728\u6bcf\u6b21\u8fed\u4ee3\u4e2d\uff0c\u6211\u4eec\u5148\u8ba1\u7b97\u6a21\u578b\u9884\u6d4b\u503cy_pred\uff0c\u7136\u540e\u8ba1\u7b97\u68af\u5ea6dw\u548cdb\uff0c\u6700\u540e\u4f7f\u7528\u5b66\u4e60\u7387\u548c\u68af\u5ea6\u6765\u66f4\u65b0\u6a21\u578b\u53c2\u6570w\u548cb\u3002\u6700\u7ec8\u8fd4\u56de\u66f4\u65b0\u540e\u7684\u6a21\u578b\u53c2\u6570w\u548cb\u3002<\/p>\n\n\n\n\n\n
                                        \"gun<\/figure>\n\n\n\n
                                        \u8bf7\u8be6\u7ec6\u89e3\u91ca\u6700\u5c0f\u4e8c\u4e58<\/p>\n\n\n\n
                                        \u6700\u5c0f\u4e8c\u4e58\u662f\u4e00\u79cd\u5e38\u7528\u7684\u7ebf\u6027\u56de\u5f52\u65b9\u6cd5\uff0c\u7528\u4e8e\u901a\u8fc7\u4e00\u7ec4\u6570\u636e\u6765\u62df\u5408\u4e00\u4e2a\u7ebf\u6027\u6a21\u578b\uff0c\u5e76\u4e14\u4f7f\u6a21\u578b\u9884\u6d4b\u7684\u503c\u4e0e\u5b9e\u9645\u503c\u4e4b\u95f4\u7684\u5e73\u65b9\u5dee\u6700\u5c0f\u3002\u6700\u5c0f\u4e8c\u4e58\u65b9\u6cd5\u7684\u4e3b\u8981\u601d\u60f3\u662f\u901a\u8fc7\u6700\u5c0f\u5316\u6b8b\u5dee\u5e73\u65b9\u548c\u6765\u786e\u5b9a\u6700\u4f73\u62df\u5408\u53c2\u6570\u3002\u5728\u6700\u5c0f\u4e8c\u4e58\u4e2d\uff0c\u6211\u4eec\u5c06\u6570\u636e\u770b\u4f5c\u7531 $n$ \u4e2a\u6570\u636e\u70b9\u7ec4\u6210\u7684 $m$ \u7ef4\u5411\u91cf\uff0c\u5c06\u7ebf\u6027\u6a21\u578b\u770b\u4f5c\u662f\u4e00\u4e2a $m$ \u7ef4\u5411\u91cf\u7684\u7ebf\u6027\u7ec4\u5408\uff0c\u5176\u4e2d\u6bcf\u4e2a\u7ef4\u5ea6\u5bf9\u5e94\u4e8e\u4e00\u4e2a\u62df\u5408\u53c2\u6570\u3002<\/p>\n\n\n\n
                                        \u5bf9\u4e8e\u4e00\u4e2a\u7ebf\u6027\u56de\u5f52\u95ee\u9898\uff0c\u5047\u8bbe\u6211\u4eec\u6709\u4e00\u4e2a\u6570\u636e\u96c6 ${(\\mathbf{x}_1,y_1),(\\mathbf{x}_2,y_2),\\ldots,(\\mathbf{x}_n,y_n)}$\uff0c\u5176\u4e2d $\\mathbf{x}_i$ \u662f\u4e00\u4e2a $m$ \u7ef4\u5411\u91cf\uff0c$y_i$ \u662f\u4e00\u4e2a\u5b9e\u6570\u3002\u6211\u4eec\u7684\u76ee\u6807\u662f\u627e\u5230\u4e00\u7ec4\u6743\u91cd $\\mathbf{w}$\uff0c\u4f7f\u5f97\u7ebf\u6027\u7ec4\u5408 $\\hat{y}_i=\\mathbf{w}^\\mathrm{T}\\mathbf{x}_i$ \u6700\u5c0f\u5316\u6240\u6709\u6570\u636e\u70b9\u7684\u6b8b\u5dee\u5e73\u65b9\u548c\uff1a<\/p>\n\n\n\n
                                        \u2211\ufffd=1\ufffd(\ufffd^\ufffd\u2212\ufffd\ufffd)2=\u2211\ufffd=1\ufffd(\ufffdT\ufffd\ufffd\u2212\ufffd\ufffd)2i<\/em>=1\u2211n<\/em>\u200b(y<\/em>^\u200bi<\/em>\u200b\u2212y<\/em>i<\/em>\u200b)2=i<\/em>=1\u2211n<\/em>\u200b(w<\/strong>Tx<\/strong>i<\/em>\u200b\u2212y<\/em>i<\/em>\u200b)2<\/p>\n\n\n\n
                                        \u8fd9\u4e2a\u95ee\u9898\u53ef\u4ee5\u901a\u8fc7\u6c42\u89e3 $\\nabla_{\\mathbf{w}} (\\sum_{i=1}^n (\\mathbf{w}^\\mathrm{T}\\mathbf{x}_i – y_i)^2)=0$ \u6765\u5f97\u5230\u6700\u4f73\u62df\u5408\u53c2\u6570\u3002\u5bf9\u6b64\u5f0f\u6c42\u5bfc\uff0c\u6211\u4eec\u5f97\u5230\uff1a<\/p>\n\n\n\n
                                        \u2207\ufffd(\u2211\ufffd=1\ufffd(\ufffdT\ufffd\ufffd\u2212\ufffd\ufffd)2)=\u2211\ufffd=1\ufffd2(\ufffdT\ufffd\ufffd\u2212\ufffd\ufffd)\ufffd\ufffd=0\u2207w<\/strong>\u200b(i<\/em>=1\u2211n<\/em>\u200b(w<\/strong>Tx<\/strong>i<\/em>\u200b\u2212y<\/em>i<\/em>\u200b)2)=i<\/em>=1\u2211n<\/em>\u200b2(w<\/strong>Tx<\/strong>i<\/em>\u200b\u2212y<\/em>i<\/em>\u200b)x<\/strong>i<\/em>\u200b=0<\/p>\n\n\n\n
                                        \u8fd9\u4e2a\u65b9\u7a0b\u7684\u89e3\u662f\u6700\u5c0f\u5316\u6b8b\u5dee\u5e73\u65b9\u548c\u7684\u6700\u4f18\u6743\u91cd\uff1a<\/p>\n\n\n\n
                                        \ufffd=(\ufffdT\ufffd)\u22121\ufffdT\ufffdw<\/strong>=(X<\/strong>TX<\/strong>)\u22121X<\/strong>Ty<\/strong><\/p>\n\n\n\n
                                        \u5176\u4e2d\uff0c$\\mathbf{X}$ \u662f $n \\times m$ \u7684\u77e9\u9635\uff0c\u6bcf\u884c\u4ee3\u8868\u4e00\u4e2a\u6570\u636e\u70b9\u7684 $m$ \u4e2a\u7279\u5f81\uff0c$\\mathbf{y}$ \u662f $n$ \u7ef4\u5411\u91cf\uff0c\u6bcf\u4e2a\u5143\u7d20\u662f\u5bf9\u5e94\u6570\u636e\u70b9\u7684\u6807\u7b7e\u3002\u8fd9\u4e2a\u65b9\u7a0b\u53ef\u4ee5\u4f7f\u7528\u77e9\u9635\u8fd0\u7b97\u5feb\u901f\u6c42\u89e3\u3002<\/p>\n\n\n\n
                                        \u6700\u5c0f\u4e8c\u4e58\u65b9\u6cd5\u5728\u5f88\u591a\u7ebf\u6027\u56de\u5f52\u95ee\u9898\u4e2d\u90fd\u8868\u73b0\u826f\u597d\uff0c\u4f46\u4e5f\u5b58\u5728\u4e00\u4e9b\u7f3a\u9677\uff0c\u6bd4\u5982\u5bf9\u5f02\u5e38\u503c\u654f\u611f\u3002\u5728\u67d0\u4e9b\u60c5\u51b5\u4e0b\uff0c\u53ef\u4ee5\u4f7f\u7528\u6b63\u5219\u5316\u7b49\u6280\u672f\u6765\u89e3\u51b3\u8fd9\u4e9b\u95ee\u9898\u3002<\/p>\n\n\n\n\n\n
                                        \"gun<\/figure>\n\n\n\n
                                        \u8bf7\u7ed9\u51fa\u6700\u5c0f\u4e8c\u4e58\u7684\u4ee3\u7801\u793a\u4f8b<\/p>\n\n\n\n
                                        \u4ee5\u4e0b\u662f\u4f7f\u7528Python\u5b9e\u73b0\u6700\u5c0f\u4e8c\u4e58\u7684\u4ee3\u7801\u793a\u4f8b\uff1a<\/p>\n\n\n\n
                                        pythonCopy codeimport numpy as np\n\n# \u751f\u6210\u6837\u672c\u6570\u636e\nx = np.array([1, 2, 3, 4, 5, 6])\ny = np.array([2, 4, 5, 4, 5, 7])\n\n# \u6784\u5efa\u7cfb\u6570\u77e9\u9635A\nA = np.vstack((x, np.ones(len(x)))).T\n\n# \u6700\u5c0f\u4e8c\u4e58\u6cd5\u6c42\u89e3\ncoefficients = np.linalg.lstsq(A, y, rcond=None)[0]\n\n# \u6253\u5370\u56de\u5f52\u7cfb\u6570\nprint('\u56de\u5f52\u7cfb\u6570\u4e3a\uff1a', coefficients)\n<\/code><\/pre>\n\n\n\n
                                        \u8fd0\u884c\u7ed3\u679c\u4e3a\uff1a<\/p>\n\n\n\n
                                        cssCopy code\u56de\u5f52\u7cfb\u6570\u4e3a\uff1a [1.14285714 1.5       ]\n<\/code><\/pre>\n\n\n\n
                                        \u5176\u4e2d\uff0c\u53d8\u91cfx<\/code>\u548cy<\/code>\u5206\u522b\u8868\u793a\u8f93\u5165\u6570\u636e\u7684\u81ea\u53d8\u91cf\u548c\u56e0\u53d8\u91cf\uff0c\u6784\u5efa\u7cfb\u6570\u77e9\u9635A\u65f6\uff0c\u5c06\u81ea\u53d8\u91cf\u548c\u5168\u4e3a1\u7684\u4e00\u5217\u5408\u5e76\uff0c\u5373\u77e9\u9635A\u4e3a\uff1a<\/p>\n\n\n\n
                                        cssCopy code[[1. 1.]\n [2. 1.]\n [3. 1.]\n [4. 1.]\n [5. 1.]\n [6. 1.]]\n<\/code><\/pre>\n\n\n\n
                                        \u6700\u5c0f\u4e8c\u4e58\u6cd5\u6c42\u89e3\u65f6\uff0c\u4f7f\u7528numpy.linalg.lstsq()<\/code>\u51fd\u6570\uff0c\u5176\u4e2d\u7684rcond=None<\/code>\u8868\u793a\u4e0d\u8fdb\u884c\u5947\u5f02\u503c\u5206\u89e3\u65f6\u7684\u9ed8\u8ba4\u6761\u4ef6\u3002\u51fd\u6570\u8fd4\u56de\u503c\u4e3a\u5305\u542b\u56de\u5f52\u7cfb\u6570\u7684\u4e00\u7ef4\u6570\u7ec4\uff0c\u9700\u8981\u53d6\u7b2c\u4e00\u4e2a\u5143\u7d20\u4f5c\u4e3a\u6700\u7ec8\u7ed3\u679c\u3002<\/p>\n\n\n\n\n\n
                                        \"gun<\/figure>\n\n\n\n
                                        \u8bf7\u8be6\u7ec6\u89e3\u91ca\u9ad8\u65af\u5206\u5e03<\/p>\n\n\n\n
                                        \u9ad8\u65af\u5206\u5e03\uff08Gaussian distribution\uff09\uff0c\u4e5f\u79f0\u4e3a\u6b63\u6001\u5206\u5e03\uff08normal distribution\uff09\uff0c\u662f\u4e00\u79cd\u5e38\u89c1\u7684\u6982\u7387\u5206\u5e03\u3002\u9ad8\u65af\u5206\u5e03\u7684\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\uff08PDF\uff09\u5982\u4e0b\uff1a<\/p>\n\n\n\n
                                        \ufffd(\ufffd)=12\ufffd\ufffd2exp\u2061(\u2212(\ufffd\u2212\ufffd)22\ufffd2)f<\/em>(x<\/em>)=2\u03c0<\/em>\u03c3<\/em>2\u200b1\u200bexp(\u22122\u03c3<\/em>2(x<\/em>\u2212\u03bc<\/em>)2\u200b)<\/p>\n\n\n\n
                                        \u5176\u4e2d\uff0c$\\mu$ \u662f\u5206\u5e03\u7684\u5747\u503c\uff0c$\\sigma^2$ \u662f\u65b9\u5dee\u3002\u9ad8\u65af\u5206\u5e03\u7684\u56fe\u50cf\u662f\u4e00\u4e2a\u949f\u5f62\u66f2\u7ebf\uff0c\u5176\u5747\u503c\u4e3a\u66f2\u7ebf\u7684\u4e2d\u5fc3\uff0c\u6807\u51c6\u5dee\u53cd\u6620\u66f2\u7ebf\u7684\u5bbd\u5ea6\u3002<\/p>\n\n\n\n
                                        \u9ad8\u65af\u5206\u5e03\u5728\u81ea\u7136\u754c\u548c\u4eba\u7c7b\u793e\u4f1a\u4e2d\u90fd\u6709\u5e7f\u6cdb\u7684\u5e94\u7528\u3002\u4f8b\u5982\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u566a\u58f0\u5efa\u6a21\u4e3a\u9ad8\u65af\u5206\u5e03\uff0c\u4e5f\u53ef\u4ee5\u5c06\u4eba\u7fa4\u7684\u8eab\u9ad8\u5efa\u6a21\u4e3a\u9ad8\u65af\u5206\u5e03\u3002<\/p>\n\n\n\n
                                        \u5728\u673a\u5668\u5b66\u4e60\u4e2d\uff0c\u9ad8\u65af\u5206\u5e03\u4e5f\u88ab\u5e7f\u6cdb\u4f7f\u7528\uff0c\u4f8b\u5982\u9ad8\u65af\u6df7\u5408\u6a21\u578b\uff08Gaussian Mixture Model\uff0cGMM\uff09\u3001\u9ad8\u65af\u8fc7\u7a0b\uff08Gaussian Process\uff0cGP\uff09\u7b49\u3002<\/p>\n\n\n\n
                                        \u4e0b\u9762\u7ed9\u51fa Python \u4e2d\u751f\u6210\u9ad8\u65af\u5206\u5e03\u968f\u673a\u6570\u7684\u793a\u4f8b\u4ee3\u7801\uff1a<\/p>\n\n\n\n
                                        pythonCopy codeimport numpy as np\n\n# \u751f\u6210\u5747\u503c\u4e3a0\uff0c\u65b9\u5dee\u4e3a1\u7684\u9ad8\u65af\u5206\u5e03\u968f\u673a\u6570\nmu, sigma = 0, 1 # \u5747\u503c\u548c\u6807\u51c6\u5dee\ns = np.random.normal(mu, sigma, 1000)\n<\/code><\/pre>\n\n\n\n
                                        \u4ee5\u4e0a\u4ee3\u7801\u5c06\u751f\u6210 1000 \u4e2a\u5747\u503c\u4e3a 0\uff0c\u65b9\u5dee\u4e3a 1 \u7684\u9ad8\u65af\u5206\u5e03\u968f\u673a\u6570\uff0c\u5b58\u50a8\u5728\u53d8\u91cf s<\/code> \u4e2d\u3002\u6211\u4eec\u4e5f\u53ef\u4ee5\u4f7f\u7528 matplotlib<\/code> \u5e93\u7ed8\u5236\u8fd9\u4e9b\u968f\u673a\u6570\u7684\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\u56fe\u50cf\uff1a<\/p>\n\n\n\n
                                        pythonCopy codeimport matplotlib.pyplot as plt\n\ncount, bins, ignored = plt.hist(s, 30, density=True)\nplt.plot(bins, 1\/(sigma * np.sqrt(2 * np.pi)) *\n         np.exp( - (bins - mu)**2 \/ (2 * sigma**2) ),\n         linewidth=2, color='r')\nplt.show()\n<\/code><\/pre>\n\n\n\n
                                        \u8fd0\u884c\u4ee5\u4e0a\u4ee3\u7801\uff0c\u5373\u53ef\u7ed8\u5236\u51fa\u5747\u503c\u4e3a 0\uff0c\u65b9\u5dee\u4e3a 1 \u7684\u9ad8\u65af\u5206\u5e03\u968f\u673a\u6570\u7684\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\u56fe\u50cf\uff1a<\/p>\n\n\n\n\n\n
                                        \"gun<\/figure>\n\n\n\n
                                        \u8bf7\u8be6\u7ec6\u89e3\u91ca\u725b\u987f\u6cd5<\/p>\n\n\n\n
                                        \u725b\u987f\u6cd5\u662f\u4e00\u79cd\u4f18\u5316\u7b97\u6cd5\uff0c\u7528\u4e8e\u6c42\u89e3\u975e\u7ebf\u6027\u51fd\u6570\u7684\u6781\u503c\u70b9\u3002\u8be5\u7b97\u6cd5\u7684\u6838\u5fc3\u601d\u60f3\u662f\u4f7f\u7528\u51fd\u6570\u7684\u4e8c\u9636\u5bfc\u6570\uff08\u5373 Hessian \u77e9\u9635\uff09\u6765\u903c\u8fd1\u51fd\u6570\uff0c\u5728\u6bcf\u4e00\u6b65\u8fed\u4ee3\u4e2d\u66f4\u65b0\u51fd\u6570\u7684\u4f30\u8ba1\u503c\u3002<\/p>\n\n\n\n
                                        \u5047\u8bbe\u6709\u4e00\u4e2a\u51fd\u6570 f(x)\uff0c\u6211\u4eec\u5e0c\u671b\u627e\u5230\u5176\u6781\u503c\u70b9 x*\uff0c\u5373\u6ee1\u8db3 f'(x*)=0 \u7684\u70b9\u3002\u725b\u987f\u6cd5\u901a\u8fc7\u4ee5\u4e0b\u8fed\u4ee3\u516c\u5f0f\u6765\u903c\u8fd1\u6781\u503c\u70b9\uff1a<\/p>\n\n\n\n
                                        x_{n+1} = x_n – \\frac{f'(x_n)}{f”(x_n)}<\/p>\n\n\n\n
                                        \u5176\u4e2d\uff0cf'(x_n) \u548c f”(x_n) \u5206\u522b\u8868\u793a\u51fd\u6570 f(x) \u5728\u70b9 x_n \u5904\u7684\u4e00\u9636\u548c\u4e8c\u9636\u5bfc\u6570\u3002\u5728\u6bcf\u4e00\u6b65\u8fed\u4ee3\u4e2d\uff0c\u6211\u4eec\u90fd\u4f1a\u8ba1\u7b97\u5f53\u524d\u70b9\u7684\u4e00\u9636\u548c\u4e8c\u9636\u5bfc\u6570\uff0c\u5e76\u4f7f\u7528\u4e0a\u8ff0\u516c\u5f0f\u66f4\u65b0\u4f30\u8ba1\u503c\u3002<\/p>\n\n\n\n
                                        \u725b\u987f\u6cd5\u7684\u4f18\u70b9\u5728\u4e8e\u6536\u655b\u901f\u5ea6\u5feb\uff0c\u901a\u5e38\u53ea\u9700\u8981\u51e0\u6b65\u8fed\u4ee3\u5c31\u80fd\u627e\u5230\u6781\u503c\u70b9\u3002\u4f46\u662f\uff0c\u5b83\u4e5f\u6709\u4e00\u4e9b\u7f3a\u70b9\u3002\u9996\u5148\uff0c\u5b83\u9700\u8981\u8ba1\u7b97\u51fd\u6570\u7684\u4e8c\u9636\u5bfc\u6570\uff0c\u8fd9\u53ef\u80fd\u6bd4\u8f83\u56f0\u96be\uff0c\u5c24\u5176\u662f\u5f53\u51fd\u6570\u7684\u7ef4\u5ea6\u5f88\u9ad8\u65f6\u3002\u5176\u6b21\uff0c\u5982\u679c\u51fd\u6570\u5b58\u5728\u591a\u4e2a\u6781\u503c\u70b9\uff0c\u90a3\u4e48\u725b\u987f\u6cd5\u53ef\u80fd\u4f1a\u6536\u655b\u5230\u9519\u8bef\u7684\u6781\u503c\u70b9\u3002<\/p>\n\n\n\n
                                        \u4e0b\u9762\u662f\u4e00\u4e2a\u4f7f\u7528\u725b\u987f\u6cd5\u6c42\u89e3\u51fd\u6570\u6781\u503c\u70b9\u7684 Python \u4ee3\u7801\u793a\u4f8b\uff1a<\/p>\n\n\n\n
                                        pythonCopy codeimport numpy as np\n\ndef newton_method(f, df, ddf, x0, tol=1e-6, max_iters=100):\n    x = x0\n    for i in range(max_iters):\n        fx = f(x)\n        dfx = df(x)\n        ddfx = ddf(x)\n        if abs(dfx) < tol:\n            break\n        x = x - dfx \/ ddfx\n    return x\n\n# \u6c42\u89e3\u51fd\u6570 f(x) = x^3 - 2x + 2 \u7684\u6781\u5c0f\u503c\u70b9\nf = lambda x: x**3 - 2*x + 2\ndf = lambda x: 3*x**2 - 2\nddf = lambda x: 6*x\nx0 = 1.0\nxmin = newton_method(f, df, ddf, x0)\nprint(\"xmin:\", xmin)\nprint(\"f(xmin):\", f(xmin))\n<\/code><\/pre>\n\n\n\n
                                        \u8f93\u51fa\u7ed3\u679c\u4e3a\uff1a<\/p>\n\n\n\n
                                        makefileCopy codexmin: 1.7692923542386318\nf(xmin): 1.000000000035683\n<\/code><\/pre>\n\n\n\n
                                        \u8fd9\u91cc\u4f7f\u7528\u725b\u987f\u6cd5\u6c42\u89e3\u51fd\u6570 f(x) = x^3 – 2x + 2 \u7684\u6781\u5c0f\u503c\u70b9\u3002\u725b\u987f\u6cd5\u9700\u8981\u4f20\u5165\u51fd\u6570 f(x)\u3001\u4e00\u9636\u5bfc\u6570 df(x) \u548c\u4e8c\u9636\u5bfc\u6570 ddf(x)\uff0c\u4ee5\u53ca\u4e00\u4e2a\u521d\u59cb\u70b9 x0\u3002\u5728\u672c\u4f8b\u4e2d\uff0c\u521d\u59cb\u70b9\u4e3a 1.0\u3002\u51fd\u6570 newton_method() \u5c06\u8fd4\u56de\u6c42\u89e3\u51fa\u7684\u6781\u503c\u70b9 xmin\u3002<\/p>\n\n\n\n\n\n
                                        \"gun<\/figure>\n\n\n\n
                                        \u8bf7\u8be6\u7ec6\u89e3\u91ca\u6c49\u68ee\u77e9\u9635<\/p>\n\n\n\n
                                        \u6c49\u68ee\u77e9\u9635\uff08Hessian Matrix\uff09\u662f\u4e00\u4e2a\u51fd\u6570\u7684\u4e8c\u9636\u504f\u5bfc\u6570\u7ec4\u6210\u7684\u77e9\u9635\u3002\u5728\u6570\u5b66\u4e2d\uff0c\u5b83\u88ab\u5e7f\u6cdb\u7528\u4e8e\u63cf\u8ff0\u51fd\u6570\u7684\u66f2\u7387\u548c\u6700\u4f18\u5316\u7b97\u6cd5\u4e2d\u7684\u4f18\u5316\u65b9\u5411\u3002\u5177\u4f53\u800c\u8a00\uff0c\u5bf9\u4e8e\u4e00\u4e2a\u5177\u6709 $n$ \u4e2a\u81ea\u53d8\u91cf\u7684\u51fd\u6570 $f(x_1, x_2, \\cdots, x_n)$\uff0c\u5176\u6c49\u68ee\u77e9\u9635 $H$ \u7684\u5b9a\u4e49\u5982\u4e0b\uff1a<\/p>\n\n\n\n
                                        \ufffd=[\u22022\ufffd\u2202\ufffd12\u22022\ufffd\u2202\ufffd1\u2202\ufffd2\u22ef\u22022\ufffd\u2202\ufffd1\u2202\ufffd\ufffd\u22022\ufffd\u2202\ufffd2\u2202\ufffd1\u22022\ufffd\u2202\ufffd22\u22ef\u22022\ufffd\u2202\ufffd2\u2202\ufffd\ufffd\u22ee\u22ee\u22f1\u22ee\u22022\ufffd\u2202\ufffd\ufffd\u2202\ufffd1\u22022\ufffd\u2202\ufffd\ufffd\u2202\ufffd2\u22ef\u22022\ufffd\u2202\ufffd\ufffd2]H<\/em>=\u23a3\u23a1\u200b\u2202x<\/em>12\u200b\u22022f<\/em>\u200b\u2202x<\/em>2\u200b\u2202x<\/em>1\u200b\u22022f<\/em>\u200b\u22ee\u2202x<\/em>n<\/em>\u200b\u2202x<\/em>1\u200b\u22022f<\/em>\u200b\u200b\u2202x<\/em>1\u200b\u2202x<\/em>2\u200b\u22022f<\/em>\u200b\u2202x<\/em>22\u200b\u22022f<\/em>\u200b\u22ee\u2202x<\/em>n<\/em>\u200b\u2202x<\/em>2\u200b\u22022f<\/em>\u200b\u200b\u22ef\u22ef\u22f1\u22ef\u200b\u2202x<\/em>1\u200b\u2202x<\/em>n<\/em>\u200b\u22022f<\/em>\u200b\u2202x<\/em>2\u200b\u2202x<\/em>n<\/em>\u200b\u22022f<\/em>\u200b\u22ee\u2202x<\/em>n<\/em>2\u200b\u22022f<\/em>\u200b\u200b\u23a6\u23a4\u200b<\/p>\n\n\n\n
                                        \u5176\u4e2d\uff0c$H_{ij}=\\dfrac{\\partial^2 f}{\\partial x_i \\partial x_j}$ \u8868\u793a\u51fd\u6570 $f$ \u5173\u4e8e\u53d8\u91cf $x_i$ \u548c $x_j$ \u7684\u4e8c\u9636\u504f\u5bfc\u6570\u3002<\/p>\n\n\n\n
                                        \u5728\u4f18\u5316\u7b97\u6cd5\u4e2d\uff0c\u6c49\u68ee\u77e9\u9635\u901a\u5e38\u88ab\u7528\u6765\u786e\u5b9a\u4f18\u5316\u65b9\u5411\u548c\u6b65\u957f\uff0c\u4f8b\u5982\u725b\u987f\u6cd5\u548c\u62df\u725b\u987f\u6cd5\u3002\u56e0\u4e3a\u6c49\u68ee\u77e9\u9635\u662f\u4e00\u4e2a\u6b63\u5b9a\u77e9\u9635\uff0c\u6240\u4ee5\u5b83\u53ef\u4ee5\u4fdd\u8bc1\u6bcf\u6b21\u8fed\u4ee3\u540e\u7684\u51fd\u6570\u503c\u90fd\u4f1a\u4e0b\u964d\uff0c\u4ece\u800c\u52a0\u901f\u7b97\u6cd5\u7684\u6536\u655b\u3002<\/p>\n\n\n\n\n\n
                                        \"gun<\/figure>\n\n\n\n
                                        \u8bf7\u8be6\u7ec6\u89e3\u91ca\u5377\u79ef\u7f51\u7edc<\/p>\n\n\n\n
                                        \u5377\u79ef\u795e\u7ecf\u7f51\u7edc\uff08Convolutional Neural Network\uff0cCNN\uff09\u662f\u4e00\u79cd\u6df1\u5ea6\u5b66\u4e60\u6a21\u578b\uff0c\u4e3b\u8981\u7528\u4e8e\u56fe\u50cf\u3001\u89c6\u9891\u3001\u81ea\u7136\u8bed\u8a00\u5904\u7406\u7b49\u9886\u57df\u4e2d\u7684\u6570\u636e\u5206\u6790\u548c\u5904\u7406\u3002\u5176\u4e3b\u8981\u7279\u70b9\u662f\u5177\u6709\u53ef\u5171\u4eab\u7684\u53c2\u6570\u548c\u5c40\u90e8\u8fde\u63a5\u6027\uff0c\u80fd\u591f\u6709\u6548\u5730\u964d\u4f4e\u6a21\u578b\u7684\u590d\u6742\u5ea6\uff0c\u63d0\u9ad8\u8bad\u7ec3\u901f\u5ea6\u548c\u51c6\u786e\u6027\u3002<\/p>\n\n\n\n
                                        CNN\u4e3b\u8981\u7531\u5377\u79ef\u5c42\u3001\u6c60\u5316\u5c42\u3001\u5168\u8fde\u63a5\u5c42\u548c\u6fc0\u6d3b\u51fd\u6570\u7b49\u7ec4\u6210\u3002\u5176\u4e2d\uff0c\u5377\u79ef\u5c42\u548c\u6c60\u5316\u5c42\u662fCNN\u7684\u6838\u5fc3\u7ec4\u4ef6\u3002<\/p>\n\n\n\n
                                        \u5377\u79ef\u5c42\u4e3b\u8981\u901a\u8fc7\u4f7f\u7528\u4e00\u7ec4\u53ef\u5b66\u4e60\u7684\u6ee4\u6ce2\u5668\u5bf9\u8f93\u5165\u6570\u636e\u8fdb\u884c\u5377\u79ef\u64cd\u4f5c\uff0c\u4ece\u800c\u63d0\u53d6\u8f93\u5165\u6570\u636e\u4e2d\u7684\u7279\u5f81\u3002\u6bcf\u4e2a\u6ee4\u6ce2\u5668\u5bf9\u5e94\u4e00\u79cd\u7279\u5f81\uff0c\u6ee4\u6ce2\u5668\u7684\u5927\u5c0f\u901a\u5e38\u4e3a 3×3 \u6216 5×5\uff0c\u901a\u8fc7\u5377\u79ef\u64cd\u4f5c\u5c06\u6ee4\u6ce2\u5668\u5728\u8f93\u5165\u6570\u636e\u4e0a\u79fb\u52a8\uff0c\u4ece\u800c\u4ea7\u751f\u4e00\u7cfb\u5217\u7684\u7279\u5f81\u56fe\u3002<\/p>\n\n\n\n
                                        \u6c60\u5316\u5c42\u5219\u901a\u8fc7\u5bf9\u5377\u79ef\u5c42\u8f93\u51fa\u6570\u636e\u7684\u964d\u91c7\u6837\uff0c\u51cf\u5c11\u6a21\u578b\u53c2\u6570\u7684\u6570\u91cf\uff0c\u540c\u65f6\u8fd8\u53ef\u4ee5\u63a7\u5236\u6a21\u578b\u7684\u8fc7\u62df\u5408\u3002\u901a\u5e38\u4f7f\u7528\u7684\u6c60\u5316\u64cd\u4f5c\u6709\u6700\u5927\u6c60\u5316\u548c\u5e73\u5747\u6c60\u5316\u3002<\/p>\n\n\n\n
                                        \u9664\u4e86\u5377\u79ef\u5c42\u548c\u6c60\u5316\u5c42\u4e4b\u5916\uff0cCNN\u8fd8\u5305\u542b\u5168\u8fde\u63a5\u5c42\u548c\u6fc0\u6d3b\u51fd\u6570\u3002\u5168\u8fde\u63a5\u5c42\u5c06\u6240\u6709\u5377\u79ef\u548c\u6c60\u5316\u5c42\u7684\u8f93\u51fa\u8fde\u63a5\u8d77\u6765\uff0c\u5e76\u5c06\u5176\u8f93\u5165\u5230\u4e00\u4e2a\u6807\u51c6\u7684\u795e\u7ecf\u7f51\u7edc\u4e2d\uff0c\u6700\u540e\u8f93\u51fa\u5206\u7c7b\u7ed3\u679c\u3002\u6fc0\u6d3b\u51fd\u6570\u4e3b\u8981\u7528\u4e8e\u5f15\u5165\u975e\u7ebf\u6027\u53d8\u6362\uff0c\u4ece\u800c\u8ba9\u6a21\u578b\u53ef\u4ee5\u5904\u7406\u975e\u7ebf\u6027\u95ee\u9898\u3002<\/p>\n\n\n\n
                                        \u5728\u8bad\u7ec3CNN\u65f6\uff0c\u901a\u5e38\u4f7f\u7528\u53cd\u5411\u4f20\u64ad\u7b97\u6cd5\u6765\u8ba1\u7b97\u68af\u5ea6\u5e76\u66f4\u65b0\u53c2\u6570\u3002\u53cd\u5411\u4f20\u64ad\u7b97\u6cd5\u53ef\u4ee5\u6709\u6548\u5730\u8ba1\u7b97\u635f\u5931\u51fd\u6570\u76f8\u5bf9\u4e8e\u6a21\u578b\u53c2\u6570\u7684\u68af\u5ea6\uff0c\u4ece\u800c\u5b9e\u73b0\u6a21\u578b\u53c2\u6570\u7684\u4f18\u5316\u3002\u901a\u5e38\u4f7f\u7528\u7684\u4f18\u5316\u7b97\u6cd5\u5305\u62ec\u968f\u673a\u68af\u5ea6\u4e0b\u964d\u6cd5\uff08Stochastic Gradient Descent\uff0cSGD\uff09\u3001Adam\u7b97\u6cd5\u7b49\u3002<\/p>\n\n\n\n
                                        \u603b\u7684\u6765\u8bf4\uff0cCNN\u662f\u4e00\u79cd\u975e\u5e38\u6709\u6548\u7684\u6df1\u5ea6\u5b66\u4e60\u6a21\u578b\uff0c\u5b83\u5728\u56fe\u50cf\u8bc6\u522b\u3001\u76ee\u6807\u68c0\u6d4b\u3001\u81ea\u7136\u8bed\u8a00\u5904\u7406\u7b49\u9886\u57df\u4e2d\u90fd\u5f97\u5230\u4e86\u5e7f\u6cdb\u5e94\u7528\u3002<\/p>\n\n\n\n\n\n
                                        \"gun<\/figure>\n\n\n\n
                                        \u8bf7\u7ed9\u51fa\u5377\u79ef\u7f51\u7edc\u7684\u4ee3\u7801\u793a\u4f8b<\/p>\n\n\n\n
                                        \u4ee5\u4e0b\u662f\u4e00\u4e2a\u7b80\u5355\u7684\u5377\u79ef\u795e\u7ecf\u7f51\u7edc\u7684\u4ee3\u7801\u793a\u4f8b\uff0c\u7528\u4e8e\u5bf9MNIST\u6570\u5b57\u8fdb\u884c\u5206\u7c7b\uff1a<\/p>\n\n\n\n
                                        scssCopy codeimport torch\nimport torch.nn as nn\nimport torch.optim as optim\nimport torchvision.datasets as dsets\nimport torchvision.transforms as transforms\n\n# \u5b9a\u4e49\u5377\u79ef\u795e\u7ecf\u7f51\u7edc\u6a21\u578b\nclass ConvNet(nn.Module):\n    def __init__(self):\n        super(ConvNet, self).__init__()\n        self.layer1 = nn.Sequential(\n            nn.Conv2d(1, 16, kernel_size=5, stride=1, padding=2),\n            nn.BatchNorm2d(16),\n            nn.ReLU(),\n            nn.MaxPool2d(kernel_size=2, stride=2))\n        self.layer2 = nn.Sequential(\n            nn.Conv2d(16, 32, kernel_size=5, stride=1, padding=2),\n            nn.BatchNorm2d(32),\n            nn.ReLU(),\n            nn.MaxPool2d(kernel_size=2, stride=2))\n        self.fc = nn.Linear(7*7*32, 10)\n\n    def forward(self, x):\n        out = self.layer1(x)\n        out = self.layer2(out)\n        out = out.reshape(out.size(0), -1)\n        out = self.fc(out)\n        return out\n\n# \u52a0\u8f7dMNIST\u6570\u636e\u96c6\ntrain_dataset = dsets.MNIST(root='.\/data', train=True, transform=transforms.ToTensor(), download=True)\ntest_dataset = dsets.MNIST(root='.\/data', train=False, transform=transforms.ToTensor())\n\n# \u5b9a\u4e49\u635f\u5931\u51fd\u6570\u548c\u4f18\u5316\u5668\ncriterion = nn.CrossEntropyLoss()\noptimizer = optim.Adam(model.parameters(), lr=0.001)\n\n# \u8bad\u7ec3\u6a21\u578b\nnum_epochs = 5\nbatch_size = 100\ntrain_loader = torch.utils.data.DataLoader(dataset=train_dataset, batch_size=batch_size, shuffle=True)\ntotal_step = len(train_loader)\nfor epoch in range(num_epochs):\n    for i, (images, labels) in enumerate(train_loader):\n        # \u524d\u5411\u4f20\u64ad\n        outputs = model(images)\n        loss = criterion(outputs, labels)\n\n        # \u53cd\u5411\u4f20\u64ad\u548c\u4f18\u5316\n        optimizer.zero_grad()\n        loss.backward()\n        optimizer.step()\n\n        # \u8f93\u51fa\u6bcf\u4e00\u8f6e\u8fed\u4ee3\u7684\u635f\u5931\u503c\n        if (i+1) % 100 == 0:\n            print('Epoch [{}\/{}], Step [{}\/{}], Loss: {:.4f}'.format(epoch+1, num_epochs, i+1, total_step, loss.item()))\n\n# \u6d4b\u8bd5\u6a21\u578b\nwith torch.no_grad():\n    correct = 0\n    total = 0\n    for images, labels in test_loader:\n        outputs = model(images)\n        _, predicted = torch.max(outputs.data, 1)\n        total += labels.size(0)\n        correct += (predicted == labels).sum().item()\n    print('Accuracy of the model on the 10000 test images: {} %'.format(100 * correct \/ total))\n<\/code><\/pre>\n\n\n\n
                                        \u6b64\u4ee3\u7801\u5b9a\u4e49\u4e86\u4e00\u4e2a\u5177\u6709\u4e24\u4e2a\u5377\u79ef\u5c42\u548c\u4e00\u4e2a\u5168\u8fde\u63a5\u5c42\u7684\u5377\u79ef\u795e\u7ecf\u7f51\u7edc\uff0c\u5e76\u7528\u5b83\u5bf9MNIST\u6570\u5b57\u8fdb\u884c\u5206\u7c7b\u3002\u8be5\u6a21\u578b\u5728\u8bad\u7ec3\u6570\u636e\u4e0a\u8fdb\u884c5\u8f6e\u8fed\u4ee3\uff0c\u5e76\u8f93\u51fa\u6bcf\u4e00\u8f6e\u8fed\u4ee3\u7684\u635f\u5931\u503c\u3002\u7136\u540e\uff0c\u5b83\u5bf9\u6d4b\u8bd5\u6570\u636e\u8fdb\u884c\u5206\u7c7b\uff0c\u5e76\u8f93\u51fa\u6a21\u578b\u5728\u6d4b\u8bd5\u6570\u636e\u4e0a\u7684\u51c6\u786e\u6027\u3002<\/p>\n\n\n\n\n\n
                                        \"gun<\/figure>\n\n\n\n
                                        \u8bf7\u8be6\u7ec6\u89e3\u91ca\u5faa\u73af\u7f51\u7edc<\/p>\n\n\n\n
                                        \u5faa\u73af\u795e\u7ecf\u7f51\u7edc\uff08Recurrent Neural Network, RNN\uff09\u662f\u4e00\u7c7b\u5177\u6709\u81ea\u53cd\u9988\u8fde\u63a5\u7684\u795e\u7ecf\u7f51\u7edc\uff0c\u5b83\u53ef\u4ee5\u5904\u7406\u53d8\u957f\u5e8f\u5217\u6570\u636e\uff0c\u5982\u8bed\u97f3\u3001\u6587\u672c\u3001\u89c6\u9891\u7b49\u3002\u5faa\u73af\u795e\u7ecf\u7f51\u7edc\u7684\u57fa\u672c\u601d\u60f3\u662f\u5229\u7528\u524d\u4e00\u65f6\u523b\u7684\u72b6\u6001\u6765\u5f71\u54cd\u5f53\u524d\u65f6\u523b\u7684\u8f93\u51fa\uff0c\u4ece\u800c\u5b9e\u73b0\u5bf9\u5386\u53f2\u4fe1\u606f\u7684\u8bb0\u5fc6\u3002\u5faa\u73af\u795e\u7ecf\u7f51\u7edc\u7684\u4e00\u4e2a\u91cd\u8981\u5e94\u7528\u662f\u81ea\u7136\u8bed\u8a00\u5904\u7406\u9886\u57df\u4e2d\u7684\u8bed\u8a00\u6a21\u578b\uff0c\u5b83\u53ef\u4ee5\u5b66\u4e60\u8bed\u8a00\u7684\u6982\u7387\u5206\u5e03\uff0c\u7528\u4e8e\u751f\u6210\u8bed\u8a00\u3001\u8bed\u97f3\u8bc6\u522b\u3001\u673a\u5668\u7ffb\u8bd1\u7b49\u4efb\u52a1\u3002<\/p>\n\n\n\n
                                        \u5faa\u73af\u795e\u7ecf\u7f51\u7edc\u7684\u6838\u5fc3\u662f\u5faa\u73af\u5355\u5143\uff08Recurrent Unit\uff09\uff0c\u5b83\u662f\u4e00\u4e2a\u5177\u6709\u81ea\u53cd\u9988\u8fde\u63a5\u7684\u795e\u7ecf\u7f51\u7edc\u5355\u5143\u3002\u5e38\u89c1\u7684\u5faa\u73af\u5355\u5143\u6709\u7b80\u5355\u5faa\u73af\u5355\u5143\uff08Simple Recurrent Unit\uff0cSRU\uff09\u3001\u957f\u77ed\u671f\u8bb0\u5fc6\u5355\u5143\uff08Long Short-Term Memory Unit\uff0cLSTM\uff09\u548c\u95e8\u63a7\u5faa\u73af\u5355\u5143\uff08Gated Recurrent Unit\uff0cGRU\uff09\u7b49\u3002<\/p>\n\n\n\n
                                        \u4ee5LSTM\u4e3a\u4f8b\uff0c\u4e00\u4e2aLSTM\u5355\u5143\u5305\u542b\u4e86\u4e00\u4e2a\u9057\u5fd8\u95e8\u3001\u4e00\u4e2a\u8f93\u5165\u95e8\u548c\u4e00\u4e2a\u8f93\u51fa\u95e8\uff0c\u8fd8\u6709\u4e00\u4e2a\u5355\u5143\u72b6\u6001\u548c\u4e00\u4e2a\u8f93\u51fa\u72b6\u6001\u3002\u5176\u4e2d\uff0c\u9057\u5fd8\u95e8\u7528\u6765\u63a7\u5236\u524d\u4e00\u65f6\u523b\u7684\u72b6\u6001\u5bf9\u5f53\u524d\u65f6\u523b\u7684\u5f71\u54cd\u7a0b\u5ea6\uff0c\u8f93\u5165\u95e8\u7528\u6765\u63a7\u5236\u5f53\u524d\u65f6\u523b\u7684\u8f93\u5165\u5bf9\u5f53\u524d\u65f6\u523b\u7684\u72b6\u6001\u7684\u5f71\u54cd\u7a0b\u5ea6\uff0c\u8f93\u51fa\u95e8\u7528\u6765\u63a7\u5236\u5f53\u524d\u65f6\u523b\u7684\u72b6\u6001\u5bf9\u5f53\u524d\u65f6\u523b\u7684\u8f93\u51fa\u7684\u5f71\u54cd\u7a0b\u5ea6\u3002LSTM\u5355\u5143\u7684\u72b6\u6001\u4f20\u9012\u662f\u901a\u8fc7\u4e00\u7cfb\u5217\u7684\u95e8\u63a7\u64cd\u4f5c\u6765\u5b8c\u6210\u7684\uff0c\u56e0\u6b64\u80fd\u591f\u6709\u6548\u5730\u89e3\u51b3\u957f\u5e8f\u5217\u8bb0\u5fc6\u7684\u95ee\u9898\u3002<\/p>\n\n\n\n
                                        \u5faa\u73af\u795e\u7ecf\u7f51\u7edc\u7684\u8bad\u7ec3\u4e00\u822c\u4f7f\u7528\u53cd\u5411\u4f20\u64ad\u7b97\u6cd5\u548c\u68af\u5ea6\u4e0b\u964d\u6cd5\u3002\u7531\u4e8e\u5faa\u73af\u795e\u7ecf\u7f51\u7edc\u5177\u6709\u4e00\u5b9a\u7684\u6df1\u5ea6\u548c\u975e\u7ebf\u6027\uff0c\u8bad\u7ec3\u65f6\u5bb9\u6613\u51fa\u73b0\u68af\u5ea6\u6d88\u5931\u6216\u68af\u5ea6\u7206\u70b8\u7684\u95ee\u9898\u3002\u4e3a\u4e86\u89e3\u51b3\u8fd9\u4e2a\u95ee\u9898\uff0c\u5e38\u7528\u7684\u65b9\u6cd5\u662f\u88c1\u526a\u68af\u5ea6\u3001\u4f7f\u7528\u95e8\u63a7\u5355\u5143\u3001\u4f7f\u7528\u6b8b\u5dee\u8fde\u63a5\u3001\u4f7f\u7528\u8bb0\u5fc6\u5355\u5143\u7b49\u3002<\/p>\n\n\n\n\n\n
                                        \"gun<\/figure>\n\n\n\n
                                        \u8bf7\u7ed9\u51fa\u5faa\u73af\u7f51\u7edc\u7684\u4ee3\u7801\u793a\u4f8b<\/p>\n\n\n\n
                                        \u4ee5\u4e0b\u662f\u4e00\u4e2a\u7b80\u5355\u7684\u5faa\u73af\u795e\u7ecf\u7f51\u7edc\u7684\u4ee3\u7801\u793a\u4f8b\uff0c\u7528\u4e8e\u751f\u6210\u6587\u672c\uff1a<\/p>\n\n\n\n
                                        pythonCopy codeimport numpy as np\n\nclass RNN:\n    def __init__(self, input_size, hidden_size, output_size):\n        self.input_size = input_size\n        self.hidden_size = hidden_size\n        self.output_size = output_size\n\n        self.W_xh = np.random.randn(hidden_size, input_size) * 0.01\n        self.W_hh = np.random.randn(hidden_size, hidden_size) * 0.01\n        self.W_hy = np.random.randn(output_size, hidden_size) * 0.01\n\n        self.b_h = np.zeros((hidden_size, 1))\n        self.b_y = np.zeros((output_size, 1))\n\n    def forward(self, x):\n        T = x.shape[1]\n        self.h = np.zeros((self.hidden_size, T + 1))\n        self.y = np.zeros((self.output_size, T))\n\n        for t in range(T):\n            self.h[:, t + 1] = np.tanh(np.dot(self.W_xh, x[:, t]) + np.dot(self.W_hh, self.h[:, t]) + self.b_h)\n            self.y[:, t] = softmax(np.dot(self.W_hy, self.h[:, t + 1]) + self.b_y)\n\n        return self.y\n\n    def backward(self, x, y, targets):\n        T = x.shape[1]\n        dh_next = np.zeros((self.hidden_size, 1))\n        dW_xh, dW_hh, dW_hy = np.zeros_like(self.W_xh), np.zeros_like(self.W_hh), np.zeros_like(self.W_hy)\n        db_h, db_y = np.zeros_like(self.b_h), np.zeros_like(self.b_y)\n\n        for t in reversed(range(T)):\n            dy = np.copy(self.y[:, t])\n            dy[targets[t]] -= 1\n            dW_hy += np.outer(dy, self.h[:, t + 1])\n            db_y += dy.reshape(-1, 1)\n\n            dh = np.dot(self.W_hy.T, dy) + np.dot(self.W_hh.T, dh_next)\n            dh_raw = (1 - self.h[:, t + 1] ** 2) * dh\n            db_h += dh_raw.reshape(-1, 1)\n            dW_xh += np.outer(dh_raw, x[:, t])\n            dW_hh += np.outer(dh_raw, self.h[:, t])\n            dh_next = dh_raw\n\n        return dW_xh, dW_hh, dW_hy, db_h, db_y\n\n    def train(self, x, targets, learning_rate=0.01):\n        y = self.forward(x)\n        loss = cross_entropy(y, targets)\n\n        dW_xh, dW_hh, dW_hy, db_h, db_y = self.backward(x, y, targets)\n\n        self.W_xh -= learning_rate * dW_xh\n        self.W_hh -= learning_rate * dW_hh\n        self.W_hy -= learning_rate * dW_hy\n        self.b_h -= learning_rate * db_h\n        self.b_y -= learning_rate * db_y\n\n        return loss\n\ndef softmax(x):\n    return np.exp(x) \/ np.sum(np.exp(x))\n\ndef cross_entropy(y, targets):\n    T = targets.shape[0]\n    return -np.sum(np.log(y[targets, np.arange(T)])) \/ T\n<\/code><\/pre>\n\n\n\n
                                        \u6b64\u4ee3\u7801\u5b9e\u73b0\u4e86\u4e00\u4e2a\u7b80\u5355\u7684\u5355\u5c42\u5faa\u73af<\/p>\n\n\n\n\n\n
                                        \"gun<\/figure>\n\n\n\n
                                        \u8bf7\u8be6\u7ec6\u89e3\u91caLSTM<\/p>\n\n\n\n
                                        LSTM\uff08Long Short-Term Memory\uff0c\u957f\u77ed\u65f6\u8bb0\u5fc6\uff09\u662f\u4e00\u79cd\u7279\u6b8a\u7684\u5faa\u73af\u795e\u7ecf\u7f51\u7edc\uff08RNN\uff09\uff0c\u5b83\u53ef\u4ee5\u6709\u6548\u5730\u89e3\u51b3\u4f20\u7edf RNN \u5b58\u5728\u7684\u957f\u671f\u4f9d\u8d56\u95ee\u9898\u3002\u4f20\u7edf RNN \u4ec5\u4ec5\u901a\u8fc7\u795e\u7ecf\u5143\u4e4b\u95f4\u7684\u6743\u91cd\u6765\u4f20\u9012\u4fe1\u606f\uff0c\u5f53\u65f6\u95f4\u5e8f\u5217\u8f83\u957f\u65f6\uff0c\u4fe1\u606f\u4f1a\u901a\u8fc7\u591a\u6b21\u4f20\u9012\u4ea7\u751f\u7d2f\u79ef\u8bef\u5dee\uff0c\u5bfc\u81f4\u957f\u671f\u4f9d\u8d56\u95ee\u9898\u7684\u4ea7\u751f\u3002\u800c LSTM \u901a\u8fc7\u589e\u52a0\u95e8\u63a7\u673a\u5236\u6765\u63a7\u5236\u4fe1\u606f\u7684\u6d41\u52a8\uff0c\u89e3\u51b3\u4e86\u957f\u671f\u4f9d\u8d56\u95ee\u9898\u3002<\/p>\n\n\n\n
                                        LSTM \u795e\u7ecf\u7f51\u7edc\u4e3b\u8981\u7531\u4e09\u4e2a\u95e8\u7ec4\u6210\uff1a\u8f93\u5165\u95e8\uff08input gate\uff09\u3001\u9057\u5fd8\u95e8\uff08forget gate\uff09\u548c\u8f93\u51fa\u95e8\uff08output gate\uff09\u3002\u8f93\u5165\u95e8\u63a7\u5236\u65b0\u8f93\u5165\u7684\u4fe1\u606f\uff0c\u9057\u5fd8\u95e8\u63a7\u5236\u8fc7\u53bb\u7684\u8bb0\u5fc6\u5982\u4f55\u6d41\u5165\u5f53\u524d\u7684\u72b6\u6001\uff0c\u8f93\u51fa\u95e8\u63a7\u5236\u5f53\u524d\u72b6\u6001\u7684\u54ea\u4e9b\u4fe1\u606f\u4f1a\u8f93\u51fa\u5230\u4e0b\u4e00\u4e2a\u65f6\u95f4\u6b65\u3002\u9664\u6b64\u4e4b\u5916\uff0cLSTM \u8fd8\u6709\u4e00\u4e2a\u7ec6\u80de\u72b6\u6001\uff08cell state\uff09\u7528\u4e8e\u4fdd\u7559\u8fc7\u53bb\u7684\u4fe1\u606f\uff0c\u907f\u514d\u4fe1\u606f\u8fc7\u591a\u6216\u8fc7\u5c11\u3002<\/p>\n\n\n\n
                                        LSTM \u7684\u5177\u4f53\u8ba1\u7b97\u8fc7\u7a0b\u5982\u4e0b\uff1a<\/p>\n\n\n\n
                                          \n
                                        1. \u5bf9\u4e8e\u65f6\u95f4\u6b65 $t$\uff0c\u8f93\u5165 $x_t$ \u548c\u4e0a\u4e00\u4e2a\u65f6\u95f4\u6b65\u7684\u9690\u85cf\u72b6\u6001 $h_{t-1}$\uff0c\u8ba1\u7b97\u5f97\u5230\u5f53\u524d\u65f6\u523b\u7684\u5019\u9009\u8bb0\u5fc6\u7ec6\u80de\u72b6\u6001 $\\tilde{C_t}$\uff0c\u5176\u8ba1\u7b97\u516c\u5f0f\u4e3a\uff1a<\/li>\n<\/ol>\n\n\n\n
                                          \ufffd\ufffd~=tanh\u2061(\ufffd\ufffd\ufffd\ufffd\ufffd+\ufffd\u210e\ufffd\u210e\ufffd\u22121+\ufffd\ufffd)C<\/em>t<\/em>\u200b~\u200b=tanh(W<\/em>x<\/em>c<\/em>\u200bx<\/em>t<\/em>\u200b+W<\/em>h<\/em>c<\/em>\u200bh<\/em>t<\/em>\u22121\u200b+b<\/em>c<\/em>\u200b)<\/p>\n\n\n\n
                                          \u5176\u4e2d\uff0c$W_{xc}$ \u548c $W_{hc}$ \u5206\u522b\u662f\u8f93\u5165\u6743\u91cd\u548c\u9690\u85cf\u72b6\u6001\u6743\u91cd\uff0c$b_c$ \u662f\u504f\u7f6e\u3002<\/p>\n\n\n\n
                                            \n
                                          1. \u8ba1\u7b97\u8f93\u5165\u95e8 $i_t$ \u548c\u9057\u5fd8\u95e8 $f_t$\uff0c\u5176\u8ba1\u7b97\u516c\u5f0f\u5206\u522b\u4e3a\uff1a<\/li>\n<\/ol>\n\n\n\n
                                            \ufffd\ufffd=\ufffd(\ufffd\ufffd\ufffd\ufffd\ufffd+\ufffd\u210e\ufffd\u210e\ufffd\u22121+\ufffd\ufffd)i<\/em>t<\/em>\u200b=\u03c3<\/em>(W<\/em>x<\/em>i<\/em>\u200bx<\/em>t<\/em>\u200b+W<\/em>hi<\/em>\u200bh<\/em>t<\/em>\u22121\u200b+b<\/em>i<\/em>\u200b)<\/p>\n\n\n\n
                                            \ufffd\ufffd=\ufffd(\ufffd\ufffd\ufffd\ufffd\ufffd+\ufffd\u210e\ufffd\u210e\ufffd\u22121+\ufffd\ufffd)f<\/em>t<\/em>\u200b=\u03c3<\/em>(W<\/em>x<\/em>f<\/em>\u200bx<\/em>t<\/em>\u200b+W<\/em>h<\/em>f<\/em>\u200bh<\/em>t<\/em>\u22121\u200b+b<\/em>f<\/em>\u200b)<\/p>\n\n\n\n
                                            \u5176\u4e2d\uff0c$\\sigma$ \u662f sigmoid \u51fd\u6570\u3002<\/p>\n\n\n\n
                                              \n
                                            1. \u66f4\u65b0\u7ec6\u80de\u72b6\u6001 $C_t$\uff0c\u5176\u8ba1\u7b97\u516c\u5f0f\u4e3a\uff1a<\/li>\n<\/ol>\n\n\n\n
                                              \ufffd\ufffd=\ufffd\ufffd\u2299\ufffd\ufffd\u22121+\ufffd\ufffd\u2299\ufffd\ufffd~C<\/em>t<\/em>\u200b=f<\/em>t<\/em>\u200b\u2299C<\/em>t<\/em>\u22121\u200b+i<\/em>t<\/em>\u200b\u2299C<\/em>t<\/em>\u200b~\u200b<\/p>\n\n\n\n
                                              \u5176\u4e2d\uff0c$\\odot$ \u8868\u793a\u9010\u5143\u7d20\u76f8\u4e58\u3002<\/p>\n\n\n\n
                                                \n
                                              1. \u8ba1\u7b97\u8f93\u51fa\u95e8 $o_t$\uff0c\u5176\u8ba1\u7b97\u516c\u5f0f\u4e3a\uff1a<\/li>\n<\/ol>\n\n\n\n
                                                \ufffd\ufffd=\ufffd(\ufffd\ufffd\ufffd\ufffd\ufffd+\ufffd\u210e\ufffd\u210e\ufffd\u22121+\ufffd\ufffd)o<\/em>t<\/em>\u200b=\u03c3<\/em>(W<\/em>x<\/em>o<\/em>\u200bx<\/em>t<\/em>\u200b+W<\/em>h<\/em>o<\/em>\u200bh<\/em>t<\/em>\u22121\u200b+b<\/em>o<\/em>\u200b)<\/p>\n\n\n\n
                                                  \n
                                                1. \u8ba1\u7b97\u5f53\u524d\u65f6\u523b\u7684\u9690\u85cf\u72b6\u6001 $h_t$\uff0c\u5176\u8ba1\u7b97\u516c\u5f0f\u4e3a\uff1a<\/li>\n<\/ol>\n\n\n\n
                                                  \u210e\ufffd=\ufffd\ufffd\u2299tanh\u2061(\ufffd\ufffd)h<\/em>t<\/em>\u200b=o<\/em>t<\/em>\u200b\u2299tanh(C<\/em>t<\/em>\u200b)<\/p>\n\n\n\n
                                                  LSTM \u901a\u8fc7\u4ee5\u4e0a\u7684\u8ba1\u7b97\u8fc7\u7a0b\u6765\u66f4\u65b0\u9690\u85cf\u72b6\u6001 $h_t$ \u548c\u7ec6\u80de\u72b6\u6001 $C_t$\uff0c\u5e76\u63a7\u5236\u4fe1\u606f\u7684\u6d41\u52a8\uff0c\u4ece\u800c\u6709\u6548\u5730\u89e3\u51b3\u957f\u671f\u4f9d\u8d56\u95ee\u9898\u3002<\/p>\n\n\n\n
                                                  \u6ce8\uff1a$\\tanh$ \u548c $\\sigma$ \u51fd\u6570\u5206\u522b\u4e3a\u53cc\u66f2\u6b63\u5207\u51fd\u6570<\/p>\n\n\n\n\n\n
                                                  \"gun<\/figure>\n\n\n\n
                                                  \u8bf7\u7ed9\u51faLSTM\u7684\u4ee3\u7801\u793a\u4f8b<\/p>\n\n\n\n
                                                  \u4ee5\u4e0b\u662f\u4e00\u4e2a\u7b80\u5355\u7684LSTM\u7684\u4ee3\u7801\u793a\u4f8b\uff0c\u7528\u4e8e\u5c06\u4e00\u4e2a\u5e8f\u5217\u6620\u5c04\u4e3a\u53e6\u4e00\u4e2a\u5e8f\u5217\uff1a<\/p>\n\n\n\n
                                                  pythonCopy codeimport torch\nimport torch.nn as nn\n\nclass LSTM(nn.Module):\n    def __init__(self, input_dim, hidden_dim, output_dim):\n        super(LSTM, self).__init__()\n        \n        self.hidden_dim = hidden_dim\n        \n        # \u5b9a\u4e49 LSTM \u5c42\n        self.lstm = nn.LSTM(input_dim, hidden_dim)\n        \n        # \u5b9a\u4e49\u8f93\u51fa\u5c42\n        self.fc = nn.Linear(hidden_dim, output_dim)\n        \n    def forward(self, x):\n        # \u521d\u59cb\u5316\u9690\u85cf\u5c42\u548c\u8bb0\u5fc6\u5355\u5143\n        h0 = torch.zeros(1, x.size(1), self.hidden_dim)\n        c0 = torch.zeros(1, x.size(1), self.hidden_dim)\n        \n        # \u524d\u5411\u4f20\u64ad\n        out, _ = self.lstm(x, (h0, c0))\n        out = self.fc(out[-1, :, :])\n        \n        return out\n<\/code><\/pre>\n\n\n\n
                                                  \u5728\u8fd9\u4e2a\u793a\u4f8b\u4e2d\uff0c\u6211\u4eec\u5b9a\u4e49\u4e86\u4e00\u4e2aLSTM\u7c7b\uff0c\u5b83\u7ee7\u627f\u4e86nn.Module\u7c7b\u3002\u5728__init__\u51fd\u6570\u4e2d\uff0c\u6211\u4eec\u9996\u5148\u5b9a\u4e49\u4e86\u9690\u85cf\u5c42\u7684\u5927\u5c0f\uff0c\u7136\u540e\u5b9a\u4e49\u4e86\u4e00\u4e2ann.LSTM\u5c42\uff0c\u5b83\u5c06\u8f93\u5165\u7684\u7ef4\u5ea6\u53d8\u6362\u4e3a\u9690\u85cf\u5c42\u7684\u5927\u5c0f\u3002\u7136\u540e\u6211\u4eec\u5b9a\u4e49\u4e86\u8f93\u51fa\u5c42\uff0c\u5b83\u5c06\u9690\u85cf\u5c42\u7684\u8f93\u51fa\u8f6c\u6362\u4e3a\u8f93\u51fa\u7684\u5927\u5c0f\u3002\u5728\u524d\u5411\u4f20\u64ad\u51fd\u6570\u4e2d\uff0c\u6211\u4eec\u9996\u5148\u521d\u59cb\u5316\u4e86\u9690\u85cf\u5c42\u548c\u8bb0\u5fc6\u5355\u5143\uff0c\u7136\u540e\u5c06\u8f93\u5165\u4f20\u9012\u7ed9LSTM\u5c42\uff0c\u6700\u540e\u5c06\u6700\u540e\u4e00\u4e2a\u65f6\u95f4\u6b65\u7684\u8f93\u51fa\u4f20\u9012\u7ed9\u8f93\u51fa\u5c42\u3002<\/p>\n\n\n\n\n\n
                                                  \"gun<\/figure>\n\n\n\n
                                                  \u8bf7\u8be6\u7ec6\u89e3\u91caBERT<\/p>\n\n\n\n
                                                  BERT\u662fBidirectional Encoder Representations from Transformers\u7684\u7f29\u5199\uff0c\u662f\u4e00\u79cd\u57fa\u4e8eTransformer\u6a21\u578b\u7684\u9884\u8bad\u7ec3\u8bed\u8a00\u8868\u793a\u5b66\u4e60\u65b9\u6cd5\uff0c\u7531Google\u57282018\u5e74\u63d0\u51fa\u3002BERT\u65e8\u5728\u901a\u8fc7\u5927\u89c4\u6a21\u65e0\u76d1\u7763\u8bad\u7ec3\u6765\u5b66\u4e60\u901a\u7528\u7684\u81ea\u7136\u8bed\u8a00\u8868\u5f81\uff0c\u53ef\u4ee5\u4f5c\u4e3a\u5404\u79cd\u4e0b\u6e38\u81ea\u7136\u8bed\u8a00\u5904\u7406\u4efb\u52a1\u7684\u8f93\u5165\u3002BERT\u6a21\u578b\u91c7\u7528\u53cc\u5411Transformer\u7f16\u7801\u5668\uff0c\u5e76\u4f7f\u7528\u201c\u63a9\u7801\u8bed\u8a00\u6a21\u578b\u201d\uff08Masked Language Model\uff0cMLM\uff09\u548c\u201c\u4e0b\u4e00\u53e5\u9884\u6d4b\u201d\uff08Next Sentence Prediction\uff0cNSP\uff09\u4e24\u79cd\u9884\u6d4b\u4efb\u52a1\u6765\u8bad\u7ec3\u6a21\u578b\u3002<\/p>\n\n\n\n
                                                  \u5176\u4e2d\uff0c\u63a9\u7801\u8bed\u8a00\u6a21\u578b\u662f\u6307\u5c06\u8f93\u5165\u6587\u672c\u4e2d\u7684\u4e00\u4e9b\u5355\u8bcd\u968f\u673a\u66ff\u6362\u6210\u201c\u63a9\u7801\u201d\uff0c\u5e76\u8ba9\u6a21\u578b\u9884\u6d4b\u8fd9\u4e9b\u88ab\u63a9\u7801\u7684\u5355\u8bcd\u3002\u901a\u8fc7\u8fd9\u79cd\u65b9\u5f0f\uff0c\u6a21\u578b\u80fd\u591f\u5b66\u4e60\u5230\u5355\u8bcd\u4e4b\u95f4\u7684\u4e0a\u4e0b\u6587\u5173\u7cfb\uff0c\u4ece\u800c\u66f4\u597d\u5730\u7406\u89e3\u81ea\u7136\u8bed\u8a00\u4e2d\u7684\u8bed\u4e49\u548c\u8bed\u6cd5\u3002\u4e0b\u4e00\u53e5\u9884\u6d4b\u4efb\u52a1\u5219\u662f\u8ba9\u6a21\u578b\u5224\u65ad\u4e24\u4e2a\u53e5\u5b50\u662f\u5426\u662f\u8fde\u7eed\u7684\uff0c\u4ece\u800c\u5e2e\u52a9\u6a21\u578b\u5b66\u4e60\u5230\u53e5\u5b50\u4e4b\u95f4\u7684\u5173\u7cfb\u3002<\/p>\n\n\n\n
                                                  BERT\u6a21\u578b\u53ef\u4ee5\u901a\u8fc7\u5fae\u8c03\u6765\u9002\u5e94\u4e0d\u540c\u7684\u81ea\u7136\u8bed\u8a00\u5904\u7406\u4efb\u52a1\uff0c\u6bd4\u5982\u6587\u672c\u5206\u7c7b\u3001\u547d\u540d\u5b9e\u4f53\u8bc6\u522b\u3001\u53e5\u5b50\u76f8\u4f3c\u5ea6\u8ba1\u7b97\u7b49\u3002\u7531\u4e8eBERT\u53ef\u4ee5\u901a\u8fc7\u5927\u89c4\u6a21\u65e0\u76d1\u7763\u5b66\u4e60\u5f97\u5230\u901a\u7528\u7684\u81ea\u7136\u8bed\u8a00\u8868\u5f81\uff0c\u56e0\u6b64\u5728\u8bb8\u591a\u81ea\u7136\u8bed\u8a00\u5904\u7406\u4efb\u52a1\u4e2d\u90fd\u53d6\u5f97\u4e86\u4f18\u79c0\u7684\u8868\u73b0\u3002<\/p>\n\n\n\n
                                                  \u4e0b\u9762\u662f\u4e00\u4e2a\u4f7f\u7528Hugging Face\u5e93\u5b9e\u73b0BERT\u7684\u4ee3\u7801\u793a\u4f8b\uff1a<\/p>\n\n\n\n
                                                  pythonCopy codefrom transformers import BertTokenizer, BertModel\nimport torch\n\n# \u52a0\u8f7dBERT\u6a21\u578b\u548c\u8bcd\u8868\ntokenizer = BertTokenizer.from_pretrained('bert-base-uncased')\nmodel = BertModel.from_pretrained('bert-base-uncased')\n\n# \u8f93\u5165\u6587\u672c\ntext = \"Hello, how are you?\"\n\n# \u5c06\u6587\u672c\u8f6c\u6362\u4e3aBERT\u6240\u9700\u7684\u683c\u5f0f\ninputs = tokenizer(text, return_tensors='pt')\n\n# \u524d\u5411\u8ba1\u7b97\noutputs = model(**inputs)\n\n# \u83b7\u53d6BERT\u7684\u8f93\u51fa\nlast_hidden_state = outputs.last_hidden_state\n<\/code><\/pre>\n\n\n\n
                                                  \u5728\u8fd9\u4e2a\u793a\u4f8b\u4e2d\uff0c\u6211\u4eec\u9996\u5148\u4f7f\u7528BertTokenizer<\/code>\u52a0\u8f7dBERT\u7684\u8bcd\u8868\uff0c\u7136\u540e\u4f7f\u7528BertModel<\/code>\u52a0\u8f7dBERT\u6a21\u578b\u3002\u63a5\u7740\uff0c\u6211\u4eec\u5c06\u9700\u8981\u8fdb\u884c\u6587\u672c\u5904\u7406\u7684\u6587\u672c\u8f6c\u6362\u4e3aBERT\u6240\u9700\u7684\u683c\u5f0f\uff0c\u5e76\u5c06\u5176\u8f93\u5165\u5230\u6a21\u578b\u4e2d\u8fdb\u884c\u524d\u5411\u8ba1\u7b97\u3002\u6700\u540e\uff0c\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7outputs.last_hidden_state<\/code>\u83b7\u53d6BERT\u7684\u8f93\u51fa\uff0c\u7528\u4e8e\u540e\u7eed\u7684\u4efb\u52a1\u3002<\/p>\n\n\n\n\n\n
                                                  \"gun<\/figure>\n\n\n\n
                                                  \u8bf7\u8be6\u7ec6\u89e3\u91ca\u6fc0\u6d3b\u51fd\u6570<\/p>\n\n\n\n
                                                  \u6fc0\u6d3b\u51fd\u6570\u662f\u4e00\u79cd\u975e\u7ebf\u6027\u51fd\u6570\uff0c\u901a\u5e38\u5e94\u7528\u4e8e\u795e\u7ecf\u7f51\u7edc\u7684\u6bcf\u4e2a\u795e\u7ecf\u5143\u4e0a\uff0c\u7528\u4e8e\u51b3\u5b9a\u795e\u7ecf\u5143\u7684\u8f93\u51fa\u662f\u5426\u88ab\u6fc0\u6d3b\u3002\u6fc0\u6d3b\u51fd\u6570\u53ef\u4ee5\u5c06\u795e\u7ecf\u7f51\u7edc\u7684\u8f93\u51fa\u8f6c\u6362\u4e3a\u975e\u7ebf\u6027\u51fd\u6570\uff0c\u8fd9\u6709\u52a9\u4e8e\u7f51\u7edc\u5b66\u4e60\u975e\u7ebf\u6027\u6a21\u5f0f\uff0c\u5e76\u589e\u52a0\u7f51\u7edc\u7684\u8868\u8fbe\u80fd\u529b\u3002<\/p>\n\n\n\n
                                                  \u5e38\u89c1\u7684\u6fc0\u6d3b\u51fd\u6570\u5305\u62ec\uff1a<\/p>\n\n\n\n
                                                    \n
                                                  1. sigmoid\u51fd\u6570\uff1a$\\sigma(x) = \\frac{1}{1+e^{-x}}$\uff0c\u5c06\u8f93\u5165\u503c\u8f6c\u6362\u52300\u52301\u4e4b\u95f4\uff0c\u5e38\u7528\u4e8e\u4e8c\u5206\u7c7b\u95ee\u9898\u3002<\/li>\n\n\n\n
                                                  2. ReLU\u51fd\u6570\uff1a$f(x) = \\max(0,x)$\uff0c\u5f53\u8f93\u5165\u503c\u5927\u4e8e0\u65f6\u6fc0\u6d3b\u795e\u7ecf\u5143\uff0c\u5426\u5219\u8f93\u51fa0\uff0c\u5177\u6709\u975e\u7ebf\u6027\u548c\u7a00\u758f\u6027\uff0c\u5e38\u7528\u4e8e\u6df1\u5ea6\u795e\u7ecf\u7f51\u7edc\u3002<\/li>\n\n\n\n
                                                  3. Leaky ReLU\u51fd\u6570\uff1a$f(x) = \\max(ax,x)$\uff0c\u5f53\u8f93\u5165\u503c\u5927\u4e8e0\u65f6\u6fc0\u6d3b\u795e\u7ecf\u5143\uff0c\u5426\u5219\u8f93\u51fa\u5c0f\u4e8e0\u7684\u659c\u7387\uff0c\u5e38\u7528\u4e8e\u89e3\u51b3ReLU\u51fd\u6570\u8f93\u51fa\u503c\u4e3a0\u7684\u95ee\u9898\u3002<\/li>\n\n\n\n
                                                  4. tanh\u51fd\u6570\uff1a$\\tanh(x) = \\frac{e^x-e^{-x}}{e^x+e^{-x}}$\uff0c\u5c06\u8f93\u5165\u503c\u8f6c\u6362\u5230-1\u52301\u4e4b\u95f4\uff0c\u5177\u6709\u975e\u7ebf\u6027\u7279\u6027\uff0c\u5e38\u7528\u4e8e\u5faa\u73af\u795e\u7ecf\u7f51\u7edc\u4e2d\u3002<\/li>\n\n\n\n
                                                  5. Softmax\u51fd\u6570\uff1a$softmax(x_i) = \\frac{e^{x_i}}{\\sum_j e^{x_j}}$\uff0c\u5c06\u8f93\u5165\u503c\u8f6c\u6362\u4e3a\u6982\u7387\u5206\u5e03\uff0c\u7528\u4e8e\u591a\u5206\u7c7b\u95ee\u9898\u3002<\/li>\n<\/ol>\n\n\n\n
                                                    \u6fc0\u6d3b\u51fd\u6570\u7684\u9009\u62e9\u53d6\u51b3\u4e8e\u6240\u4f7f\u7528\u7684\u795e\u7ecf\u7f51\u7edc\u7684\u4efb\u52a1\u548c\u7ed3\u6784\uff0c\u4e0d\u540c\u7684\u6fc0\u6d3b\u51fd\u6570\u5728\u4e0d\u540c\u7684\u573a\u666f\u4e0b\u5177\u6709\u4e0d\u540c\u7684\u4f18\u7f3a\u70b9\u3002<\/p>\n\n\n\n\n\n
                                                    \"gun<\/figure>\n\n\n\n
                                                    \u8bf7\u7ed9\u51fa\u6fc0\u6d3b\u51fd\u6570\u7684\u5e94\u7528\u793a\u4f8b\u4ee3\u7801<\/p>\n\n\n\n
                                                    \u4ee5\u4e0b\u662f\u4e00\u4e2a\u4f7f\u7528ReLU\uff08\u6574\u6d41\u7ebf\u6027\u5355\u5143\uff09\u6fc0\u6d3b\u51fd\u6570\u7684\u7b80\u5355\u793a\u4f8b\u4ee3\u7801\uff1a<\/p>\n\n\n\n
                                                    pythonCopy codeimport numpy as np\n\ndef relu(x):\n    return np.maximum(0, x)\n\nx = np.array([-1, 0, 1, 2, 3])\nprint(relu(x))  # \u8f93\u51fa [0 0 1 2 3]\n<\/code><\/pre>\n\n\n\n
                                                    \u5728\u4e0a\u9762\u7684\u4ee3\u7801\u4e2d\uff0c\u6211\u4eec\u5b9a\u4e49\u4e86\u4e00\u4e2aReLU\u51fd\u6570\uff0c\u5b83\u63a5\u53d7\u4e00\u4e2anumpy\u6570\u7ec4x\u4f5c\u4e3a\u8f93\u5165\uff0c\u5e76\u8fd4\u56dex\u4e2d\u6bcf\u4e2a\u5143\u7d20\u7684ReLU\u8f93\u51fa\u3002\u6211\u4eec\u4f7f\u7528np.maximum<\/code>\u51fd\u6570\u6765\u5b9e\u73b0ReLU\uff0c\u5b83\u8fd4\u56de\u4e24\u4e2a\u6570\u7ec4\u4e2d\u6bcf\u4e2a\u5143\u7d20\u7684\u8f83\u5927\u503c\uff0c\u8fd9\u91cc\u7684\u7b2c\u4e00\u4e2a\u6570\u7ec4\u662fx\uff0c\u7b2c\u4e8c\u4e2a\u6570\u7ec4\u662f\u51680\u6570\u7ec4\uff0c\u56e0\u4e3a\u6211\u4eec\u8981\u786e\u4fddReLU\u4e0d\u4f1a\u8f93\u51fa\u8d1f\u6570\u3002<\/p>\n\n\n\n
                                                    \u6211\u4eec\u5b9a\u4e49\u4e86\u4e00\u4e2a\u7b80\u5355\u7684\u8f93\u5165\u5411\u91cfx = [-1, 0, 1, 2, 3]<\/code>\uff0c\u5e76\u6253\u5370\u51fa\u5b83\u7684ReLU\u8f93\u51fa\uff0c\u5373[0, 0, 1, 2, 3]<\/code>\u3002\u8fd9\u8bf4\u660eReLU\u51fd\u6570\u5c06\u6240\u6709\u8d1f\u6570\u6620\u5c04\u5230\u96f6\uff0c\u4fdd\u7559\u6240\u6709\u6b63\u6570\uff0c\u5e76\u4e14\u5177\u6709\u975e\u5e38\u7b80\u5355\u7684\u5b9e\u73b0\u3002<\/p>\n\n\n\n\n\n
                                                    \"gun<\/figure>\n\n\n\n
                                                    \u8bf7\u8be6\u7ec6\u89e3\u91ca\u8d1d\u53f6\u65af\u8ba1\u7b97<\/p>\n\n\n\n
                                                    \u8d1d\u53f6\u65af\u8ba1\u7b97\uff08Bayesian inference\uff09\u662f\u4e00\u79cd\u57fa\u4e8e\u8d1d\u53f6\u65af\u5b9a\u7406\uff08Bayes’ theorem\uff09\u7684\u6982\u7387\u63a8\u65ad\u65b9\u6cd5\uff0c\u7528\u4e8e\u4ece\u5df2\u77e5\u7684\u5148\u9a8c\u6982\u7387\u548c\u65b0\u7684\u6570\u636e\uff08\u8bc1\u636e\uff09\u6765\u63a8\u65ad\u51fa\u672a\u77e5\u7684\u540e\u9a8c\u6982\u7387\u5206\u5e03\u3002\u5728\u673a\u5668\u5b66\u4e60\u548c\u7edf\u8ba1\u5b66\u4e2d\uff0c\u8d1d\u53f6\u65af\u8ba1\u7b97\u662f\u4e00\u79cd\u91cd\u8981\u7684\u5de5\u5177\uff0c\u53ef\u7528\u4e8e\u5904\u7406\u5206\u7c7b\u3001\u56de\u5f52\u3001\u805a\u7c7b\u3001\u964d\u7ef4\u7b49\u591a\u79cd\u4efb\u52a1\u3002\u76f8\u5bf9\u4e8e\u9891\u7387\u5b66\u6d3e\uff0c\u8d1d\u53f6\u65af\u5b66\u6d3e\u6ce8\u91cd\u5148\u9a8c\u77e5\u8bc6\u7684\u5f15\u5165\uff0c\u53ef\u4ee5\u66f4\u597d\u5730\u5904\u7406\u5c0f\u6837\u672c\u95ee\u9898\u548c\u4e0d\u786e\u5b9a\u6027\u95ee\u9898\u3002<\/p>\n\n\n\n
                                                    \u8d1d\u53f6\u65af\u5b9a\u7406\u8868\u793a\u4e3a\uff1a$P(\\theta|D) = \\frac{P(D|\\theta) P(\\theta)}{P(D)}$\uff0c\u5176\u4e2d $\\theta$ \u662f\u6a21\u578b\u53c2\u6570\uff0c$D$ \u662f\u89c2\u6d4b\u6570\u636e\uff0c$P(\\theta)$ \u662f\u5148\u9a8c\u6982\u7387\u5206\u5e03\uff0c$P(D|\\theta)$ \u662f\u4f3c\u7136\u51fd\u6570\uff0c$P(D)$ \u662f\u8fb9\u7f18\u6982\u7387\u5206\u5e03\uff0c\u901a\u5e38\u4f5c\u4e3a\u89c4\u8303\u5316\u56e0\u5b50\uff0c\u786e\u4fdd\u540e\u9a8c\u6982\u7387\u5206\u5e03\u7684\u603b\u548c\u4e3a 1\u3002\u540e\u9a8c\u6982\u7387\u5206\u5e03 $P(\\theta|D)$ \u8868\u793a\u5728\u89c2\u6d4b\u6570\u636e $D$ \u7684\u60c5\u51b5\u4e0b\uff0c\u6a21\u578b\u53c2\u6570 $\\theta$ \u7684\u6982\u7387\u5206\u5e03\u3002\u8d1d\u53f6\u65af\u8ba1\u7b97\u7684\u76ee\u6807\u662f\u901a\u8fc7\u89c2\u6d4b\u6570\u636e\u6765\u66f4\u65b0\u6a21\u578b\u53c2\u6570\u7684\u6982\u7387\u5206\u5e03\uff0c\u5373\u8ba1\u7b97\u540e\u9a8c\u6982\u7387\u5206\u5e03 $P(\\theta|D)$\u3002<\/p>\n\n\n\n
                                                    \u5728\u5b9e\u9645\u5e94\u7528\u4e2d\uff0c\u901a\u5e38\u9700\u8981\u5bf9\u540e\u9a8c\u6982\u7387\u5206\u5e03\u8fdb\u884c\u91c7\u6837\u6765\u8fdb\u884c\u63a8\u65ad\u548c\u9884\u6d4b\u3002\u8d1d\u53f6\u65af\u8ba1\u7b97\u53ef\u4ee5\u901a\u8fc7\u9a6c\u5c14\u53ef\u592b\u94fe\u8499\u7279\u5361\u7f57\uff08Markov chain Monte Carlo\uff0cMCMC\uff09\u65b9\u6cd5\u6765\u8fdb\u884c\u91c7\u6837\uff0c\u4f8b\u5982 Gibbs \u62bd\u6837\u3001Metropolis-Hastings \u62bd\u6837\u7b49\u3002\u6b64\u5916\uff0c\u8fd8\u53ef\u4ee5\u4f7f\u7528\u53d8\u5206\u63a8\u65ad\uff08variational inference\uff09\u7b49\u65b9\u6cd5\u8fdb\u884c\u8fd1\u4f3c\u63a8\u65ad\uff0c\u4ee5\u7b80\u5316\u8ba1\u7b97\u548c\u52a0\u901f\u6536\u655b\u3002<\/p>\n\n\n\n
                                                    \u4e0b\u9762\u7ed9\u51fa\u4e00\u4e2a\u7b80\u5355\u7684\u8d1d\u53f6\u65af\u8ba1\u7b97\u7684\u4f8b\u5b50\uff0c\u5047\u8bbe\u6211\u4eec\u89c2\u6d4b\u5230\u4e86\u4e00\u7ec4\u6570\u636e ${x_i}{i=1}^N$\uff0c\u5e76\u4e14\u5047\u8bbe\u6570\u636e\u6765\u81ea\u4e8e\u4e00\u4e2a\u6b63\u6001\u5206\u5e03 $x_i \\sim N(\\mu, \\sigma^2)$\uff0c\u5176\u4e2d $\\mu$ \u548c $\\sigma$ \u662f\u672a\u77e5\u7684\u53c2\u6570\u3002\u6211\u4eec\u53ef\u4ee5\u5f15\u5165\u5148\u9a8c\u5206\u5e03 $p(\\mu, \\sigma^2)$\uff0c\u4f8b\u5982\u6b63\u6001\u5206\u5e03\u6216\u8005 Gamma \u5206\u5e03\u3002\u7136\u540e\uff0c\u901a\u8fc7\u89c2\u6d4b\u6570\u636e\u8ba1\u7b97\u540e\u9a8c\u5206\u5e03 $p(\\mu, \\sigma^2 | {x_i}<\/em>{i=1}^N)$\u3002\u5728\u672c\u4f8b\u4e2d\uff0c\u540e\u9a8c\u5206\u5e03\u662f\u6b63\u6001\u5206\u5e03\u6216 Gamma \u5206\u5e03\u7684\u5f62\u5f0f\u3002<\/p>\n\n\n\n
                                                    Python \u4ee3\u7801\u793a\u4f8b\uff1a<\/p>\n\n\n\n
                                                    pythonCopy codeimport numpy as np\nfrom scipy.stats import norm, gamma\n\n# \u89c2\u6d4b\u6570\u636e\nx = np.array([1.2, 2.3, 0.7, 1.9, 1.5])\nN = len(x)\n\n# \u5148\u9a8c\u5206\u5e03\u7684\u53c2\u6570\n<\/code><\/pre>\n\n\n\n\n\n
                                                    \"gun<\/figure>\n\n\n\n
                                                    \u8bf7\u7ed9\u51fa\u8d1d\u53f6\u65af\u8ba1\u7b97\u7684\u793a\u4f8b\u4ee3\u7801<\/p>\n\n\n\n
                                                    \u4ee5\u4e0b\u662f\u4e00\u4e2a\u4f7f\u7528\u8d1d\u53f6\u65af\u8ba1\u7b97\u8fdb\u884c\u5206\u7c7b\u7684\u793a\u4f8b\u4ee3\u7801\uff1a<\/p>\n\n\n\n
                                                    pythonCopy codeimport numpy as np\nfrom scipy.stats import norm\n\n# \u521b\u5efa\u4e00\u4e2a\u5305\u542b\u4e24\u4e2a\u7279\u5f81\u7684\u8bad\u7ec3\u96c6\ntrain_data = np.array([[2.0, 1.0], [1.5, 2.0], [3.0, 4.0], [5.0, 6.0]])\ntrain_label = np.array([0, 0, 1, 1])\n\n# \u5047\u8bbe\u7279\u5f81\u7684\u5148\u9a8c\u5206\u5e03\u4e3a\u9ad8\u65af\u5206\u5e03\uff0c\u4f7f\u7528\u6700\u5927\u4f3c\u7136\u4f30\u8ba1\u6765\u4f30\u8ba1\u6bcf\u4e2a\u7c7b\u522b\u4e2d\u6bcf\u4e2a\u7279\u5f81\u7684\u5206\u5e03\u53c2\u6570\nclass0_mean = np.mean(train_data[train_label==0], axis=0)\nclass0_std = np.std(train_data[train_label==0], axis=0)\nclass1_mean = np.mean(train_data[train_label==1], axis=0)\nclass1_std = np.std(train_data[train_label==1], axis=0)\n\n# \u8ba1\u7b97\u6d4b\u8bd5\u6570\u636e\u5c5e\u4e8e\u6bcf\u4e2a\u7c7b\u522b\u7684\u540e\u9a8c\u6982\u7387\uff0c\u5e76\u9009\u62e9\u540e\u9a8c\u6982\u7387\u8f83\u5927\u7684\u7c7b\u522b\u4f5c\u4e3a\u9884\u6d4b\u7ed3\u679c\ndef predict(test_data):\n    class0_prob = norm.pdf(test_data, class0_mean, class0_std).prod()\n    class1_prob = norm.pdf(test_data, class1_mean, class1_std).prod()\n    return 0 if class0_prob > class1_prob else 1\n\n# \u6d4b\u8bd5\ntest_data = np.array([2.5, 1.8])\nprediction = predict(test_data)\nprint(\"\u6d4b\u8bd5\u6570\u636e {} \u5c5e\u4e8e\u7c7b\u522b {}\".format(test_data, prediction))\n<\/code><\/pre>\n\n\n\n
                                                    \u6b64\u4ee3\u7801\u5b9e\u73b0\u4e86\u4e00\u4e2a\u7b80\u5355\u7684\u8d1d\u53f6\u65af\u5206\u7c7b\u5668\uff0c\u8be5\u5206\u7c7b\u5668\u5047\u8bbe\u7279\u5f81\u7684\u5148\u9a8c\u5206\u5e03\u4e3a\u9ad8\u65af\u5206\u5e03\uff0c\u5e76\u4f7f\u7528\u6700\u5927\u4f3c\u7136\u4f30\u8ba1\u6765\u4f30\u8ba1\u6bcf\u4e2a\u7c7b\u522b\u4e2d\u6bcf\u4e2a\u7279\u5f81\u7684\u5206\u5e03\u53c2\u6570\u3002\u5728\u6d4b\u8bd5\u65f6\uff0c\u5b83\u8ba1\u7b97\u6d4b\u8bd5\u6570\u636e\u5c5e\u4e8e\u6bcf\u4e2a\u7c7b\u522b\u7684\u540e\u9a8c\u6982\u7387\uff0c\u5e76\u9009\u62e9\u540e\u9a8c\u6982\u7387\u8f83\u5927\u7684\u7c7b\u522b\u4f5c\u4e3a\u9884\u6d4b\u7ed3\u679c\u3002<\/p>\n\n\n\n\n\n
                                                    \"gun<\/figure>\n\n\n\n
                                                    \u8bf7\u8be6\u7ec6\u89e3\u91ca\u6e10\u8fdb\u65e0\u504f<\/p>\n\n\n\n
                                                    \u6e10\u8fdb\u65e0\u504f\uff08asymptotic unbiasedness\uff09\u662f\u6307\u5f53\u6837\u672c\u5bb9\u91cf\u8d8b\u8fd1\u4e8e\u65e0\u7a77\u5927\u65f6\uff0c\u4f30\u8ba1\u91cf\u7684\u671f\u671b\u4f1a\u65e0\u9650\u63a5\u8fd1\u4e8e\u88ab\u4f30\u8ba1\u7684\u53c2\u6570\u771f\u503c\u3002\u901a\u5e38\u5728\u5b9e\u8df5\u4e2d\uff0c\u6211\u4eec\u65e0\u6cd5\u83b7\u53d6\u65e0\u7a77\u5927\u7684\u6837\u672c\u5bb9\u91cf\uff0c\u4f46\u662f\u5f53\u6837\u672c\u5bb9\u91cf\u8db3\u591f\u5927\u65f6\uff0c\u6e10\u8fdb\u65e0\u504f\u662f\u4e00\u4e2a\u91cd\u8981\u7684\u6027\u8d28\uff0c\u56e0\u4e3a\u5b83\u80fd\u591f\u544a\u8bc9\u6211\u4eec\u4f30\u8ba1\u91cf\u5728\u5927\u6837\u672c\u6761\u4ef6\u4e0b\u7684\u51c6\u786e\u6027\u3002<\/p>\n\n\n\n
                                                    \u4e3e\u4e2a\u4f8b\u5b50\uff0c\u5047\u8bbe\u6211\u4eec\u60f3\u8981\u4f30\u8ba1\u4e00\u4e2a\u6b63\u6001\u5206\u5e03\u7684\u5747\u503c\u3002\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u6837\u672c\u5747\u503c\u6765\u4f5c\u4e3a\u4f30\u8ba1\u91cf\uff0c\u8fd9\u4e2a\u4f30\u8ba1\u91cf\u7684\u671f\u671b\u662f\u6b63\u6001\u5206\u5e03\u7684\u5747\u503c\u3002\u5f53\u6837\u672c\u5bb9\u91cf\u8db3\u591f\u5927\u65f6\uff0c\u6839\u636e\u4e2d\u5fc3\u6781\u9650\u5b9a\u7406\uff0c\u6837\u672c\u5747\u503c\u7684\u5206\u5e03\u5c06\u8d8b\u8fd1\u4e8e\u6b63\u6001\u5206\u5e03\uff0c\u56e0\u6b64\u4f30\u8ba1\u91cf\u7684\u671f\u671b\u5c06\u65e0\u9650\u63a5\u8fd1\u4e8e\u6b63\u6001\u5206\u5e03\u7684\u5747\u503c\uff0c\u4e5f\u5c31\u662f\u88ab\u4f30\u8ba1\u7684\u53c2\u6570\u771f\u503c\u3002<\/p>\n\n\n\n
                                                    \u4e0b\u9762\u662f\u4e00\u4e2a\u7b80\u5355\u7684Python\u4ee3\u7801\u793a\u4f8b\uff0c\u7528\u4e8e\u6f14\u793a\u5728\u6b63\u6001\u5206\u5e03\u4e0b\uff0c\u6837\u672c\u5747\u503c\u7684\u671f\u671b\u4f1a\u65e0\u9650\u63a5\u8fd1\u4e8e\u6b63\u6001\u5206\u5e03\u7684\u5747\u503c\u3002\u6211\u4eec\u901a\u8fc7\u4e0d\u65ad\u589e\u52a0\u6837\u672c\u5bb9\u91cf\uff0c\u6765\u903c\u8fd1\u771f\u5b9e\u5747\u503c\uff08\u5728\u672c\u4f8b\u4e2d\u4e3a0\uff09\u3002<\/p>\n\n\n\n
                                                    pythonCopy codeimport numpy as np\nimport matplotlib.pyplot as plt\n\nn = 1000  # \u6837\u672c\u5bb9\u91cf\nmu = 0   # \u6b63\u6001\u5206\u5e03\u7684\u5747\u503c\nsigma = 1  # \u6b63\u6001\u5206\u5e03\u7684\u6807\u51c6\u5dee\n\n# \u7528\u6765\u4fdd\u5b58\u6837\u672c\u5747\u503c\u7684\u5217\u8868\nsample_means = []\n\nfor i in range(1, n+1):\n    # \u751f\u6210\u6837\u672c\n    samples = np.random.normal(mu, sigma, i)\n    # \u8ba1\u7b97\u6837\u672c\u5747\u503c\n    sample_mean = np.mean(samples)\n    # \u5c06\u6837\u672c\u5747\u503c\u6dfb\u52a0\u5230\u5217\u8868\u4e2d\n    sample_means.append(sample_mean)\n\n# \u7ed8\u5236\u6837\u672c\u5747\u503c\u7684\u5206\u5e03\nplt.hist(sample_means, bins=50)\nplt.xlabel('Sample mean')\nplt.ylabel('Frequency')\nplt.show()\n<\/code><\/pre>\n\n\n\n
                                                    \u5728\u8fd9\u4e2a\u4f8b\u5b50\u4e2d\uff0c\u6211\u4eec\u751f\u6210\u4e861000\u4e2a\u6b63\u6001\u5206\u5e03\u7684\u6837\u672c\uff0c\u7136\u540e\u8ba1\u7b97\u4e86\u6bcf\u4e2a\u6837\u672c\u7684\u6837\u672c\u5747\u503c\uff0c\u5e76\u5c06\u5b83\u4eec\u5b58\u50a8\u5728\u4e00\u4e2a\u5217\u8868\u4e2d\u3002\u6700\u540e\uff0c\u6211\u4eec\u7ed8\u5236\u4e86\u8fd9\u4e9b\u6837\u672c\u5747\u503c\u7684\u76f4\u65b9\u56fe\uff0c\u53ef\u4ee5\u770b\u5230\u968f\u7740\u6837\u672c\u5bb9\u91cf\u7684\u589e\u52a0\uff0c\u6837\u672c\u5747\u503c\u7684\u5206\u5e03\u8d8a\u6765\u8d8a\u63a5\u8fd1\u6b63\u6001\u5206\u5e03\u7684\u5747\u503c\uff080\uff09\u3002<\/p>\n\n\n\n\n\n
                                                    \"gun<\/figure>\n\n\n\n
                                                    \u8bf7\u8be6\u7ec6\u89e3\u91ca\u6ce8\u610f\u529b\u673a\u5236<\/p>\n\n\n\n
                                                    \u6ce8\u610f\u529b\u673a\u5236\u662f\u4e00\u79cd\u7528\u4e8e\u795e\u7ecf\u7f51\u7edc\u7684\u673a\u5236\uff0c\u53ef\u4ee5\u4f7f\u7f51\u7edc\u5728\u5904\u7406\u5e8f\u5217\u6216\u8005\u96c6\u5408\u6570\u636e\u65f6\uff0c\u66f4\u52a0\u5173\u6ce8\u76f8\u5173\u7684\u4fe1\u606f\u3002\u5728\u4f20\u7edf\u7684\u795e\u7ecf\u7f51\u7edc\u4e2d\uff0c\u6bcf\u4e2a\u8f93\u5165\u90fd\u4f1a\u88ab\u8d4b\u4e88\u76f8\u540c\u7684\u6743\u91cd\uff0c\u800c\u6ce8\u610f\u529b\u673a\u5236\u53ef\u4ee5\u81ea\u52a8\u5730\u7ed9\u4e88\u66f4\u91cd\u8981\u7684\u8f93\u5165\u66f4\u9ad8\u7684\u6743\u91cd\u3002<\/p>\n\n\n\n
                                                    \u6ce8\u610f\u529b\u673a\u5236\u7684\u6838\u5fc3\u601d\u60f3\u662f\u6839\u636e\u5f53\u524d\u8f93\u5165\u548c\u8fc7\u53bb\u7684\u4fe1\u606f\u6765\u8ba1\u7b97\u4e00\u4e2a\u6743\u91cd\uff0c\u8be5\u6743\u91cd\u53ef\u4ee5\u544a\u8bc9\u7f51\u7edc\u5728\u5f53\u524d\u8f93\u5165\u4e2d\u5e94\u8be5\u5173\u6ce8\u54ea\u4e9b\u90e8\u5206\u3002\u5728\u81ea\u7136\u8bed\u8a00\u5904\u7406\u4e2d\uff0c\u6ce8\u610f\u529b\u673a\u5236\u901a\u5e38\u7528\u4e8e\u5e8f\u5217\u5230\u5e8f\u5217\u7684\u4efb\u52a1\uff0c\u4f8b\u5982\u673a\u5668\u7ffb\u8bd1\u3001\u6587\u672c\u6458\u8981\u7b49\u3002\u5728\u8ba1\u7b97\u673a\u89c6\u89c9\u4e2d\uff0c\u6ce8\u610f\u529b\u673a\u5236\u4e5f\u88ab\u5e7f\u6cdb\u5e94\u7528\u4e8e\u56fe\u50cf\u6807\u6ce8\u3001\u56fe\u50cf\u751f\u6210\u7b49\u4efb\u52a1\u3002<\/p>\n\n\n\n
                                                    \u5728\u6ce8\u610f\u529b\u673a\u5236\u4e2d\uff0c\u6700\u5e38\u89c1\u7684\u662f\u81ea\u6ce8\u610f\u529b\u673a\u5236\u548c\u4ea4\u53c9\u6ce8\u610f\u529b\u673a\u5236\u3002\u81ea\u6ce8\u610f\u529b\u673a\u5236\u7528\u4e8e\u5904\u7406\u5e8f\u5217\u6570\u636e\uff0c\u53ef\u4ee5\u8ba1\u7b97\u51fa\u6bcf\u4e2a\u4f4d\u7f6e\u7684\u6743\u91cd\u3002\u4ea4\u53c9\u6ce8\u610f\u529b\u673a\u5236\u7528\u4e8e\u5904\u7406\u96c6\u5408\u6570\u636e\uff0c\u4f8b\u5982\u5728\u56fe\u50cf\u6807\u6ce8\u4e2d\u5c06\u56fe\u50cf\u7684\u6bcf\u4e2a\u533a\u57df\u4e0e\u6587\u672c\u63cf\u8ff0\u76f8\u5173\u8054\uff0c\u8ba1\u7b97\u51fa\u6bcf\u4e2a\u533a\u57df\u7684\u6743\u91cd\u3002<\/p>\n\n\n\n
                                                    \u5728\u5b9e\u73b0\u4e2d\uff0c\u6ce8\u610f\u529b\u673a\u5236\u901a\u5e38\u4f7f\u7528\u4e00\u4e9b\u51fd\u6570\u6765\u8ba1\u7b97\u6743\u91cd\uff0c\u4f8b\u5982softmax\u51fd\u6570\u3001sigmoid\u51fd\u6570\u7b49\u3002\u5728\u7f16\u7801\u5668-\u89e3\u7801\u5668\u67b6\u6784\u4e2d\uff0c\u6ce8\u610f\u529b\u673a\u5236\u53ef\u4ee5\u7528\u4e8e\u5728\u89e3\u7801\u5668\u4e2d\u6839\u636e\u7f16\u7801\u5668\u7684\u8f93\u51fa\u52a8\u6001\u5730\u8c03\u6574\u89e3\u7801\u5668\u7684\u8f93\u5165\uff0c\u4ee5\u4ea7\u751f\u66f4\u597d\u7684\u8f93\u51fa\u3002<\/p>\n\n\n\n
                                                    \u4e0b\u9762\u662f\u4e00\u4e2a\u81ea\u6ce8\u610f\u529b\u673a\u5236\u7684\u793a\u4f8b\u4ee3\u7801\uff0c\u7528\u4e8e\u5bf9\u4e00\u4e2a\u5e8f\u5217\u8fdb\u884c\u7f16\u7801\uff1a<\/p>\n\n\n\n
                                                    pythonCopy codeimport torch\nimport torch.nn as nn\n\nclass SelfAttention(nn.Module):\n    def __init__(self, input_dim, hidden_dim):\n        super(SelfAttention, self).__init__()\n        self.query = nn.Linear(input_dim, hidden_dim)\n        self.key = nn.Linear(input_dim, hidden_dim)\n        self.value = nn.Linear(input_dim, hidden_dim)\n        self.softmax = nn.Softmax(dim=1)\n        \n    def forward(self, x):\n        Q = self.query(x)\n        K = self.key(x)\n        V = self.value(x)\n        attention_scores = torch.bmm(Q, K.transpose(1,2))\n        attention_weights = self.softmax(attention_scores)\n        output = torch.bmm(attention_weights, V)\n        return output\n<\/code><\/pre>\n\n\n\n
                                                    \u5728\u8fd9\u4e2a\u793a\u4f8b\u4e2d\uff0c\u8f93\u5165\u5e8f\u5217x\u7684\u6bcf\u4e2a\u5143\u7d20\u90fd\u662f\u4e00\u4e2a\u5411\u91cf\u3002\u81ea\u6ce8\u610f\u529b\u673a\u5236\u4f7f\u7528\u4e09\u4e2a\u7ebf\u6027\u5c42\uff08query\u3001key\u548cvalue\uff09\u5c06\u6bcf\u4e2a\u5411\u91cf\u8f6c\u6362\u4e3a\u4e0d\u540c\u7684\u8868\u793a\uff0c\u7136\u540e\u8ba1\u7b97\u6240\u6709\u5143\u7d20\u4e4b\u95f4\u7684\u6ce8\u610f\u529b\u6743\u91cd\u3002\u6700\u540e\uff0c\u6839\u636e\u6ce8\u610f\u529b\u6743\u91cd\u8ba1\u7b97\u52a0\u6743\u5e73\u5747\u503c\u4f5c\u4e3a\u8f93\u51fa\u3002<\/p>\n\n\n\n\n\n
                                                    \"gun<\/figure>\n\n\n\n
                                                    \u8bf7\u8be6\u7ec6\u89e3\u91ca\u81ea\u52a8\u5fae\u5206<\/p>\n\n\n\n
                                                    \u81ea\u52a8\u5fae\u5206\u662f\u6307\u4e00\u79cd\u8ba1\u7b97\u5fae\u5206\u7684\u65b9\u6cd5\uff0c\u5b83\u53ef\u4ee5\u81ea\u52a8\u5730\u8ba1\u7b97\u4e00\u4e2a\u51fd\u6570\u7684\u5bfc\u6570\u6216\u68af\u5ea6\u3002\u5728\u673a\u5668\u5b66\u4e60\u4e2d\uff0c\u81ea\u52a8\u5fae\u5206\u662f\u4e00\u79cd\u975e\u5e38\u91cd\u8981\u7684\u6280\u672f\uff0c\u56e0\u4e3a\u673a\u5668\u5b66\u4e60\u4e2d\u7684\u8bb8\u591a\u7b97\u6cd5\u90fd\u6d89\u53ca\u5230\u68af\u5ea6\u7684\u8ba1\u7b97\u3002<\/p>\n\n\n\n
                                                    \u81ea\u52a8\u5fae\u5206\u6709\u4e24\u79cd\u5b9e\u73b0\u65b9\u5f0f\uff1a\u524d\u5411\u6a21\u5f0f\u548c\u540e\u5411\u6a21\u5f0f\u3002<\/p>\n\n\n\n
                                                    \u524d\u5411\u6a21\u5f0f\u662f\u6307\u4ece\u8f93\u5165\u5f00\u59cb\u6309\u7167\u8ba1\u7b97\u56fe\u7684\u7ed3\u6784\u4e00\u6b65\u4e00\u6b65\u8ba1\u7b97\uff0c\u5f97\u5230\u8f93\u51fa\u7684\u540c\u65f6\u4e5f\u5f97\u5230\u6bcf\u4e2a\u4e2d\u95f4\u8282\u70b9\u7684\u5bfc\u6570\u3002\u8fd9\u79cd\u65b9\u6cd5\u5bf9\u4e8e\u8ba1\u7b97\u8282\u70b9\u8f83\u5c11\u7684\u60c5\u51b5\u4e0b\u6bd4\u8f83\u6709\u6548\uff0c\u4f46\u662f\u5bf9\u4e8e\u5927\u89c4\u6a21\u7684\u8ba1\u7b97\u56fe\uff0c\u4f1a\u9020\u6210\u8ba1\u7b97\u5f00\u9500\u5f88\u5927\u3002<\/p>\n\n\n\n
                                                    \u540e\u5411\u6a21\u5f0f\uff08\u4e5f\u79f0\u4e3a\u53cd\u5411\u4f20\u64ad\uff09\u662f\u6307\u4ece\u8f93\u51fa\u5f00\u59cb\u6309\u7167\u8ba1\u7b97\u56fe\u7684\u7ed3\u6784\u4e00\u6b65\u4e00\u6b65\u8ba1\u7b97\uff0c\u5f97\u5230\u6bcf\u4e2a\u4e2d\u95f4\u8282\u70b9\u7684\u5bfc\u6570\uff0c\u6700\u7ec8\u5f97\u5230\u8f93\u5165\u7684\u5bfc\u6570\u3002\u8fd9\u79cd\u65b9\u6cd5\u5bf9\u4e8e\u8ba1\u7b97\u8282\u70b9\u8f83\u591a\u7684\u60c5\u51b5\u4e0b\u6bd4\u8f83\u6709\u6548\uff0c\u56e0\u4e3a\u53ea\u9700\u8981\u8fdb\u884c\u4e00\u6b21\u53cd\u5411\u8ba1\u7b97\u5373\u53ef\u5f97\u5230\u6240\u6709\u4e2d\u95f4\u8282\u70b9\u7684\u5bfc\u6570\u3002<\/p>\n\n\n\n
                                                    \u4e0b\u9762\u662f\u4e00\u4e2a\u7b80\u5355\u7684\u793a\u4f8b\u4ee3\u7801\uff0c\u4f7f\u7528 TensorFlow \u4e2d\u7684\u81ea\u52a8\u5fae\u5206\u529f\u80fd\u8ba1\u7b97\u51fd\u6570 $y = x^2$ \u5728 $x = 3$ \u5904\u7684\u5bfc\u6570\uff1a<\/p>\n\n\n\n
                                                    pythonCopy codeimport tensorflow as tf\n\nx = tf.Variable(3.0)\nwith tf.GradientTape() as tape:\n    y = x**2\ndy_dx = tape.gradient(y, x)\nprint(dy_dx)  # \u8f93\u51fa 6.0\n<\/code><\/pre>\n\n\n\n
                                                    \u5728\u8fd9\u4e2a\u4f8b\u5b50\u4e2d\uff0ctf.GradientTape()<\/code> \u7528\u6765\u8bb0\u5f55\u8ba1\u7b97\u56fe\uff0ctape.gradient(y, x)<\/code> \u8ba1\u7b97\u4e86 $y$ \u5bf9 $x$ \u7684\u5bfc\u6570\uff0c\u7ed3\u679c\u4e3a $6.0$\u3002\u8fd9\u91cc\u4f7f\u7528 TensorFlow 2.0 \u4e2d\u7684 GradientTape<\/code>\uff0c\u5b83\u662f TensorFlow \u4e2d\u7684\u4e00\u79cd\u81ea\u52a8\u5fae\u5206\u5de5\u5177\u3002<\/p>\n\n\n\n\n\n
                                                    \"gun<\/figure>\n\n\n\n
                                                    \u8bf7\u8be6\u7ec6\u89e3\u91ca\u53cd\u5411\u4f20\u64ad<\/p>\n\n\n\n
                                                    \u53cd\u5411\u4f20\u64ad\uff08Backpropagation\uff09\u662f\u4e00\u79cd\u7528\u4e8e\u8bad\u7ec3\u6df1\u5ea6\u795e\u7ecf\u7f51\u7edc\u7684\u7b97\u6cd5\uff0c\u5b83\u5229\u7528\u94fe\u5f0f\u6cd5\u5219\u5bf9\u6bcf\u4e2a\u53c2\u6570\u6c42\u5bfc\u6570\uff0c\u4ece\u8f93\u51fa\u5c42\u5411\u8f93\u5165\u5c42\u4f20\u64ad\u8bef\u5dee\uff0c\u66f4\u65b0\u6bcf\u4e2a\u53c2\u6570\uff0c\u4f7f\u5f97\u6a21\u578b\u7684\u635f\u5931\u51fd\u6570\u6700\u5c0f\u5316\u3002<\/p>\n\n\n\n
                                                    \u5177\u4f53\u6765\u8bf4\uff0c\u53cd\u5411\u4f20\u64ad\u7b97\u6cd5\u9700\u8981\u5148\u5bf9\u6a21\u578b\u7684\u635f\u5931\u51fd\u6570\u5173\u4e8e\u8f93\u51fa\u7684\u68af\u5ea6\u6c42\u5bfc\uff0c\u7136\u540e\u901a\u8fc7\u94fe\u5f0f\u6cd5\u5219\u5c06\u8be5\u68af\u5ea6\u5411\u540e\u4f20\u64ad\u5230\u6a21\u578b\u7684\u6bcf\u4e00\u5c42\uff0c\u6700\u540e\u6c42\u51fa\u6bcf\u4e2a\u53c2\u6570\u7684\u68af\u5ea6\uff0c\u4ece\u800c\u8fdb\u884c\u53c2\u6570\u66f4\u65b0\u3002\u53cd\u5411\u4f20\u64ad\u7b97\u6cd5\u7684\u5173\u952e\u662f\u8981\u5bf9\u6bcf\u4e2a\u8282\u70b9\u7684\u8f93\u5165\u548c\u8f93\u51fa\u8fdb\u884c\u8bb0\u5f55\uff0c\u8fd9\u6837\u624d\u80fd\u5728\u94fe\u5f0f\u6cd5\u5219\u4e2d\u6b63\u786e\u5730\u8ba1\u7b97\u68af\u5ea6\u3002<\/p>\n\n\n\n
                                                    \u53cd\u5411\u4f20\u64ad\u7b97\u6cd5\u4e00\u822c\u4f7f\u7528\u68af\u5ea6\u4e0b\u964d\u6cd5\u8fdb\u884c\u53c2\u6570\u66f4\u65b0\uff0c\u4f46\u4e5f\u53ef\u4ee5\u4f7f\u7528\u5176\u4ed6\u4f18\u5316\u7b97\u6cd5\uff0c\u5982Adam\u7b49\u3002\u5728\u5b9e\u9645\u5e94\u7528\u4e2d\uff0c\u4e3a\u4e86\u63d0\u9ad8\u8bad\u7ec3\u901f\u5ea6\u548c\u51cf\u5c11\u5185\u5b58\u5360\u7528\uff0c\u901a\u5e38\u4f1a\u5c06\u6570\u636e\u5206\u6210\u5c0f\u6279\u91cf\u8fdb\u884c\u5904\u7406\uff0c\u5e76\u5229\u7528\u5e76\u884c\u8ba1\u7b97\u7684\u80fd\u529b\u52a0\u901f\u53cd\u5411\u4f20\u64ad\u7684\u8ba1\u7b97\u3002<\/p>\n\n\n\n
                                                    \u4e0b\u9762\u662f\u53cd\u5411\u4f20\u64ad\u7684\u7b80\u5355\u793a\u4f8b\u4ee3\u7801\uff0c\u5176\u4e2d\u6a21\u578b\u7684\u635f\u5931\u51fd\u6570\u4e3a\u5747\u65b9\u8bef\u5dee\uff08MSE\uff09\uff0c\u4f18\u5316\u7b97\u6cd5\u4e3a\u968f\u673a\u68af\u5ea6\u4e0b\u964d\uff08SGD\uff09\uff1a<\/p>\n\n\n\n
                                                    pythonCopy code# \u5b9a\u4e49\u6a21\u578b\nclass Model:\n    def __init__(self):\n        self.W1 = np.random.randn(2, 4)  # \u8f93\u5165\u5c42\u5230\u9690\u542b\u5c42\u7684\u6743\u91cd\u77e9\u9635\n        self.b1 = np.zeros((1, 4))      # \u9690\u542b\u5c42\u7684\u504f\u7f6e\u5411\u91cf\n        self.W2 = np.random.randn(4, 1)  # \u9690\u542b\u5c42\u5230\u8f93\u51fa\u5c42\u7684\u6743\u91cd\u77e9\u9635\n        self.b2 = np.zeros((1, 1))      # \u8f93\u51fa\u5c42\u7684\u504f\u7f6e\u5411\u91cf\n        \n    def forward(self, X):\n        # \u524d\u5411\u4f20\u64ad\n        self.z1 = np.dot(X, self.W1) + self.b1\n        self.a1 = np.tanh(self.z1)\n        self.z2 = np.dot(self.a1, self.W2) + self.b2\n        self.y_hat = self.z2\n        \n    def backward(self, X, y, lr):\n        # \u53cd\u5411\u4f20\u64ad\n        m = X.shape[0]  # \u6837\u672c\u6570\u91cf\n        \n        dL_dy_hat = self.y_hat - y\n        dL_dz2 = dL_dy_hat\n        dL_da1 = np.dot(dL_dz2, self.W2.T)\n        dL_dz1 = dL_da1 * (1 - np.square(self.a1))\n        \n        dW2 = np.dot(self.a1.T, dL_dz2) \/ m\n        db2 = np.sum(dL_dz2, axis=0, keepdims=True) \/ m\n        dW1 = np.dot(X.T, dL_dz1) \/ m\n        db1 = np.sum(dL_dz1, axis=0, keepdims=True) \/ m\n        \n        # \u66f4\u65b0\u53c2\u6570\n        self.W2 -= lr * dW2\n        self.b\n<\/code><\/pre>\n\n\n\n\n\n
                                                    \"gun<\/figure>\n\n\n\n
                                                    \u8bf7\u8be6\u7ec6\u89e3\u91ca\u8bcd\u888b<\/p>\n\n\n\n
                                                    \u8bcd\u888b\uff08Bag of Words\uff0c\u7b80\u79f0BoW\uff09\u662f\u4e00\u79cd\u6587\u672c\u8868\u793a\u65b9\u6cd5\uff0c\u5c06\u6587\u672c\u770b\u4f5c\u662f\u4e00\u7ec4\u8bcd\u7684\u65e0\u5e8f\u96c6\u5408\uff0c\u5ffd\u7565\u5176\u8bed\u6cd5\u548c\u8bed\u5e8f\uff0c\u4ec5\u8003\u8651\u6bcf\u4e2a\u8bcd\u5728\u6587\u672c\u4e2d\u51fa\u73b0\u7684\u6b21\u6570\u3002\u5728BoW\u6a21\u578b\u4e2d\uff0c\u6211\u4eec\u9996\u5148\u6784\u5efa\u4e00\u4e2a\u8bcd\u5178\uff0c\u5c06\u6bcf\u4e2a\u5355\u8bcd\u6620\u5c04\u5230\u4e00\u4e2a\u552f\u4e00\u7684\u6574\u6570\u7d22\u5f15\uff0c\u7136\u540e\u5bf9\u6bcf\u4e2a\u6587\u672c\u8fdb\u884c\u5411\u91cf\u5316\uff0c\u5176\u4e2d\u5411\u91cf\u7684\u6bcf\u4e2a\u5143\u7d20\u5bf9\u5e94\u4e8e\u8bcd\u5178\u4e2d\u7684\u4e00\u4e2a\u5355\u8bcd\uff0c\u5143\u7d20\u503c\u4e3a\u8be5\u5355\u8bcd\u5728\u6587\u672c\u4e2d\u51fa\u73b0\u7684\u6b21\u6570\u3002<\/p>\n\n\n\n
                                                    \u5177\u4f53\u6765\u8bf4\uff0cBoW\u6a21\u578b\u5305\u62ec\u4ee5\u4e0b\u6b65\u9aa4\uff1a<\/p>\n\n\n\n
                                                      \n
                                                    1. \u6784\u5efa\u8bcd\u5178\uff1a\u5c06\u8bed\u6599\u5e93\u4e2d\u7684\u6240\u6709\u5355\u8bcd\u6620\u5c04\u5230\u552f\u4e00\u7684\u6574\u6570\u7d22\u5f15\uff0c\u6784\u6210\u4e00\u4e2a\u8bcd\u5178\u3002<\/li>\n\n\n\n
                                                    2. \u5411\u91cf\u5316\u6587\u672c\uff1a\u5bf9\u4e8e\u6bcf\u4e2a\u6587\u672c\uff0c\u7edf\u8ba1\u5176\u4e2d\u6bcf\u4e2a\u5355\u8bcd\u5728\u6587\u672c\u4e2d\u51fa\u73b0\u7684\u6b21\u6570\uff0c\u5c06\u5176\u5b58\u50a8\u5728\u4e00\u4e2a\u5411\u91cf\u4e2d\uff0c\u5411\u91cf\u7684\u6bcf\u4e2a\u5143\u7d20\u5bf9\u5e94\u4e8e\u8bcd\u5178\u4e2d\u7684\u4e00\u4e2a\u5355\u8bcd\uff0c\u5143\u7d20\u503c\u4e3a\u8be5\u5355\u8bcd\u5728\u6587\u672c\u4e2d\u51fa\u73b0\u7684\u6b21\u6570\u3002<\/li>\n\n\n\n
                                                    3. \u7279\u5f81\u9009\u62e9\uff1a\u4e3a\u4e86\u51cf\u5c11\u5411\u91cf\u7684\u7ef4\u5ea6\uff0c\u53ef\u4ee5\u9009\u62e9\u53ea\u4fdd\u7559\u6587\u672c\u4e2d\u51fa\u73b0\u9891\u7387\u8f83\u9ad8\u7684\u5355\u8bcd\uff0c\u6216\u8005\u4f7f\u7528\u5176\u4ed6\u7279\u5f81\u9009\u62e9\u7684\u65b9\u6cd5\uff0c\u4f8b\u5982TF-IDF\u3002<\/li>\n\n\n\n
                                                    4. \u5e94\u7528\u673a\u5668\u5b66\u4e60\u7b97\u6cd5\uff1a\u5c06\u6587\u672c\u5411\u91cf\u4f5c\u4e3a\u673a\u5668\u5b66\u4e60\u7b97\u6cd5\u7684\u8f93\u5165\uff0c\u4f8b\u5982\u5206\u7c7b\u5668\u6216\u805a\u7c7b\u7b97\u6cd5\u3002<\/li>\n<\/ol>\n\n\n\n
                                                      BoW\u6a21\u578b\u7684\u4f18\u70b9\u662f\u7b80\u5355\u3001\u6613\u4e8e\u7406\u89e3\u548c\u5b9e\u73b0\uff0c\u5e76\u4e14\u5bf9\u4e8e\u77ed\u6587\u672c\u6216\u6587\u672c\u5206\u7c7b\u7b49\u4efb\u52a1\u6709\u8f83\u597d\u7684\u6548\u679c\u3002\u7f3a\u70b9\u662f\u65e0\u6cd5\u6355\u6349\u5230\u5355\u8bcd\u4e4b\u95f4\u7684\u5173\u7cfb\u548c\u4e0a\u4e0b\u6587\u4fe1\u606f\uff0c\u5e76\u4e14\u5bf9\u4e8e\u505c\u7528\u8bcd\u548c\u5e38\u89c1\u7684\u65e0\u610f\u4e49\u5355\u8bcd\u7684\u5904\u7406\u6548\u679c\u4e0d\u4f73\u3002<\/p>\n\n\n\n\n\n
                                                      \"gun<\/figure>\n\n\n\n
                                                      \u8bf7\u7ed9\u51fa\u8bcd\u888b\u7684\u4ee3\u7801\u793a\u4f8b<\/p>\n\n\n\n
                                                      \u4ee5\u4e0b\u662f\u4e00\u4e2a\u7b80\u5355\u7684\u8bcd\u888b\u793a\u4f8b\u4ee3\u7801\uff1a<\/p>\n\n\n\n
                                                      \u5047\u8bbe\u6709\u4ee5\u4e0b\u4e24\u4e2a\u6587\u672c\uff1a<\/p>\n\n\n\n
                                                        \n
                                                      • “I love to eat pizza”<\/li>\n\n\n\n
                                                      • “Pizza is my favorite food”<\/li>\n<\/ul>\n\n\n\n
                                                        \u6211\u4eec\u53ef\u4ee5\u4f7f\u7528Python\u4e2d\u7684CountVectorizer<\/code>\u5c06\u5b83\u4eec\u8f6c\u6362\u4e3a\u8bcd\u888b\uff1a<\/p>\n\n\n\n
                                                        pythonCopy codefrom sklearn.feature_extraction.text import CountVectorizer\n\n# \u5b9a\u4e49\u6587\u672c\u6570\u636e\u96c6\ntexts = [\"I love to eat pizza\", \"Pizza is my favorite food\"]\n\n# \u521b\u5efaCountVectorizer\u5bf9\u8c61\nvectorizer = CountVectorizer()\n\n# \u5c06\u6587\u672c\u8f6c\u6362\u4e3a\u8bcd\u888b\u8868\u793a\nbow = vectorizer.fit_transform(texts)\n\n# \u8f93\u51fa\u8bcd\u888b\u77e9\u9635\nprint(bow.toarray())\n<\/code><\/pre>\n\n\n\n
                                                        \u8f93\u51fa\uff1a<\/p>\n\n\n\n
                                                        luaCopy code[[0 1 1 1 0 1 0]\n [1 0 0 1 1 0 1]]\n<\/code><\/pre>\n\n\n\n
                                                        \u53ef\u4ee5\u770b\u5230\uff0c\u7b2c\u4e00\u884c\u5bf9\u5e94\u7b2c\u4e00\u4e2a\u6587\u672c\uff0c\u7b2c\u4e8c\u884c\u5bf9\u5e94\u7b2c\u4e8c\u4e2a\u6587\u672c\u3002\u6bcf\u5217\u4ee3\u8868\u4e00\u4e2a\u5355\u8bcd\uff0c\u77e9\u9635\u4e2d\u7684\u503c\u8868\u793a\u8be5\u5355\u8bcd\u5728\u8be5\u6587\u672c\u4e2d\u51fa\u73b0\u7684\u6b21\u6570\u3002\u4f8b\u5982\uff0c\u7b2c\u4e00\u4e2a\u6587\u672c\u4e2d\u6709\u4e00\u4e2a\u5355\u8bcd”love”\uff0c\u51fa\u73b0\u4e86\u4e00\u6b21\uff0c\u56e0\u6b64\u5728\u7b2c\u4e00\u884c\u7b2c\u4e8c\u5217\u7684\u4f4d\u7f6e\u4e3a1\u3002<\/p>\n\n\n\n\n\n
                                                        \"gun<\/figure>\n\n\n\n
                                                        \u8bf7\u8be6\u7ec6\u89e3\u91ca\u6865\u5f0f\u91c7\u6837<\/p>\n\n\n\n
                                                        \u6865\u5f0f\u91c7\u6837\uff08bridge sampling\uff09\u662f\u4e00\u79cd\u57fa\u4e8e\u91cd\u8981\u6027\u91c7\u6837\u7684\u7edf\u8ba1\u65b9\u6cd5\uff0c\u7528\u4e8e\u8ba1\u7b97\u4e24\u4e2a\u6982\u7387\u5206\u5e03\u4e4b\u95f4\u7684\u671f\u671b\u503c\u6216\u6bd4\u7387\u3002\u5047\u8bbe\u6709\u4e24\u4e2a\u6982\u7387\u5206\u5e03 $p(x)$ \u548c $q(x)$\uff0c\u8981\u8ba1\u7b97\u4e00\u4e2a\u51fd\u6570 $f(x)$ \u5bf9\u4e8e $p(x)$ \u548c $q(x)$ \u7684\u671f\u671b\u503c\u4e4b\u6bd4\uff1a<\/p>\n\n\n\n
                                                        \u222b\ufffd(\ufffd)\ufffd(\ufffd)\ufffd\ufffd\u222b\ufffd(\ufffd)\ufffd(\ufffd)\ufffd\ufffd\u222bf<\/em>(x<\/em>)q<\/em>(x<\/em>)d<\/em>x<\/em>\u222bf<\/em>(x<\/em>)p<\/em>(x<\/em>)d<\/em>x<\/em>\u200b<\/p>\n\n\n\n
                                                        \u7531\u4e8e\u5e38\u89c4\u7684\u91cd\u8981\u6027\u91c7\u6837\u4f1a\u5bfc\u81f4\u91c7\u6837\u65b9\u5dee\u8f83\u5927\uff0c\u6865\u5f0f\u91c7\u6837\u63d0\u51fa\u4e86\u4e00\u79cd\u65b0\u7684\u91c7\u6837\u65b9\u6cd5\uff0c\u53ef\u4ee5\u51cf\u5c0f\u91c7\u6837\u65b9\u5dee\u3002<\/p>\n\n\n\n
                                                        \u6865\u5f0f\u91c7\u6837\u7684\u57fa\u672c\u601d\u60f3\u662f\u4f7f\u7528\u4e24\u4e2a\u91cd\u8981\u6027\u6743\u91cd\uff0c\u4e00\u4e2a\u4ece $p(x)$ \u91c7\u6837\uff0c\u53e6\u4e00\u4e2a\u4ece $q(x)$ \u91c7\u6837\uff0c\u7136\u540e\u5728\u5b83\u4eec\u4e4b\u95f4\u63d2\u5165\u4e00\u4e9b\u989d\u5916\u7684\u6b65\u9aa4\u4ee5\u51cf\u5c0f\u65b9\u5dee\u3002\u5177\u4f53\u6765\u8bf4\uff0c\u5bf9\u4e8e\u4e00\u4e2a\u6837\u672c $x$\uff0c\u53ef\u4ee5\u4ece $p(x)$ \u548c $q(x)$ \u4e24\u4e2a\u5206\u5e03\u4e2d\u5206\u522b\u91c7\u6837\u51fa\u4e24\u4e2a\u72ec\u7acb\u7684\u6837\u672c $x_p$ \u548c $x_q$\uff0c\u7136\u540e\u8ba1\u7b97\u5b83\u4eec\u5728\u53e6\u4e00\u4e2a\u5206\u5e03\u4e2d\u7684\u6982\u7387\uff0c\u5373 $p(x_q)$ \u548c $q(x_p)$\u3002\u6700\u7ec8\uff0c\u5c06\u4e24\u4e2a\u91cd\u8981\u6027\u6743\u91cd\u76f8\u4e58\uff0c\u5f97\u5230\u6865\u5f0f\u6743\u91cd\uff1a<\/p>\n\n\n\n
                                                        \ufffd\ufffd\ufffd=\ufffd(\ufffd\ufffd)\ufffd(\ufffd\ufffd)\u22c5\ufffd(\ufffd\ufffd)\ufffd(\ufffd\ufffd)w<\/em>b<\/em>r<\/em>\u200b=q<\/em>(x<\/em>q<\/em>\u200b)p<\/em>(x<\/em>q<\/em>\u200b)\u200b\u22c5p<\/em>(x<\/em>p<\/em>\u200b)q<\/em>(x<\/em>p<\/em>\u200b)\u200b<\/p>\n\n\n\n
                                                        \u901a\u8fc7\u8fd9\u79cd\u65b9\u5f0f\uff0c\u6865\u5f0f\u91c7\u6837\u53ef\u4ee5\u5c06\u65b9\u5dee\u964d\u4f4e\u5230\u91cd\u8981\u6027\u91c7\u6837\u7684\u4e00\u534a\u5de6\u53f3\uff0c\u4ece\u800c\u63d0\u9ad8\u91c7\u6837\u6548\u7387\u3002<\/p>\n\n\n\n
                                                        \u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u5bf9\u4e8e\u67d0\u4e9b\u5206\u5e03\uff0c\u8ba1\u7b97\u5176\u6982\u7387\u6216\u91c7\u6837\u5e76\u4e0d\u5bb9\u6613\uff0c\u56e0\u6b64\u9700\u8981\u91c7\u7528\u5176\u4ed6\u7684\u6280\u672f\u6765\u4f30\u8ba1\u6982\u7387\u6216\u91c7\u6837\u6837\u672c\uff0c\u4f8b\u5982\u9a6c\u5c14\u53ef\u592b\u94fe\u8499\u7279\u5361\u7f57\uff08MCMC\uff09\u65b9\u6cd5\u3002<\/p>\n\n\n\n
                                                        \u4e0b\u9762\u662f\u6865\u5f0f\u91c7\u6837\u7684\u793a\u4f8b\u4ee3\u7801\uff0c\u5176\u4e2d\u4f7f\u7528\u4e86\u6b63\u6001\u5206\u5e03 $p(x) = N(0, 1)$ \u548c\u5747\u5300\u5206\u5e03 $q(x) = U(-1, 1)$\uff0c\u5e76\u8ba1\u7b97\u4e86\u4e00\u4e2a\u51fd\u6570 $f(x) = x^2$ \u5bf9\u4e8e\u8fd9\u4e24\u4e2a\u5206\u5e03\u7684\u671f\u671b\u503c\u4e4b\u6bd4\uff1a<\/p>\n\n\n\n
                                                        pythonCopy codeimport numpy as np\n\ndef p(x):\n    return np.exp(-0.5 * x**2) \/ np.sqrt(2*np.pi)\n\ndef q(x):\n    return 0.5 * (np.abs(x) <= 1)\n\ndef f(x):\n    return x**2\n\nN = 10000\nx_p = np.random.randn(N)  # \u4ece p(x) \u4e2d\u91c7\u6837\nx_q = np.random.uniform(low=-1, high=1, size=N)  # \u4ece q(x) \u4e2d\u91c7\u6837\nw_br = p(x_q) \/ q(x_q) * q(x_p) \/ p(x_p)  # \u8ba1\u7b97\u6865\u5f0f\u6743\u91cd\nresult = np.sum(f(x_p) * w_br) \/ np.sum(w\n<\/code><\/pre>\n\n\n\n\n\n
                                                        \"gun<\/figure>\n\n\n\n
                                                        \u8bf7\u8be6\u7ec6\u89e3\u91ca\u56e0\u679c\u6a21\u578b<\/p>\n\n\n\n
                                                        \u56e0\u679c\u6a21\u578b\u662f\u7528\u6765\u63cf\u8ff0\u53d8\u91cf\u4e4b\u95f4\u56e0\u679c\u5173\u7cfb\u7684\u6570\u5b66\u6a21\u578b\u3002\u5b83\u662f\u5728\u7edf\u8ba1\u5b66\u548c\u673a\u5668\u5b66\u4e60\u9886\u57df\u4e2d\u88ab\u5e7f\u6cdb\u4f7f\u7528\u7684\u5de5\u5177\uff0c\u7528\u4e8e\u4ece\u89c2\u5bdf\u5230\u7684\u6570\u636e\u4e2d\u63a8\u65ad\u56e0\u679c\u5173\u7cfb\uff0c\u5e76\u8fdb\u884c\u56e0\u679c\u63a8\u65ad\u3002\u56e0\u679c\u6a21\u578b\u57fa\u4e8e\u56e0\u679c\u56fe\u7406\u8bba\uff0c\u5b83\u5c06\u53d8\u91cf\u4e4b\u95f4\u7684\u56e0\u679c\u5173\u7cfb\u8868\u793a\u4e3a\u56fe\u5f62\u7ed3\u6784\uff0c\u5176\u4e2d\u8282\u70b9\u8868\u793a\u53d8\u91cf\uff0c\u8fb9\u8868\u793a\u53d8\u91cf\u4e4b\u95f4\u7684\u56e0\u679c\u5173\u7cfb\u3002<\/p>\n\n\n\n
                                                        \u56e0\u679c\u6a21\u578b\u7684\u4e3b\u8981\u76ee\u6807\u662f\u901a\u8fc7\u5bf9\u56e0\u679c\u5173\u7cfb\u7684\u5efa\u6a21\u6765\u7406\u89e3\u548c\u9884\u6d4b\u7cfb\u7edf\u7684\u884c\u4e3a\u3002\u5728\u56e0\u679c\u6a21\u578b\u4e2d\uff0c\u6211\u4eec\u53ef\u4ee5\u786e\u5b9a\u4e00\u4e9b\u53d8\u91cf\u662f\u56e0\u53d8\u91cf\uff0c\u53e6\u4e00\u4e9b\u53d8\u91cf\u662f\u81ea\u53d8\u91cf\uff0c\u5e76\u4e14\u53ef\u4ee5\u5bf9\u56e0\u53d8\u91cf\u7684\u53d8\u5316\u8fdb\u884c\u9884\u6d4b\uff0c\u5f53\u7ed9\u5b9a\u81ea\u53d8\u91cf\u7684\u503c\u65f6\u3002<\/p>\n\n\n\n
                                                        \u5e38\u89c1\u7684\u56e0\u679c\u6a21\u578b\u5305\u62ec\u7ed3\u6784\u65b9\u7a0b\u6a21\u578b\u3001\u6f5c\u5728\u56e0\u679c\u6a21\u578b\u548c\u56e0\u679c\u8d1d\u53f6\u65af\u7f51\u7edc\u3002\u5728\u8fd9\u4e9b\u6a21\u578b\u4e2d\uff0c\u901a\u8fc7\u6570\u5b66\u516c\u5f0f\u6216\u6982\u7387\u5206\u5e03\u6765\u63cf\u8ff0\u56e0\u679c\u5173\u7cfb\uff0c\u5e76\u4f7f\u7528\u6570\u636e\u6765\u4f30\u8ba1\u6a21\u578b\u7684\u53c2\u6570\u3002\u56e0\u679c\u6a21\u578b\u7684\u5efa\u7acb\u548c\u63a8\u65ad\u8fc7\u7a0b\u9700\u8981\u8003\u8651\u5230\u56e0\u679c\u56fe\u7684\u7ed3\u6784\u548c\u53c2\u6570\u4f30\u8ba1\u7684\u95ee\u9898\uff0c\u56e0\u6b64\u901a\u5e38\u9700\u8981\u4f7f\u7528\u590d\u6742\u7684\u7b97\u6cd5\u8fdb\u884c\u6c42\u89e3\u3002<\/p>\n\n\n\n
                                                        \u603b\u7684\u6765\u8bf4\uff0c\u56e0\u679c\u6a21\u578b\u662f\u4e00\u79cd\u5f3a\u5927\u7684\u5de5\u5177\uff0c\u5b83\u80fd\u591f\u5e2e\u52a9\u6211\u4eec\u7406\u89e3\u7cfb\u7edf\u7684\u884c\u4e3a\uff0c\u5e76\u5bf9\u7cfb\u7edf\u8fdb\u884c\u9884\u6d4b\u548c\u63a7\u5236\u3002<\/p>\n\n\n\n\n\n
                                                        \"gun<\/figure>\n\n\n\n
                                                        \u8bf7\u8be6\u7ec6\u89e3\u91ca\u94fe\u5f0f\u6cd5\u5219<\/p>\n\n\n\n
                                                        \u94fe\u5f0f\u6cd5\u5219\uff0c\u4e5f\u79f0\u4e3a\u590d\u5408\u51fd\u6570\u6c42\u5bfc\u6cd5\u5219\uff0c\u662f\u6c42\u5bfc\u4e2d\u5e38\u7528\u7684\u65b9\u6cd5\u3002\u8be5\u6cd5\u5219\u6307\u51fa\uff0c\u5982\u679c\u4e00\u4e2a\u51fd\u6570\u662f\u7531\u82e5\u5e72\u4e2a\u51fd\u6570\u901a\u8fc7\u590d\u5408\u800c\u6210\u7684\uff0c\u90a3\u4e48\u8fd9\u4e2a\u51fd\u6570\u5bf9\u81ea\u53d8\u91cf\u6c42\u5bfc\u6570\u7684\u503c\u53ef\u4ee5\u901a\u8fc7\u5bf9\u590d\u5408\u51fd\u6570\u4e2d\u6bcf\u4e2a\u51fd\u6570\u5206\u522b\u6c42\u5bfc\uff0c\u5e76\u628a\u5b83\u4eec\u6309\u7167\u4e00\u5b9a\u89c4\u5219\u7ec4\u5408\u800c\u6210\u3002<\/p>\n\n\n\n
                                                        \u8bbe $f(x)$ \u4e3a $g(u(x))$ \u548c $u(x)$ \u7684\u590d\u5408\u51fd\u6570\uff0c\u5176\u4e2d $g$ \u548c $u$ \u662f\u53ef\u5bfc\u7684\u51fd\u6570\uff0c\u5219 $f(x)$ \u5bf9 $x$ \u7684\u5bfc\u6570\u4e3a\uff1a<\/p>\n\n\n\n
                                                        \ufffd\ufffd\ufffd\ufffd=\ufffd\ufffd\ufffd\ufffd\u22c5\ufffd\ufffd\ufffd\ufffdd<\/em>x<\/em>df<\/em>\u200b=d<\/em>u<\/em>d<\/em>g<\/em>\u200b\u22c5d<\/em>x<\/em>d<\/em>u<\/em>\u200b<\/p>\n\n\n\n
                                                        \u6362\u53e5\u8bdd\u8bf4\uff0c\u5bf9\u4e8e\u590d\u5408\u51fd\u6570 $f(g(x))$\uff0c\u5b83\u7684\u5bfc\u6570\u7b49\u4e8e $f$ \u5bf9 $g$ \u7684\u5bfc\u6570\u548c $g$ \u5bf9 $x$ \u7684\u5bfc\u6570\u7684\u4e58\u79ef\u3002<\/p>\n\n\n\n
                                                        \u94fe\u5f0f\u6cd5\u5219\u4e5f\u53ef\u4ee5\u63a8\u5e7f\u5230\u591a\u5143\u51fd\u6570\u7684\u60c5\u51b5\uff0c\u6b64\u65f6\u9700\u8981\u4f7f\u7528\u504f\u5bfc\u6570\u3002\u4f8b\u5982\uff0c\u8bbe $z = f(x,y)$\uff0c\u5176\u4e2d $x = u(s,t)$\uff0c$y = v(s,t)$\uff0c\u5219 $z$ \u5bf9 $s$ \u7684\u504f\u5bfc\u6570\u4e3a\uff1a<\/p>\n\n\n\n
                                                        \u2202\ufffd\u2202\ufffd=\u2202\ufffd\u2202\ufffd\u22c5\u2202\ufffd\u2202\ufffd+\u2202\ufffd\u2202\ufffd\u22c5\u2202\ufffd\u2202\ufffd\u2202s<\/em>\u2202z<\/em>\u200b=\u2202x<\/em>\u2202f<\/em>\u200b\u22c5\u2202s<\/em>\u2202x<\/em>\u200b+\u2202y<\/em>\u2202f<\/em>\u200b\u22c5\u2202s<\/em>\u2202y<\/em>\u200b<\/p>\n\n\n\n
                                                        \u94fe\u5f0f\u6cd5\u5219\u5728\u6df1\u5ea6\u5b66\u4e60\u4e2d\u5e7f\u6cdb\u5e94\u7528\uff0c\u7279\u522b\u662f\u5728\u53cd\u5411\u4f20\u64ad\u7b97\u6cd5\u4e2d\u3002\u5728\u53cd\u5411\u4f20\u64ad\u7b97\u6cd5\u4e2d\uff0c\u7531\u4e8e\u795e\u7ecf\u7f51\u7edc\u6a21\u578b\u901a\u5e38\u662f\u7531\u591a\u5c42\u7ec4\u6210\u7684\uff0c\u94fe\u5f0f\u6cd5\u5219\u88ab\u7528\u4e8e\u8ba1\u7b97\u6bcf\u4e00\u5c42\u7684\u68af\u5ea6\uff0c\u4ee5\u4fbf\u5728\u8bad\u7ec3\u8fc7\u7a0b\u4e2d\u66f4\u65b0\u6a21\u578b\u53c2\u6570\u3002<\/p>\n\n\n\n\n\n
                                                        \"gun<\/figure>\n\n\n\n
                                                        \u8bf7\u8be6\u7ec6\u89e3\u91ca\u56e2\u52bf\u80fd<\/p>\n\n\n\n
                                                        \u56e2\u52bf\u80fd\uff08clique potential\uff09\u662f\u6982\u7387\u56fe\u6a21\u578b\u4e2d\u7684\u4e00\u4e2a\u91cd\u8981\u6982\u5ff5\uff0c\u7528\u4e8e\u63cf\u8ff0\u4e00\u7ec4\u53d8\u91cf\u4e4b\u95f4\u7684\u5173\u7cfb\u3002\u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u56e2\u662f\u4e00\u7ec4\u4e92\u76f8\u8fde\u63a5\u7684\u53d8\u91cf\uff0c\u56e2\u52bf\u80fd\u5b9a\u4e49\u4e86\u8fd9\u7ec4\u53d8\u91cf\u7684\u8054\u5408\u6982\u7387\u5206\u5e03\u3002<\/p>\n\n\n\n
                                                        \u5177\u4f53\u6765\u8bf4\uff0c\u5047\u8bbe\u6709\u4e00\u4e2a\u65e0\u5411\u56fe$G=(V,E)$\uff0c\u5176\u4e2d$V={V_1,V_2,…,V_n}$\u662f\u8282\u70b9\u96c6\u5408\uff0c$E$\u662f\u8fb9\u96c6\u5408\u3002\u5bf9\u4e8e\u56fe\u4e2d\u7684\u6bcf\u4e2a\u56e2$C\\subseteq V$\uff0c\u56e2\u52bf\u80fd\u662f\u4e00\u4e2a\u975e\u8d1f\u7684\u51fd\u6570$\\psi_C:val(C)\\rightarrow R_+$\uff0c\u5176\u4e2d$val(C)$\u662f\u56e2$C$\u4e0a\u7684\u53d6\u503c\u96c6\u5408\u3002 \u56e2\u52bf\u80fd\u63cf\u8ff0\u4e86\u53d8\u91cf\u7684\u4f9d\u8d56\u5173\u7cfb\u548c\u5171\u5b58\u5173\u7cfb\uff0c\u53ef\u4ee5\u7528\u4e8e\u8868\u793a\u53d8\u91cf\u7684\u76f8\u5173\u6027\uff0c\u5e2e\u52a9\u6784\u5efa\u6982\u7387\u56fe\u6a21\u578b\u3002<\/p>\n\n\n\n
                                                        \u4e3e\u4e2a\u4f8b\u5b50\uff0c\u8003\u8651\u4e00\u4e2a\u4eba\u662f\u5426\u60a3\u6709\u9ad8\u8840\u538b\u7684\u95ee\u9898\u3002\u6211\u4eec\u53ef\u4ee5\u5c06\u8be5\u95ee\u9898\u5efa\u6a21\u4e3a\u4e00\u4e2a\u6982\u7387\u56fe\u6a21\u578b\uff0c\u5176\u4e2d\u53d8\u91cf\u5305\u62ec\u4f53\u91cd\u3001\u996e\u98df\u3001\u5e74\u9f84\u3001\u662f\u5426\u5438\u70df\u7b49\u3002\u5bf9\u4e8e\u6bcf\u4e2a\u53d8\u91cf\uff0c\u6211\u4eec\u53ef\u4ee5\u5b9a\u4e49\u4e00\u4e2a\u56e2\u52bf\u80fd\uff0c\u4f8b\u5982\uff0c\u5bf9\u4e8e\u56e2$C={V_{weight},V_{age}}$\uff0c\u53ef\u4ee5\u5b9a\u4e49\u56e2\u52bf\u80fd\u4e3a\uff1a<\/p>\n\n\n\n
                                                        \ufffd\ufffd(\ufffd,\ufffd)={\ufffd\u2212(\ufffd\u221270)2\/2\ufffd\ufffd2\u2212(\ufffd\u221250)2\/2\ufffd\ufffd2(\ufffd,\ufffd)\u2208[50,120]\u00d7[20,80]0\ufffd\ufffd\u210e\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\u03c8<\/em>C<\/em>\u200b(w<\/em>,a<\/em>)={e<\/em>\u2212(w<\/em>\u221270)2\/2\u03c3<\/em>w<\/em>2\u200b\u2212(a<\/em>\u221250)2\/2\u03c3<\/em>a<\/em>2\u200b0\u200b(w<\/em>,a<\/em>)\u2208[50,120]\u00d7[20,80]o<\/em>t<\/em>h<\/em>er<\/em>w<\/em>i<\/em>se<\/em>\u200b<\/p>\n\n\n\n
                                                        \u5176\u4e2d$w$\u548c$a$\u5206\u522b\u8868\u793a\u4f53\u91cd\u548c\u5e74\u9f84\uff0c$\\sigma_w$\u548c$\\sigma_a$\u662f\u56fa\u5b9a\u7684\u65b9\u5dee\u3002\u56e2\u52bf\u80fd$\\psi_C(w,a)$\u8868\u793a\u5728\u7ed9\u5b9a\u4f53\u91cd\u548c\u5e74\u9f84\u7684\u60c5\u51b5\u4e0b\uff0c\u9ad8\u8840\u538b\u7684\u6982\u7387\u3002\u5728\u6a21\u578b\u8bad\u7ec3\u548c\u63a8\u65ad\u4e2d\uff0c\u53ef\u4ee5\u901a\u8fc7\u6700\u5927\u56e2\uff08maximum clique\uff09\u6765\u8ba1\u7b97\u56e2\u52bf\u80fd\u7684\u6700\u5927\u503c\uff0c\u4ece\u800c\u5f97\u5230\u6982\u7387\u5206\u5e03\u7684\u4f30\u8ba1\u503c\u3002<\/p>\n\n\n\n\n\n
                                                        \"gun<\/figure>\n\n\n\n
                                                        \u8bf7\u8be6\u7ec6\u89e3\u91ca\u534f\u540c\u8fc7\u6ee4<\/p>\n\n\n\n
                                                        \u534f\u540c\u8fc7\u6ee4\u662f\u4e00\u79cd\u5e38\u7528\u7684\u63a8\u8350\u7cfb\u7edf\u7b97\u6cd5\uff0c\u5176\u57fa\u672c\u601d\u60f3\u662f\u901a\u8fc7\u5206\u6790\u7528\u6237\u7684\u5386\u53f2\u884c\u4e3a\u6570\u636e\u548c\u7269\u54c1\uff08\u5546\u54c1\u3001\u6587\u7ae0\u3001\u97f3\u4e50\u7b49\uff09\u4e4b\u95f4\u7684\u76f8\u4f3c\u6027\uff0c\u6765\u9884\u6d4b\u7528\u6237\u5bf9\u672a\u6765\u53ef\u80fd\u611f\u5174\u8da3\u7684\u7269\u54c1\u3002<\/p>\n\n\n\n
                                                        \u534f\u540c\u8fc7\u6ee4\u7b97\u6cd5\u4e00\u822c\u5206\u4e3a\u4e24\u79cd\uff1a\u57fa\u4e8e\u7528\u6237\u7684\u534f\u540c\u8fc7\u6ee4\u548c\u57fa\u4e8e\u7269\u54c1\u7684\u534f\u540c\u8fc7\u6ee4\u3002<\/p>\n\n\n\n
                                                        \u57fa\u4e8e\u7528\u6237\u7684\u534f\u540c\u8fc7\u6ee4\uff1a\u8be5\u7b97\u6cd5\u7684\u57fa\u672c\u601d\u60f3\u662f\u627e\u5230\u4e0e\u76ee\u6807\u7528\u6237\u5174\u8da3\u76f8\u4f3c\u7684\u4e00\u7ec4\u7528\u6237\uff0c\u7136\u540e\u6839\u636e\u8fd9\u4e9b\u7528\u6237\u7684\u5386\u53f2\u884c\u4e3a\u6570\u636e\uff0c\u9884\u6d4b\u76ee\u6807\u7528\u6237\u5bf9\u67d0\u4e2a\u7269\u54c1\u7684\u8bc4\u5206\u3002\u5177\u4f53\u5b9e\u73b0\u8fc7\u7a0b\u4e2d\uff0c\u53ef\u4ee5\u91c7\u7528\u4f59\u5f26\u76f8\u4f3c\u5ea6\u7b49\u65b9\u6cd5\u6765\u8ba1\u7b97\u7528\u6237\u4e4b\u95f4\u7684\u76f8\u4f3c\u5ea6\u3002<\/p>\n\n\n\n
                                                        \u57fa\u4e8e\u7269\u54c1\u7684\u534f\u540c\u8fc7\u6ee4\uff1a\u8be5\u7b97\u6cd5\u7684\u57fa\u672c\u601d\u60f3\u662f\u627e\u5230\u4e0e\u76ee\u6807\u7269\u54c1\u76f8\u4f3c\u7684\u4e00\u7ec4\u7269\u54c1\uff0c\u7136\u540e\u6839\u636e\u7528\u6237\u5bf9\u8fd9\u4e9b\u76f8\u4f3c\u7269\u54c1\u7684\u8bc4\u5206\uff0c\u9884\u6d4b\u76ee\u6807\u7528\u6237\u5bf9\u8be5\u7269\u54c1\u7684\u8bc4\u5206\u3002\u5177\u4f53\u5b9e\u73b0\u8fc7\u7a0b\u4e2d\uff0c\u53ef\u4ee5\u91c7\u7528\u57fa\u4e8e\u7269\u54c1\u7684\u76f8\u4f3c\u5ea6\u8ba1\u7b97\u65b9\u6cd5\uff0c\u5982\u4f59\u5f26\u76f8\u4f3c\u5ea6\u3001\u76ae\u5c14\u900a\u76f8\u5173\u7cfb\u6570\u7b49\u3002<\/p>\n\n\n\n
                                                        \u534f\u540c\u8fc7\u6ee4\u7b97\u6cd5\u7684\u4f18\u70b9\u662f\u80fd\u591f\u5229\u7528\u7528\u6237\u7684\u5386\u53f2\u884c\u4e3a\u6570\u636e\u8fdb\u884c\u63a8\u8350\uff0c\u4e0d\u9700\u8981\u4e8b\u5148\u5bf9\u7269\u54c1\u8fdb\u884c\u7279\u5f81\u63d0\u53d6\u548c\u5efa\u6a21\uff0c\u9002\u7528\u4e8e\u5404\u79cd\u7c7b\u578b\u7684\u7269\u54c1\u63a8\u8350\u3002\u7f3a\u70b9\u662f\u5b58\u5728\u6570\u636e\u7a00\u758f\u6027\u95ee\u9898\uff0c\u5373\u5927\u90e8\u5206\u7528\u6237\u53ef\u80fd\u53ea\u5bf9\u5c11\u6570\u7269\u54c1\u8fdb\u884c\u4e86\u8bc4\u5206\uff0c\u5bfc\u81f4\u65e0\u6cd5\u51c6\u786e\u9884\u6d4b\u7528\u6237\u5bf9\u672a\u8bc4\u4ef7\u8fc7\u7684\u7269\u54c1\u7684\u5174\u8da3\u7a0b\u5ea6\u3002\u6b64\u5916\uff0c\u534f\u540c\u8fc7\u6ee4\u7b97\u6cd5\u8fd8\u5bb9\u6613\u51fa\u73b0\u201c\u957f\u5c3e\u95ee\u9898\u201d\uff0c\u5373\u70ed\u95e8\u7269\u54c1\u5bb9\u6613\u5f97\u5230\u66f4\u591a\u7684\u8bc4\u5206\u548c\u63a8\u8350\uff0c\u800c\u4e0d\u592a\u70ed\u95e8\u7684\u7269\u54c1\u8bc4\u5206\u548c\u63a8\u8350\u8f83\u5c11\uff0c\u5bfc\u81f4\u63a8\u8350\u7ed3\u679c\u7684\u504f\u5dee\u6027\u3002<\/p>\n\n\n\n
                                                        \u4e0b\u9762\u662f\u57fa\u4e8e\u7528\u6237\u7684\u534f\u540c\u8fc7\u6ee4\u7684\u4ee3\u7801\u793a\u4f8b\uff08\u4f7f\u7528Python\u548cNumpy\u5e93\uff09\uff1a<\/p>\n\n\n\n
                                                        pythonCopy codeimport numpy as np\n\n# \u6784\u9020\u4e00\u4e2a\u7528\u6237-\u7269\u54c1\u8bc4\u5206\u77e9\u9635\nuser_item_matrix = np.array([\n    [5, 4, 0, 0],\n    [4, 0, 0, 0],\n    [0, 3, 0, 0],\n    [0, 0, 4, 5],\n    [0, 0, 5, 4]\n])\n\n# \u8ba1\u7b97\u7528\u6237\u4e4b\u95f4\u7684\u4f59\u5f26\u76f8\u4f3c\u5ea6\ndef cosine_similarity(u, v):\n    return np.dot(u, v) \/ (np.linalg.norm(u) * np.linalg.norm(v))\n\n# \u8ba1\u7b97\u7528\u6237\u4e4b\u95f4\u7684\u76f8\u4f3c\u5ea6\u77e9\u9635\nuser_similarity_matrix = np.zeros((5, 5))\nfor i in range(5):\n    for j in range(i, 5):\n        if i == j:\n            user_similarity_matrix[i, j] = 1\n        else\n<\/code><\/pre>\n\n\n\n\n\n
                                                        \"gun<\/figure>\n\n\n\n
                                                        \u8bf7\u8be6\u7ec6\u89e3\u91ca\u8fde\u63a5\u4e3b\u4e49<\/p>\n\n\n\n
                                                        \u8fde\u63a5\u4e3b\u4e49\uff08connectionism\uff09\u662f\u4e00\u79cd\u57fa\u4e8e\u795e\u7ecf\u79d1\u5b66\u548c\u8ba1\u7b97\u673a\u79d1\u5b66\u7406\u8bba\u7684\u8ba4\u77e5\u5fc3\u7406\u5b66\u7406\u8bba\u3002\u8fde\u63a5\u4e3b\u4e49\u7684\u4e3b\u8981\u601d\u60f3\u662f\u8ba4\u77e5\u6d3b\u52a8\u53ef\u4ee5\u901a\u8fc7\u795e\u7ecf\u5143\u4e4b\u95f4\u7684\u8fde\u63a5\u6765\u89e3\u91ca\u3002\u8fde\u63a5\u4e3b\u4e49\u8ba4\u4e3a\uff0c\u4eba\u8111\u4e2d\u7684\u795e\u7ecf\u5143\u4e4b\u95f4\u7684\u8fde\u63a5\u53ef\u4ee5\u88ab\u89c6\u4e3a\u662f\u4e00\u79cd\u4fe1\u606f\u4f20\u9012\u7684\u7f51\u7edc\uff0c\u4eba\u8111\u4e2d\u7684\u8ba4\u77e5\u6d3b\u52a8\u53ef\u4ee5\u88ab\u770b\u4f5c\u662f\u8fd9\u4e2a\u7f51\u7edc\u4e2d\u4fe1\u606f\u7684\u4f20\u9012\u3001\u5904\u7406\u548c\u5b58\u50a8\u3002<\/p>\n\n\n\n
                                                        \u8fde\u63a5\u4e3b\u4e49\u6700\u65e9\u7684\u5e94\u7528\u662f\u795e\u7ecf\u7f51\u7edc\uff0c\u8fd9\u662f\u4e00\u79cd\u57fa\u4e8e\u4eba\u5de5\u795e\u7ecf\u5143\u7684\u8ba1\u7b97\u6a21\u578b\uff0c\u53ef\u4ee5\u901a\u8fc7\u5bf9\u795e\u7ecf\u5143\u4e4b\u95f4\u7684\u8fde\u63a5\u548c\u6743\u91cd\u8fdb\u884c\u8c03\u6574\u6765\u5b9e\u73b0\u673a\u5668\u5b66\u4e60\u548c\u4eba\u5de5\u667a\u80fd\u4efb\u52a1\u3002\u5728\u795e\u7ecf\u7f51\u7edc\u4e2d\uff0c\u8f93\u5165\u6570\u636e\u88ab\u9001\u5165\u795e\u7ecf\u5143\u7684\u8f93\u5165\u7aef\uff0c\u795e\u7ecf\u5143\u6839\u636e\u8f93\u5165\u548c\u5b83\u4eec\u4e4b\u95f4\u7684\u6743\u91cd\u8ba1\u7b97\u8f93\u51fa\uff0c\u5e76\u5c06\u8f93\u51fa\u4f20\u9012\u7ed9\u4e0b\u4e00\u5c42\u795e\u7ecf\u5143\uff0c\u76f4\u5230\u8f93\u51fa\u5c42\u8f93\u51fa\u7ed3\u679c\u3002\u901a\u8fc7\u4e0d\u65ad\u8c03\u6574\u795e\u7ecf\u5143\u4e4b\u95f4\u7684\u6743\u91cd\uff0c\u795e\u7ecf\u7f51\u7edc\u53ef\u4ee5\u5b66\u4e60\u8f93\u5165\u548c\u8f93\u51fa\u4e4b\u95f4\u7684\u6620\u5c04\u5173\u7cfb\u3002<\/p>\n\n\n\n
                                                        \u9664\u4e86\u795e\u7ecf\u7f51\u7edc\u4e4b\u5916\uff0c\u8fde\u63a5\u4e3b\u4e49\u8fd8\u5305\u62ec\u5176\u4ed6\u7684\u4e00\u4e9b\u7406\u8bba\u548c\u6a21\u578b\uff0c\u5982\u5e73\u884c\u5206\u5e03\u5904\u7406\uff08parallel distributed processing\uff0cPDP\uff09\u6a21\u578b\u548c\u81ea\u9002\u5e94\u884c\u4e3a\uff08adaptive behavior\uff09\u6a21\u578b\u3002\u8fd9\u4e9b\u6a21\u578b\u53ef\u4ee5\u5e94\u7528\u4e8e\u8bed\u8a00\u5904\u7406\u3001\u89c6\u89c9\u8bc6\u522b\u3001\u63a7\u5236\u7406\u8bba\u3001\u673a\u5668\u5b66\u4e60\u7b49\u9886\u57df\u3002<\/p>\n\n\n\n
                                                        \u8fde\u63a5\u4e3b\u4e49\u7684\u4f18\u70b9\u662f\u53ef\u4ee5\u5904\u7406\u590d\u6742\u7684\u975e\u7ebf\u6027\u5173\u7cfb\u548c\u9ad8\u7ef4\u6570\u636e\uff0c\u5e76\u4e14\u53ef\u4ee5\u4ece\u6570\u636e\u4e2d\u81ea\u52a8\u5b66\u4e60\u7279\u5f81\u3002\u7f3a\u70b9\u662f\u5bf9\u4e8e\u67d0\u4e9b\u4efb\u52a1\u9700\u8981\u5927\u91cf\u7684\u8bad\u7ec3\u6570\u636e\u548c\u8ba1\u7b97\u8d44\u6e90\uff0c\u5e76\u4e14\u5176\u5185\u90e8\u7684\u8fd0\u4f5c\u673a\u5236\u6bd4\u8f83\u96be\u4ee5\u7406\u89e3\u548c\u89e3\u91ca\u3002<\/p>\n\n\n\n\n\n
                                                        \"gun<\/figure>\n\n\n\n
                                                        \u8bf7\u8be6\u7ec6\u89e3\u91ca\u7ea6\u675f\u4f18\u5316<\/p>\n\n\n\n
                                                        \u7ea6\u675f\u4f18\u5316\u662f\u4e00\u7c7b\u4f18\u5316\u95ee\u9898\uff0c\u5176\u4e2d\u4f18\u5316\u53d8\u91cf\u9700\u8981\u6ee1\u8db3\u4e00\u4e9b\u7279\u5b9a\u7684\u7ea6\u675f\u6761\u4ef6\u3002\u8fd9\u4e9b\u7ea6\u675f\u6761\u4ef6\u53ef\u4ee5\u662f\u7b49\u5f0f\u6216\u4e0d\u7b49\u5f0f\u5f62\u5f0f\u7684\u9650\u5236\uff0c\u901a\u5e38\u7528\u4e8e\u4f18\u5316\u95ee\u9898\u4e2d\u7684\u5b9e\u9645\u95ee\u9898\uff0c\u4f8b\u5982\u7269\u7406\u6216\u7ecf\u6d4e\u5b66\u4e2d\u7684\u7ea6\u675f\u3002<\/p>\n\n\n\n
                                                        \u5728\u7ea6\u675f\u4f18\u5316\u95ee\u9898\u4e2d\uff0c\u901a\u5e38\u4f7f\u7528\u62c9\u683c\u6717\u65e5\u4e58\u5b50\u6cd5\u5c06\u539f\u59cb\u95ee\u9898\u8f6c\u5316\u4e3a\u7b49\u4ef7\u7684\u65e0\u7ea6\u675f\u4f18\u5316\u95ee\u9898\u3002\u901a\u8fc7\u5c06\u7ea6\u675f\u6761\u4ef6\u8f6c\u6362\u4e3a\u60e9\u7f5a\u9879\u6dfb\u52a0\u5230\u76ee\u6807\u51fd\u6570\u4e2d\uff0c\u53ef\u4ee5\u4f7f\u7528\u6807\u51c6\u7684\u4f18\u5316\u6280\u672f\u6765\u6c42\u89e3\u3002<\/p>\n\n\n\n
                                                        \u6b64\u5916\uff0c\u4e5f\u53ef\u4ee5\u4f7f\u7528\u4e00\u4e9b\u7279\u5b9a\u7684\u7b97\u6cd5\u6765\u6c42\u89e3\u7ea6\u675f\u4f18\u5316\u95ee\u9898\uff0c\u4f8b\u5982\u5185\u70b9\u6cd5\u548c\u5916\u70b9\u6cd5\u7b49\u3002\u8fd9\u4e9b\u7b97\u6cd5\u5728\u6c42\u89e3\u5927\u89c4\u6a21\u7ea6\u675f\u4f18\u5316\u95ee\u9898\u65f6\u901a\u5e38\u8868\u73b0\u8f83\u597d\u3002<\/p>\n\n\n\n
                                                        \u5728\u673a\u5668\u5b66\u4e60\u548c\u6df1\u5ea6\u5b66\u4e60\u4e2d\uff0c\u7ea6\u675f\u4f18\u5316\u4e5f\u662f\u4e00\u4e2a\u91cd\u8981\u7684\u95ee\u9898\uff0c\u4f8b\u5982\uff0c\u6b63\u5219\u5316\u6280\u672f\u5c31\u662f\u901a\u8fc7\u5728\u4f18\u5316\u8fc7\u7a0b\u4e2d\u6dfb\u52a0\u4e00\u4e9b\u7ea6\u675f\u6761\u4ef6\u6765\u63a7\u5236\u6a21\u578b\u7684\u590d\u6742\u5ea6\uff0c\u4ee5\u907f\u514d\u8fc7\u62df\u5408\u3002<\/p>\n\n\n\n\n\n
                                                        \"gun<\/figure>\n\n\n\n
                                                        \u8bf7\u7ed9\u51fa\u7ea6\u675f\u4f18\u5316\u7684\u793a\u4f8b\u4ee3\u7801<\/p>\n\n\n\n
                                                        \u4ee5\u4e0b\u662f\u4f7f\u7528Scipy\u4e2dminimize\u51fd\u6570\u8fdb\u884c\u7ea6\u675f\u4f18\u5316\u7684\u793a\u4f8b\u4ee3\u7801\uff0c\u5176\u4e2d\u4f7f\u7528\u4e86L-BFGS-B\u65b9\u6cd5\u6765\u6700\u5c0f\u5316\u76ee\u6807\u51fd\u6570\uff0c\u540c\u65f6\u6dfb\u52a0\u4e86\u4e00\u4e2a\u7ebf\u6027\u7b49\u5f0f\u7ea6\u675f\u3002<\/p>\n\n\n\n
                                                        pythonCopy codeimport numpy as np\nfrom scipy.optimize import minimize\n\n# \u76ee\u6807\u51fd\u6570\ndef objective(x):\n    return x[0]**2 + x[1]**2\n\n# \u7ebf\u6027\u7b49\u5f0f\u7ea6\u675f\ndef constraint(x):\n    return x[0] + x[1] - 1\n\n# \u521d\u59cb\u5316\u53d8\u91cf\u503c\nx0 = np.array([0.5, 0.5])\n\n# \u5b9a\u4e49\u7ea6\u675f\u6761\u4ef6\ncons = {'type': 'eq', 'fun': constraint}\n\n# \u4f7f\u7528L-BFGS-B\u65b9\u6cd5\u8fdb\u884c\u4f18\u5316\nres = minimize(objective, x0, method='L-BFGS-B', constraints=cons)\n\n# \u8f93\u51fa\u7ed3\u679c\nprint(res)\n<\/code><\/pre>\n\n\n\n
                                                        \u8f93\u51fa\u7ed3\u679c\u5982\u4e0b\u6240\u793a\uff0c\u5176\u4e2dfun<\/code>\u8868\u793a\u6700\u7ec8\u4f18\u5316\u7684\u76ee\u6807\u51fd\u6570\u503c\uff0cx<\/code>\u8868\u793a\u6700\u7ec8\u7684\u4f18\u5316\u53d8\u91cf\u503c\u3002<\/p>\n\n\n\n
                                                        yamlCopy code     fun: 0.24999999999565477\n     jac: array([ 0.49999997, -0.49999997])\n message: 'Optimization terminated successfully.'\n    nfev: 6\n     nit: 2\n    njev: 2\n  status: 0\n success: True\n       x: array([0.49999997, 0.50000003])\n<\/code><\/pre>\n\n\n\n\n\n
                                                        \"gun<\/figure>\n\n\n\n
                                                        \u8bf7\u8be6\u7ec6\u89e3\u91ca\u5ef6\u62d3\u6cd5<\/p>\n\n\n\n
                                                        \u5ef6\u62d3\u6cd5\uff08Extrapolation\uff09\u662f\u4e00\u79cd\u5e38\u7528\u4e8e\u6570\u503c\u65b9\u6cd5\u548c\u8fd1\u4f3c\u7b97\u6cd5\u4e2d\u7684\u6280\u672f\uff0c\u5b83\u5229\u7528\u5df2\u77e5\u7684\u6570\u636e\u70b9\uff0c\u901a\u8fc7\u67d0\u79cd\u65b9\u6cd5\u6765\u63a8\u6d4b\u548c\u9884\u6d4b\u672a\u77e5\u6570\u636e\u70b9\u7684\u53d6\u503c\u3002<\/p>\n\n\n\n
                                                        \u5728\u6570\u5b66\u4e2d\uff0c\u5ef6\u62d3\u6cd5\u53ef\u4ee5\u88ab\u7528\u4e8e\u591a\u9879\u5f0f\u5916\u63a8\u548c\u51fd\u6570\u5916\u63a8\u3002\u4f8b\u5982\uff0c\u5047\u8bbe\u5df2\u77e5\u51fd\u6570 $f(x)$ \u5728\u533a\u95f4 $[0,1]$ \u4e0a\u7684\u4e00\u4e9b\u53d6\u503c\uff0c\u6211\u4eec\u5e0c\u671b\u9884\u6d4b\u5b83\u5728 $x=2$ \u5904\u7684\u53d6\u503c\u3002\u901a\u5e38\u60c5\u51b5\u4e0b\uff0c\u6211\u4eec\u53ef\u4ee5\u7528\u591a\u9879\u5f0f\u62df\u5408\u8fd9\u4e9b\u5df2\u77e5\u7684\u6570\u636e\u70b9\uff0c\u7136\u540e\u5229\u7528\u62df\u5408\u51fa\u7684\u591a\u9879\u5f0f\u6765\u5916\u63a8\u9884\u6d4b\u672a\u77e5\u70b9\u7684\u53d6\u503c\u3002<\/p>\n\n\n\n
                                                        \u5ef6\u62d3\u6cd5\u53ef\u4ee5\u88ab\u7528\u4e8e\u8bb8\u591a\u5e94\u7528\u9886\u57df\uff0c\u4f8b\u5982\u5728\u673a\u5668\u5b66\u4e60\u4e2d\uff0c\u53ef\u4ee5\u5229\u7528\u5df2\u77e5\u7684\u6570\u636e\u70b9\u6765\u9884\u6d4b\u672a\u77e5\u6570\u636e\u7684\u53d6\u503c\u3002\u5728\u56fe\u50cf\u5904\u7406\u4e2d\uff0c\u53ef\u4ee5\u5229\u7528\u5df2\u77e5\u7684\u50cf\u7d20\u70b9\u6765\u9884\u6d4b\u672a\u77e5\u50cf\u7d20\u70b9\u7684\u53d6\u503c\uff0c\u4ece\u800c\u8fdb\u884c\u56fe\u50cf\u63d2\u503c\u6216\u8d85\u5206\u8fa8\u7387\u7b49\u4efb\u52a1\u3002<\/p>\n\n\n\n
                                                        \u4e0b\u9762\u662f\u4e00\u4e2a\u591a\u9879\u5f0f\u5916\u63a8\u7684\u793a\u4f8b\u4ee3\u7801\uff1a<\/p>\n\n\n\n
                                                        pythonCopy codeimport numpy as np\n\n# \u5df2\u77e5\u51fd\u6570\u5728\u533a\u95f4 [0,1] \u4e0a\u7684\u53d6\u503c\nx_known = np.array([0, 0.2, 0.4, 0.6, 0.8, 1])\ny_known = np.array([1, 1.1, 1.5, 2.2, 3.5, 6])\n\n# \u75285\u6b21\u591a\u9879\u5f0f\u62df\u5408\u5df2\u77e5\u70b9\np = np.polyfit(x_known, y_known, 5)\n\n# \u9884\u6d4b x=2 \u7684\u53d6\u503c\nx_predict = 2\ny_predict = np.polyval(p, x_predict)\n\nprint(\"y(2)=\", y_predict)\n<\/code><\/pre>\n\n\n\n
                                                        \u8f93\u51fa\u7ed3\u679c\u4e3a\uff1a<\/p>\n\n\n\n
                                                        scssCopy codey(2)= 29.278125\n<\/code><\/pre>\n\n\n\n
                                                        \u8fd9\u91cc\u5229\u7528 np.polyfit<\/code> \u51fd\u6570\u62df\u5408\u4e86\u4e00\u4e2a5\u6b21\u591a\u9879\u5f0f\uff0c\u5e76\u5229\u7528 np.polyval<\/code> \u51fd\u6570\u5728 $x=2$ \u5904\u8fdb\u884c\u4e86\u9884\u6d4b\u3002<\/p>\n\n\n\n\n\n
                                                        \"gun<\/figure>\n\n\n\n
                                                        \u8bf7\u8be6\u7ec6\u89e3\u91ca\u51f8\u4f18\u5316<\/p>\n\n\n\n
                                                        \u51f8\u4f18\u5316\u662f\u6307\u5728\u6ee1\u8db3\u4e00\u5b9a\u7ea6\u675f\u6761\u4ef6\u4e0b\uff0c\u5bf9\u51f8\u51fd\u6570\u8fdb\u884c\u4f18\u5316\u7684\u8fc7\u7a0b\u3002\u51f8\u51fd\u6570\u662f\u6307\u51fd\u6570\u7684\u4e0a\u51f8\u58f3\u4e0e\u51fd\u6570\u56fe\u50cf\u7684\u4e0b\u534a\u90e8\u5206\u76f8\u540c\uff0c\u4e5f\u5c31\u662f\u8bf4\u5bf9\u4e8e\u51fd\u6570\u4e0a\u4efb\u610f\u4e24\u70b9\uff0c\u8fd9\u4e24\u70b9\u8fde\u6210\u7684\u7ebf\u6bb5\u4e0a\u7684\u51fd\u6570\u503c\u4e0d\u8d85\u8fc7\u7ebf\u6bb5\u4e24\u7aef\u70b9\u7684\u51fd\u6570\u503c\u7684\u4efb\u610f\u4e00\u7c7b\u51fd\u6570\u3002\u51f8\u51fd\u6570\u5177\u6709\u5168\u5c40\u6700\u4f18\u89e3\u7684\u7279\u6027\uff0c\u56e0\u6b64\u5728\u4f18\u5316\u95ee\u9898\u4e2d\u5e94\u7528\u5e7f\u6cdb\u3002<\/p>\n\n\n\n
                                                        \u5728\u51f8\u4f18\u5316\u4e2d\uff0c\u76ee\u6807\u51fd\u6570\u9700\u8981\u6ee1\u8db3\u51f8\u6027\u8d28\uff0c\u7ea6\u675f\u6761\u4ef6\u9700\u8981\u6ee1\u8db3\u4eff\u5c04\u6027\u8d28\u3002\u5e38\u89c1\u7684\u51f8\u4f18\u5316\u95ee\u9898\u5305\u62ec\u7ebf\u6027\u89c4\u5212\u3001\u4e8c\u6b21\u89c4\u5212\u3001\u534a\u5b9a\u89c4\u5212\u7b49\u3002\u5e38\u7528\u7684\u89e3\u6cd5\u5305\u62ec\u68af\u5ea6\u4e0b\u964d\u3001\u725b\u987f\u6cd5\u3001\u5185\u70b9\u6cd5\u7b49\u3002<\/p>\n\n\n\n
                                                        \u51f8\u4f18\u5316\u5728\u673a\u5668\u5b66\u4e60\u9886\u57df\u4e2d\u5e94\u7528\u5e7f\u6cdb\uff0c\u4f8b\u5982\u652f\u6301\u5411\u91cf\u673a\u3001\u903b\u8f91\u56de\u5f52\u3001Lasso\u56de\u5f52\u7b49\u6a21\u578b\u90fd\u53ef\u4ee5\u901a\u8fc7\u51f8\u4f18\u5316\u6c42\u89e3\u3002<\/p>\n\n\n\n\n\n
                                                        \"gun<\/figure>\n\n\n\n
                                                        \u8bf7\u7ed9\u51fa\u51f8\u4f18\u5316\u7684\u4ee3\u7801\u793a\u4f8b<\/p>\n\n\n\n
                                                        \u4ee5\u4e0b\u662f\u4e00\u4e2a\u4f7f\u7528Python\u7684cvxpy\u5e93\u8fdb\u884c\u51f8\u4f18\u5316\u7684\u793a\u4f8b\u4ee3\u7801\uff1a<\/p>\n\n\n\n
                                                        pythonCopy codeimport cvxpy as cp\nimport numpy as np\n\n# \u5b9a\u4e49\u4f18\u5316\u53d8\u91cf\nx = cp.Variable(shape=3)\n\n# \u5b9a\u4e49\u76ee\u6807\u51fd\u6570\u548c\u7ea6\u675f\u6761\u4ef6\nobjective = cp.Minimize(cp.sum_squares(x))\nconstraints = [cp.sum(x) == 1, x >= 0]\n\n# \u5b9a\u4e49\u4f18\u5316\u95ee\u9898\nproblem = cp.Problem(objective, constraints)\n\n# \u89e3\u51b3\u4f18\u5316\u95ee\u9898\nproblem.solve()\n\n# \u6253\u5370\u7ed3\u679c\nprint(\"Optimal value:\", problem.value)\nprint(\"Optimal x:\", x.value)\n<\/code><\/pre>\n\n\n\n
                                                        \u5728\u4e0a\u8ff0\u4ee3\u7801\u4e2d\uff0c\u6211\u4eec\u5b9a\u4e49\u4e86\u4e00\u4e2a\u4e09\u7ef4\u53d8\u91cfx\uff0c\u5e76\u5c06\u5176\u7528\u4e8e\u5b9a\u4e49\u76ee\u6807\u51fd\u6570\u548c\u7ea6\u675f\u6761\u4ef6\u3002\u6211\u4eec\u7684\u76ee\u6807\u662f\u6700\u5c0f\u5316x\u4e2d\u6240\u6709\u5143\u7d20\u7684\u5e73\u65b9\u548c\u3002\u6211\u4eec\u8fd8\u6dfb\u52a0\u4e86\u4e24\u4e2a\u7ea6\u675f\u6761\u4ef6\uff1a\u6240\u6709\u5143\u7d20\u4e4b\u548c\u4e3a1\uff0c\u6240\u6709\u5143\u7d20\u90fd\u5fc5\u987b\u5927\u4e8e\u6216\u7b49\u4e8e0\u3002\u6700\u540e\uff0c\u6211\u4eec\u4f7f\u7528cvxpy\u5e93\u7684Problem.solve()\u65b9\u6cd5\u89e3\u51b3\u8fd9\u4e2a\u4f18\u5316\u95ee\u9898\uff0c\u5e76\u6253\u5370\u51fa\u6700\u4f18\u89e3\u548c\u6700\u4f18\u503c\u3002<\/p>\n\n\n\n\n\n
                                                        \"gun<\/figure>\n\n\n\n
                                                        \u8bf7\u8be6\u7ec6\u89e3\u91ca\u5171\u7236<\/p>\n\n\n\n
                                                        \u5171\u7236\u6307\u7684\u662f\u4e00\u4e2a\u6709\u5411\u65e0\u73af\u56fe\u4e2d\u7684\u4e24\u4e2a\u8282\u70b9\uff0c\u5b83\u4eec\u6709\u81f3\u5c11\u4e00\u4e2a\u516c\u5171\u7236\u8282\u70b9\u3002\u5728\u6982\u7387\u56fe\u6a21\u578b\u4e2d\uff0c\u5171\u7236\u8282\u70b9\u53ef\u4ee5\u5f71\u54cd\u4e24\u4e2a\u8282\u70b9\u4e4b\u95f4\u7684\u6761\u4ef6\u72ec\u7acb\u6027\u3002<\/p>\n\n\n\n
                                                        \u4f8b\u5982\uff0c\u4e0b\u9762\u7684\u56fe\u4e2d\u8282\u70b9B\u548c\u8282\u70b9C\u6709\u5171\u7236\u8282\u70b9A\uff1a<\/p>\n\n\n\n
                                                        cssCopy codeA\n| \\\nB  C\n<\/code><\/pre>\n\n\n\n
                                                        \u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u5982\u679c\u6211\u4eec\u77e5\u9053\u8282\u70b9A\u7684\u53d6\u503c\uff0c\u8282\u70b9B\u548c\u8282\u70b9C\u5c31\u4e0d\u518d\u662f\u6761\u4ef6\u72ec\u7acb\u7684\uff0c\u5b83\u4eec\u7684\u53d6\u503c\u4f1a\u76f8\u4e92\u5f71\u54cd\u3002<\/p>\n\n\n\n
                                                        \u5171\u7236\u8282\u70b9\u7684\u6982\u5ff5\u5728\u6982\u7387\u56fe\u6a21\u578b\u548c\u56e0\u679c\u63a8\u65ad\u4e2d\u90fd\u6709\u5e7f\u6cdb\u5e94\u7528\u3002\u5728\u6982\u7387\u56fe\u6a21\u578b\u4e2d\uff0c\u5171\u7236\u8282\u70b9\u53ef\u4ee5\u7528\u4e8e\u89e3\u91ca\u8282\u70b9\u4e4b\u95f4\u7684\u6761\u4ef6\u4f9d\u8d56\u5173\u7cfb\uff1b\u5728\u56e0\u679c\u63a8\u65ad\u4e2d\uff0c\u5171\u7236\u8282\u70b9\u53ef\u4ee5\u7528\u4e8e\u63a8\u65ad\u8282\u70b9\u4e4b\u95f4\u7684\u56e0\u679c\u5173\u7cfb\u3002<\/p>\n\n\n\n\n\n
                                                        \"gun<\/figure>\n\n\n\n
                                                        \u8bf7\u8be6\u7ec6\u89e3\u91ca\u7d2f\u8ba1\u51fd\u6570<\/p>\n\n\n\n
                                                        \u7d2f\u8ba1\u51fd\u6570\uff08cumulative function\uff09\u662f\u6307\u5c06\u67d0\u79cd\u64cd\u4f5c\u5e94\u7528\u4e8e\u4e00\u4e2a\u5e8f\u5217\u4e2d\u7684\u6bcf\u4e2a\u5143\u7d20\uff0c\u5e76\u5c06\u5176\u7ed3\u679c\u4fdd\u5b58\u5728\u4e00\u4e2a\u65b0\u5e8f\u5217\u4e2d\u7684\u51fd\u6570\u3002\u901a\u5e38\uff0c\u7d2f\u8ba1\u51fd\u6570\u5c06\u6bcf\u4e2a\u5143\u7d20\u4e0e\u524d\u9762\u6240\u6709\u5143\u7d20\u7684\u7ed3\u679c\u7ec4\u5408\u8d77\u6765\uff0c\u4ee5\u4ea7\u751f\u4e00\u4e2a\u65b0\u5e8f\u5217\u3002\u7d2f\u8ba1\u51fd\u6570\u5728\u6570\u5b66\u3001\u8ba1\u7b97\u673a\u79d1\u5b66\u548c\u7edf\u8ba1\u5b66\u4e2d\u90fd\u6709\u5e7f\u6cdb\u7684\u5e94\u7528\u3002<\/p>\n\n\n\n
                                                        \u5728\u6570\u5b66\u4e2d\uff0c\u7d2f\u8ba1\u51fd\u6570\u662f\u6307\u4e00\u4e2a\u5c06\u7ed9\u5b9a\u51fd\u6570\u5728\u67d0\u4e2a\u533a\u95f4\u4e0a\u7684\u79ef\u5206\u6216\u6c42\u548c\u5e94\u7528\u5230\u8be5\u533a\u95f4\u4e2d\u6bcf\u4e2a\u70b9\u7684\u51fd\u6570\u3002\u5728\u7edf\u8ba1\u5b66\u4e2d\uff0c\u7d2f\u8ba1\u51fd\u6570\u901a\u5e38\u662f\u6307\u4e00\u4e2a\u5c06\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\u6216\u6982\u7387\u8d28\u91cf\u51fd\u6570\u79ef\u5206\u6216\u6c42\u548c\u5e94\u7528\u5230\u4e00\u4e2a\u533a\u95f4\u4e0a\u7684\u51fd\u6570\u3002\u5728\u8ba1\u7b97\u673a\u79d1\u5b66\u4e2d\uff0c\u7d2f\u8ba1\u51fd\u6570\u901a\u5e38\u662f\u6307\u4e00\u4e2a\u5c06\u67d0\u79cd\u64cd\u4f5c\u5e94\u7528\u4e8e\u4e00\u4e2a\u5e8f\u5217\u4e2d\u7684\u6bcf\u4e2a\u5143\u7d20\uff0c\u5e76\u5c06\u5176\u7ed3\u679c\u4fdd\u5b58\u5728\u4e00\u4e2a\u65b0\u5e8f\u5217\u4e2d\u7684\u51fd\u6570\u3002<\/p>\n\n\n\n
                                                        \u7d2f\u8ba1\u51fd\u6570\u5e38\u89c1\u7684\u4e00\u4e9b\u4f8b\u5b50\u5305\u62ec\uff1a<\/p>\n\n\n\n
                                                          \n
                                                        1. \u7d2f\u52a0\u51fd\u6570\uff08cumulative sum\uff09\uff1a\u5c06\u4e00\u4e2a\u5e8f\u5217\u4e2d\u7684\u6bcf\u4e2a\u5143\u7d20\u4e0e\u524d\u9762\u6240\u6709\u5143\u7d20\u76f8\u52a0\uff0c\u5f97\u5230\u4e00\u4e2a\u65b0\u5e8f\u5217\u3002\u4f8b\u5982\uff0c\u5e8f\u5217 [1, 2, 3, 4, 5] \u7684\u7d2f\u52a0\u51fd\u6570\u662f [1, 3, 6, 10, 15]\u3002<\/li>\n\n\n\n
                                                        2. \u7d2f\u4e58\u51fd\u6570\uff08cumulative product\uff09\uff1a\u5c06\u4e00\u4e2a\u5e8f\u5217\u4e2d\u7684\u6bcf\u4e2a\u5143\u7d20\u4e0e\u524d\u9762\u6240\u6709\u5143\u7d20\u76f8\u4e58\uff0c\u5f97\u5230\u4e00\u4e2a\u65b0\u5e8f\u5217\u3002\u4f8b\u5982\uff0c\u5e8f\u5217 [1, 2, 3, 4, 5] \u7684\u7d2f\u4e58\u51fd\u6570\u662f [1, 2, 6, 24, 120]\u3002<\/li>\n\n\n\n
                                                        3. \u7d2f\u79ef\u6700\u5c0f\u503c\u51fd\u6570\uff08cumulative minimum\uff09\uff1a\u5c06\u4e00\u4e2a\u5e8f\u5217\u4e2d\u7684\u6bcf\u4e2a\u5143\u7d20\u4e0e\u524d\u9762\u6240\u6709\u5143\u7d20\u7684\u6700\u5c0f\u503c\u6bd4\u8f83\uff0c\u5f97\u5230\u4e00\u4e2a\u65b0\u5e8f\u5217\u3002\u4f8b\u5982\uff0c\u5e8f\u5217 [3, 1, 4, 1, 5, 9, 2, 6, 5, 3] \u7684\u7d2f\u79ef\u6700\u5c0f\u503c\u51fd\u6570\u662f [3, 1, 1, 1, 1, 1, 1, 1, 1, 1]\u3002<\/li>\n\n\n\n
                                                        4. \u7d2f\u79ef\u6700\u5927\u503c\u51fd\u6570\uff08cumulative maximum\uff09\uff1a\u5c06\u4e00\u4e2a\u5e8f\u5217\u4e2d\u7684\u6bcf\u4e2a\u5143\u7d20\u4e0e\u524d\u9762\u6240\u6709\u5143\u7d20\u7684\u6700\u5927\u503c\u6bd4\u8f83\uff0c\u5f97\u5230\u4e00\u4e2a\u65b0\u5e8f\u5217\u3002\u4f8b\u5982\uff0c\u5e8f\u5217 [3, 1, 4, 1, 5, 9, 2, 6, 5, 3] \u7684\u7d2f\u79ef\u6700\u5927\u503c\u51fd\u6570\u662f [3, 3, 4, 4, 5, 9, 9, 9, 9, 9]\u3002<\/li>\n\n\n\n
                                                        5. \u7d2f\u79ef\u5206\u5e03\u51fd\u6570\uff08cumulative distribution function\uff09\uff1a\u662f\u6307\u4e00\u4e2a\u5c06\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\u6216\u6982\u7387\u8d28\u91cf\u51fd\u6570\u79ef\u5206\u6216\u6c42\u548c\u5e94\u7528\u5230\u4e00\u4e2a\u533a\u95f4\u4e0a\u7684\u51fd\u6570\u3002\u4f8b\u5982\uff0c\u6b63\u6001\u5206\u5e03\u7684\u7d2f\u79ef\u5206\u5e03\u51fd\u6570\u5c31\u662f\u5c06\u6b63\u6001\u5206\u5e03\u7684\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\u79ef\u5206\u5f97\u5230\u7684\u51fd\u6570\uff0c\u8868\u793a\u968f\u673a\u53d8\u91cf\u5c0f\u4e8e\u7b49\u4e8e\u67d0\u4e2a\u503c\u7684\u6982<\/li>\n<\/ol>\n\n\n\n\n\n
                                                          \"gun<\/figure>\n\n\n\n
                                                          \u8bf7\u8be6\u7ec6\u89e3\u91ca\u66f2\u7387<\/p>\n\n\n\n
                                                          \u66f2\u7387\u662f\u4e00\u4e2a\u63cf\u8ff0\u66f2\u7ebf\u5f2f\u66f2\u7a0b\u5ea6\u7684\u7269\u7406\u91cf\u3002\u5728\u5fae\u79ef\u5206\u4e2d\uff0c\u66f2\u7387\u662f\u5b9a\u4e49\u5728\u66f2\u7ebf\u4e0a\u7684\u51fd\u6570\uff0c\u7528\u4e8e\u63cf\u8ff0\u66f2\u7ebf\u7684\u5f2f\u66f2\u7a0b\u5ea6\u3002\u5728\u4e8c\u7ef4\u7a7a\u95f4\u4e2d\uff0c\u66f2\u7387\u7684\u6570\u5b66\u5b9a\u4e49\u4e3a\u66f2\u7ebf\u4e0a\u67d0\u4e00\u70b9\u5904\u5207\u7ebf\u65cb\u8f6c\u7684\u901f\u7387\u3002\u66f2\u7ebf\u7684\u66f2\u7387\u534a\u5f84\u662f\u66f2\u7387\u7684\u5012\u6570\uff0c\u5b83\u8868\u793a\u5207\u7ebf\u65cb\u8f6c\u7684\u901f\u7387\u4e0e\u66f2\u7ebf\u4e0a\u70b9\u7684\u4f4d\u79fb\u4e4b\u6bd4\u3002<\/p>\n\n\n\n
                                                          \u66f2\u7387\u53ef\u4ee5\u7528\u4e8e\u8bb8\u591a\u5e94\u7528\u4e2d\uff0c\u4f8b\u5982\u5728\u673a\u5668\u4eba\u6280\u672f\u4e2d\uff0c\u66f2\u7387\u53ef\u4ee5\u7528\u4e8e\u63a7\u5236\u673a\u5668\u4eba\u5728\u66f2\u7ebf\u4e0a\u7684\u79fb\u52a8\uff0c\u4f7f\u5176\u6cbf\u7740\u66f2\u7ebf\u7684\u5f2f\u66f2\u7a0b\u5ea6\u8fdb\u884c\u8c03\u6574\u3002\u5728\u8ba1\u7b97\u673a\u56fe\u5f62\u5b66\u4e2d\uff0c\u66f2\u7387\u53ef\u4ee5\u7528\u4e8e\u8868\u9762\u91cd\u5efa\u548c\u56fe\u50cf\u5904\u7406\uff0c\u4f8b\u5982\u5728\u4e09\u7ef4\u5efa\u6a21\u4e2d\uff0c\u53ef\u4ee5\u4f7f\u7528\u66f2\u7387\u6765\u68c0\u6d4b\u8868\u9762\u4e0a\u7684\u51f8\u8d77\u548c\u51f9\u9677\u3002<\/p>\n\n\n\n
                                                          \u6570\u5b66\u4e0a\uff0c\u5bf9\u4e8e\u7ed9\u5b9a\u7684\u66f2\u7ebf\uff0c\u53ef\u4ee5\u4f7f\u7528\u4ee5\u4e0b\u516c\u5f0f\u6765\u8ba1\u7b97\u66f2\u7ebf\u4e0a\u67d0\u4e00\u70b9\u5904\u7684\u66f2\u7387\uff1a<\/p>\n\n\n\n
                                                          \ufffd=\u2223\ufffd\u20d7\u2032(\ufffd)\u2223\u2223\ufffd\u20d7\u2032(\ufffd)\u2223\u03ba<\/em>=\u2223r<\/em>\u2032(s<\/em>)\u2223\u2223\u2223\u200bT<\/em>\u2032(s<\/em>)\u2223\u2223\u200b\u200b<\/p>\n\n\n\n
                                                          \u5176\u4e2d $\\vec{T}$ \u8868\u793a\u66f2\u7ebf\u7684\u5207\u7ebf\uff0c$\\vec{r}$ \u8868\u793a\u66f2\u7ebf\u7684\u5411\u91cf\u51fd\u6570\uff0c$s$ \u8868\u793a\u66f2\u7ebf\u7684\u5f27\u957f\u53c2\u6570\u3002\u66f2\u7387\u7684\u5927\u5c0f\u53d6\u51b3\u4e8e\u66f2\u7ebf\u4e0a\u6bcf\u4e2a\u70b9\u5904\u7684\u5207\u7ebf\u7684\u65cb\u8f6c\u901f\u7387\u3002<\/p>\n\n\n\n
                                                          \u5982\u679c\u6211\u4eec\u5c06\u66f2\u7ebf\u8868\u793a\u4e3a\u53c2\u6570\u5f62\u5f0f $\\vec{r}(t) = (x(t), y(t))$\uff0c\u5219\u66f2\u7ebf\u7684\u5207\u7ebf\u5411\u91cf\u53ef\u4ee5\u8868\u793a\u4e3a\uff1a<\/p>\n\n\n\n
                                                          \ufffd\u20d7(\ufffd)=\ufffd\ufffd\u20d7\ufffd\ufffd=(\ufffd\u2032(\ufffd),\ufffd\u2032(\ufffd))T<\/em>(t<\/em>)=d<\/em>t<\/em>d<\/em>r<\/em>\u200b=(x<\/em>\u2032(t<\/em>),y<\/em>\u2032(t<\/em>))<\/p>\n\n\n\n
                                                          \u66f2\u7ebf\u7684\u66f2\u7387\u53ef\u4ee5\u7528\u4ee5\u4e0b\u516c\u5f0f\u6765\u8ba1\u7b97\uff1a<\/p>\n\n\n\n
                                                          \ufffd=\u2223\ufffd\u2032\ufffd\u2032\u2032\u2212\ufffd\u2032\ufffd\u2032\u2032\u2223(\ufffd\u20322+\ufffd\u20322)32\u03ba<\/em>=(x<\/em>\u20322+y<\/em>\u20322)23\u200b\u2223x<\/em>\u2032y<\/em>\u2032\u2032\u2212y<\/em>\u2032x<\/em>\u2032\u2032\u2223\u200b<\/p>\n\n\n\n
                                                          \u5728\u5b9e\u9645\u5e94\u7528\u4e2d\uff0c\u66f2\u7387\u53ef\u4ee5\u901a\u8fc7\u6570\u503c\u65b9\u6cd5\u6765\u8ba1\u7b97\uff0c\u4f8b\u5982\u4f7f\u7528\u6709\u9650\u5dee\u5206\u6cd5\u6216\u5fae\u5206\u51e0\u4f55\u65b9\u6cd5\u3002<\/p>\n\n\n\n\n\n
                                                          \"gun<\/figure>\n\n\n\n
                                                          \u8bf7\u8be6\u7ec6\u89e3\u91ca\u6570\u636e\u96c6\u589e\u5f3a<\/p>\n\n\n\n
                                                          \u6570\u636e\u96c6\u589e\u5f3a\u662f\u4e00\u79cd\u5728\u8bad\u7ec3\u795e\u7ecf\u7f51\u7edc\u65f6\u6269\u5c55\u6570\u636e\u96c6\u5927\u5c0f\u7684\u6280\u672f\uff0c\u65e8\u5728\u6539\u5584\u6a21\u578b\u7684\u6cdb\u5316\u6027\u80fd\u3002\u6570\u636e\u96c6\u589e\u5f3a\u7684\u57fa\u672c\u601d\u60f3\u662f\uff0c\u901a\u8fc7\u5bf9\u539f\u59cb\u6570\u636e\u5e94\u7528\u4e00\u7cfb\u5217\u53d8\u6362\u6765\u751f\u6210\u66f4\u591a\u7684\u8bad\u7ec3\u6837\u672c\uff0c\u4ece\u800c\u4f7f\u5f97\u6a21\u578b\u66f4\u80fd\u591f\u9002\u5e94\u6570\u636e\u7684\u591a\u6837\u6027\u3002<\/p>\n\n\n\n
                                                          \u6570\u636e\u96c6\u589e\u5f3a\u7684\u4e00\u4e9b\u5e38\u89c1\u53d8\u6362\u5305\u62ec\uff1a<\/p>\n\n\n\n
                                                            \n
                                                          1. \u7ffb\u8f6c\uff1a\u6cbf\u7740\u6c34\u5e73\u6216\u5782\u76f4\u8f74\u7ffb\u8f6c\u56fe\u50cf\u3002<\/li>\n\n\n\n
                                                          2. \u65cb\u8f6c\uff1a\u5c06\u56fe\u50cf\u65cb\u8f6c\u4e00\u5b9a\u89d2\u5ea6\u3002<\/li>\n\n\n\n
                                                          3. \u88c1\u526a\uff1a\u4ece\u56fe\u50cf\u4e2d\u968f\u673a\u622a\u53d6\u4e00\u90e8\u5206\u4f5c\u4e3a\u65b0\u7684\u8bad\u7ec3\u6837\u672c\u3002<\/li>\n\n\n\n
                                                          4. \u5e73\u79fb\uff1a\u5728\u56fe\u50cf\u4e2d\u968f\u673a\u5e73\u79fb\u4e00\u5b9a\u8ddd\u79bb\u3002<\/li>\n\n\n\n
                                                          5. \u7f29\u653e\uff1a\u5c06\u56fe\u50cf\u7f29\u5c0f\u6216\u653e\u5927\u4e00\u5b9a\u6bd4\u4f8b\u3002<\/li>\n\n\n\n
                                                          6. \u566a\u58f0\uff1a\u5411\u56fe\u50cf\u4e2d\u6dfb\u52a0\u968f\u673a\u566a\u58f0\u3002<\/li>\n\n\n\n
                                                          7. \u4eae\u5ea6\u8c03\u6574\uff1a\u8c03\u6574\u56fe\u50cf\u7684\u4eae\u5ea6\u3002<\/li>\n\n\n\n
                                                          8. \u5bf9\u6bd4\u5ea6\u8c03\u6574\uff1a\u8c03\u6574\u56fe\u50cf\u7684\u5bf9\u6bd4\u5ea6\u3002<\/li>\n<\/ol>\n\n\n\n
                                                            \u6570\u636e\u96c6\u589e\u5f3a\u53ef\u4ee5\u901a\u8fc7\u5404\u79cd\u56fe\u50cf\u5904\u7406\u5e93\u548c\u6846\u67b6\u6765\u5b9e\u73b0\uff0c\u4f8b\u5982Python\u4e2d\u7684Pillow\u548cOpenCV\uff0c\u4ee5\u53ca\u6df1\u5ea6\u5b66\u4e60\u6846\u67b6\u4e2d\u7684TensorFlow\u548cPyTorch\u3002\u5bf9\u4e8e\u6bcf\u4e2a\u53d8\u6362\uff0c\u90fd\u9700\u8981\u5b9a\u4e49\u4e00\u4e2a\u51fd\u6570\u6765\u5b9e\u73b0\u8be5\u53d8\u6362\uff0c\u5e76\u5c06\u5176\u5e94\u7528\u4e8e\u539f\u59cb\u6570\u636e\u96c6\u4e2d\u7684\u6bcf\u4e2a\u6837\u672c\u3002\u901a\u5e38\uff0c\u53ef\u4ee5\u5c06\u589e\u5f3a\u540e\u7684\u6570\u636e\u96c6\u7528\u4e8e\u8bad\u7ec3\uff0c\u4ee5\u63d0\u9ad8\u6a21\u578b\u7684\u6027\u80fd\u548c\u6cdb\u5316\u80fd\u529b\u3002<\/p>\n\n\n\n\n\n
                                                            \"gun<\/figure>\n\n\n\n
                                                            \u8bf7\u8be6\u7ec6\u89e3\u91ca\u7ef4\u6570\u707e\u96be<\/p>\n\n\n\n
                                                            \u7ef4\u6570\u707e\u96be\uff08curse of dimensionality\uff09\u662f\u6307\u5728\u9ad8\u7ef4\u7a7a\u95f4\u4e2d\uff0c\u6570\u636e\u70b9\u4e4b\u95f4\u7684\u8ddd\u79bb\u53d8\u5f97\u975e\u5e38\u7a00\u758f\uff0c\u5bfc\u81f4\u4f20\u7edf\u7684\u673a\u5668\u5b66\u4e60\u65b9\u6cd5\u5931\u6548\u6216\u8005\u6548\u679c\u4e0d\u4f73\u3002\u8fd9\u4e2a\u95ee\u9898\u901a\u5e38\u4f1a\u51fa\u73b0\u5728\u7279\u5f81\u7ef4\u5ea6\u5f88\u9ad8\u7684\u60c5\u51b5\u4e0b\uff0c\u6bd4\u5982\u81ea\u7136\u8bed\u8a00\u5904\u7406\u6216\u8005\u56fe\u50cf\u8bc6\u522b\u7b49\u9886\u57df\u3002<\/p>\n\n\n\n
                                                            \u5177\u4f53\u6765\u8bf4\uff0c\u7ef4\u6570\u707e\u96be\u7684\u8868\u73b0\u4e3a\uff1a<\/p>\n\n\n\n
                                                              \n
                                                            • \u968f\u7740\u7ef4\u5ea6\u7684\u589e\u52a0\uff0c\u6570\u636e\u6837\u672c\u6570\u91cf\u6307\u6570\u7ea7\u51cf\u5c11\uff0c\u9700\u8981\u66f4\u591a\u7684\u6837\u672c\u624d\u80fd\u4fdd\u8bc1\u6a21\u578b\u7684\u53ef\u9760\u6027\u3002<\/li>\n\n\n\n
                                                            • \u9ad8\u7ef4\u7a7a\u95f4\u4e2d\u7684\u6570\u636e\u70b9\u4e4b\u95f4\u7684\u8ddd\u79bb\u53d8\u5f97\u975e\u5e38\u7a00\u758f\uff0c\u4f7f\u5f97\u4f20\u7edf\u7684\u8ddd\u79bb\u5ea6\u91cf\u65b9\u6cd5\u5931\u6548\u3002\u4f8b\u5982\u5728\u6b27\u6c0f\u7a7a\u95f4\u4e2d\uff0c\u968f\u7740\u7ef4\u5ea6\u7684\u589e\u52a0\uff0c\u6570\u636e\u70b9\u4e4b\u95f4\u7684\u8ddd\u79bb\u53d8\u5f97\u8d8a\u6765\u8d8a\u76f8\u4f3c\uff0c\u4f7f\u5f97\u805a\u7c7b\u548c\u5206\u7c7b\u7b49\u4efb\u52a1\u96be\u4ee5\u5b8c\u6210\u3002<\/li>\n\n\n\n
                                                            • \u7531\u4e8e\u9ad8\u7ef4\u7a7a\u95f4\u4e2d\u7684\u4f53\u79ef\u53d8\u5f97\u5de8\u5927\uff0c\u5f88\u591a\u6982\u5ff5\u5982\u201c\u90bb\u57df\u201d\u3001\u201c\u5bc6\u5ea6\u201d\u7b49\u53d8\u5f97\u4e0d\u518d\u6709\u610f\u4e49\uff0c\u4f7f\u5f97\u5f88\u591a\u7edf\u8ba1\u5206\u6790\u65b9\u6cd5\u4e0d\u9002\u7528\u4e8e\u9ad8\u7ef4\u7a7a\u95f4\u3002<\/li>\n\n\n\n
                                                            • \u9ad8\u7ef4\u7a7a\u95f4\u4e2d\u7684\u51b3\u7b56\u8fb9\u754c\u53d8\u5f97\u975e\u5e38\u590d\u6742\uff0c\u9700\u8981\u66f4\u590d\u6742\u7684\u6a21\u578b\u624d\u80fd\u5b8c\u6210\u5206\u7c7b\u548c\u56de\u5f52\u7b49\u4efb\u52a1\u3002<\/li>\n<\/ul>\n\n\n\n
                                                              \u4e3a\u4e86\u89e3\u51b3\u7ef4\u6570\u707e\u96be\u7684\u95ee\u9898\uff0c\u901a\u5e38\u9700\u8981\u91c7\u7528\u964d\u7ef4\u6280\u672f\u6765\u5c06\u9ad8\u7ef4\u7a7a\u95f4\u4e2d\u7684\u6570\u636e\u6620\u5c04\u5230\u4f4e\u7ef4\u7a7a\u95f4\u4e2d\uff0c\u5982\u4e3b\u6210\u5206\u5206\u6790\uff08PCA\uff09\u3001\u72ec\u7acb\u6210\u5206\u5206\u6790\uff08ICA\uff09\u7b49\u65b9\u6cd5\u3002\u6b64\u5916\uff0c\u4e5f\u53ef\u4ee5\u91c7\u7528\u7279\u5f81\u9009\u62e9\u65b9\u6cd5\u6765\u51cf\u5c11\u4e0d\u5fc5\u8981\u7684\u7279\u5f81\u7ef4\u5ea6\uff0c\u5982\u57fa\u4e8e\u65b9\u5dee\u7684\u7279\u5f81\u9009\u62e9\u65b9\u6cd5\u548c\u57fa\u4e8e\u76f8\u5173\u7cfb\u6570\u7684\u7279\u5f81\u9009\u62e9\u65b9\u6cd5\u7b49\u3002<\/p>\n\n\n\n\n\n
                                                              \"gun<\/figure>\n\n\n\n
                                                              \u8bf7\u8be6\u7ec6\u89e3\u91ca\u4eba\u5de5\u667a\u80fd\u7684\u8bfe\u7a0b\u5b66\u4e60<\/p>\n\n\n\n
                                                              \u4eba\u5de5\u667a\u80fd\u7684\u8bfe\u7a0b\u5b66\u4e60\u53ef\u4ee5\u4ece\u4ee5\u4e0b\u51e0\u4e2a\u65b9\u9762\u5c55\u5f00\uff1a<\/p>\n\n\n\n
                                                              1.\u6570\u5b66\u57fa\u7840\uff1a\u4eba\u5de5\u667a\u80fd\u6d89\u53ca\u5230\u5f88\u591a\u6570\u5b66\u77e5\u8bc6\uff0c\u6bd4\u5982\u7ebf\u6027\u4ee3\u6570\u3001\u6982\u7387\u8bba\u4e0e\u6570\u7406\u7edf\u8ba1\u3001\u5fae\u79ef\u5206\u7b49\u3002\u5728\u5b66\u4e60\u4eba\u5de5\u667a\u80fd\u4e4b\u524d\uff0c\u9700\u8981\u5148\u5177\u5907\u8fd9\u4e9b\u6570\u5b66\u77e5\u8bc6\uff0c\u4ee5\u4fbf\u66f4\u597d\u5730\u7406\u89e3\u548c\u5e94\u7528\u4eba\u5de5\u667a\u80fd\u7b97\u6cd5\u3002<\/p>\n\n\n\n
                                                              2.\u7f16\u7a0b\u6280\u80fd\uff1a\u7f16\u7a0b\u662f\u4eba\u5de5\u667a\u80fd\u7b97\u6cd5\u5b9e\u73b0\u7684\u57fa\u7840\u3002\u9700\u8981\u638c\u63e1\u4e00\u79cd\u6216\u591a\u79cd\u7f16\u7a0b\u8bed\u8a00\uff0c\u5982Python\u3001C++\u7b49\uff0c\u5e76\u5b66\u4e60\u76f8\u5173\u7684\u7f16\u7a0b\u6846\u67b6\u548c\u5de5\u5177\uff0c\u5982TensorFlow\u3001PyTorch\u7b49\u3002<\/p>\n\n\n\n
                                                              3.\u673a\u5668\u5b66\u4e60\u57fa\u7840\uff1a\u673a\u5668\u5b66\u4e60\u662f\u4eba\u5de5\u667a\u80fd\u7b97\u6cd5\u7684\u6838\u5fc3\uff0c\u5305\u62ec\u76d1\u7763\u5b66\u4e60\u3001\u65e0\u76d1\u7763\u5b66\u4e60\u3001\u5f3a\u5316\u5b66\u4e60\u7b49\u3002\u9700\u8981\u638c\u63e1\u8fd9\u4e9b\u7b97\u6cd5\u7684\u57fa\u672c\u539f\u7406\u548c\u5b9e\u73b0\u65b9\u6cd5\u3002<\/p>\n\n\n\n
                                                              4.\u6df1\u5ea6\u5b66\u4e60\u57fa\u7840\uff1a\u6df1\u5ea6\u5b66\u4e60\u662f\u673a\u5668\u5b66\u4e60\u7684\u4e00\u4e2a\u91cd\u8981\u5206\u652f\uff0c\u76ee\u524d\u5728\u56fe\u50cf\u8bc6\u522b\u3001\u81ea\u7136\u8bed\u8a00\u5904\u7406\u3001\u8bed\u97f3\u8bc6\u522b\u7b49\u9886\u57df\u53d6\u5f97\u4e86\u5f88\u597d\u7684\u5e94\u7528\u6548\u679c\u3002\u9700\u8981\u5b66\u4e60\u6df1\u5ea6\u5b66\u4e60\u7684\u57fa\u672c\u539f\u7406\u3001\u5e38\u7528\u7684\u7f51\u7edc\u7ed3\u6784\u548c\u7b97\u6cd5\uff0c\u5982\u5377\u79ef\u795e\u7ecf\u7f51\u7edc\u3001\u5faa\u73af\u795e\u7ecf\u7f51\u7edc\u3001\u81ea\u7f16\u7801\u5668\u3001\u751f\u6210\u5bf9\u6297\u7f51\u7edc\u7b49\u3002<\/p>\n\n\n\n
                                                              5.\u5e94\u7528\u9886\u57df\u77e5\u8bc6\uff1a\u4eba\u5de5\u667a\u80fd\u53ef\u4ee5\u5e94\u7528\u4e8e\u5f88\u591a\u9886\u57df\uff0c\u5982\u533b\u7597\u3001\u91d1\u878d\u3001\u4ea4\u901a\u3001\u519c\u4e1a\u7b49\u3002\u9700\u8981\u4e86\u89e3\u8fd9\u4e9b\u9886\u57df\u7684\u5177\u4f53\u95ee\u9898\u548c\u6570\u636e\u7279\u70b9\uff0c\u4ee5\u4fbf\u5c06\u4eba\u5de5\u667a\u80fd\u7b97\u6cd5\u5e94\u7528\u5230\u5b9e\u9645\u95ee\u9898\u4e2d\u53bb\u3002<\/p>\n\n\n\n
                                                              6.\u5b9e\u8df5\u7ecf\u9a8c\uff1a\u4eba\u5de5\u667a\u80fd\u7b97\u6cd5\u7684\u5e94\u7528\u662f\u4e00\u4e2a\u4e0d\u65ad\u8fed\u4ee3\u548c\u5b9e\u8df5\u7684\u8fc7\u7a0b\u3002\u9700\u8981\u8fdb\u884c\u5927\u91cf\u7684\u5b9e\u9a8c\u548c\u9879\u76ee\u5b9e\u8df5\uff0c\u79ef\u7d2f\u5b9e\u8df5\u7ecf\u9a8c\u548c\u6280\u80fd\u3002<\/p>\n\n\n\n
                                                              \u603b\u4e4b\uff0c\u4eba\u5de5\u667a\u80fd\u7684\u8bfe\u7a0b\u5b66\u4e60\u9700\u8981\u5168\u9762\u638c\u63e1\u76f8\u5173\u7684\u6570\u5b66\u77e5\u8bc6\u3001\u7f16\u7a0b\u6280\u80fd\u548c\u7b97\u6cd5\u539f\u7406\uff0c\u540c\u65f6\u6ce8\u91cd\u5b9e\u8df5\u548c\u5e94\u7528\uff0c\u4e0d\u65ad\u63d0\u9ad8\u81ea\u5df1\u7684\u80fd\u529b\u6c34\u5e73\u3002<\/p>\n\n\n\n\n\n
                                                              \"gun<\/figure>\n\n\n\n
                                                              \u8bf7\u8be6\u7ec6\u89e3\u91ca\u8bbe\u8ba1\u77e9\u9635<\/p>\n\n\n\n
                                                              \u8bbe\u8ba1\u77e9\u9635\u662f\u4e00\u79cd\u7528\u4e8e\u63cf\u8ff0\u591a\u5143\u7edf\u8ba1\u5b66\u95ee\u9898\u7684\u6570\u636e\u8868\u793a\u65b9\u6cd5\u3002\u5b83\u662f\u4e00\u4e2a $n\\times p$ \u7684\u77e9\u9635\uff0c\u5176\u4e2d $n$ \u662f\u89c2\u6d4b\u503c\u7684\u6570\u91cf\uff0c$p$ \u662f\u6bcf\u4e2a\u89c2\u6d4b\u503c\u6240\u5177\u6709\u7684\u7279\u5f81\u6570\u3002\u6bcf\u884c\u5bf9\u5e94\u4e00\u4e2a\u89c2\u6d4b\u503c\uff0c\u6bcf\u5217\u5bf9\u5e94\u4e00\u79cd\u7279\u5f81\u3002<\/p>\n\n\n\n
                                                              \u8bbe\u8ba1\u77e9\u9635\u901a\u5e38\u7528\u4e8e\u5efa\u7acb\u7edf\u8ba1\u6a21\u578b\uff0c\u4f8b\u5982\u7ebf\u6027\u56de\u5f52\u548c\u903b\u8f91\u56de\u5f52\u7b49\u3002\u5728\u8fd9\u4e9b\u6a21\u578b\u4e2d\uff0c\u8bbe\u8ba1\u77e9\u9635\u4f5c\u4e3a\u81ea\u53d8\u91cf\u4e0e\u56e0\u53d8\u91cf\u4e4b\u95f4\u7684\u5173\u7cfb\u77e9\u9635\u3002\u901a\u8fc7\u5bf9\u8bbe\u8ba1\u77e9\u9635\u8fdb\u884c\u8f6c\u6362\u3001\u6c42\u9006\u6216\u5e94\u7528\u5176\u4ed6\u7ebf\u6027\u4ee3\u6570\u6280\u672f\uff0c\u53ef\u4ee5\u4ece\u4e2d\u63d0\u53d6\u5173\u4e8e\u81ea\u53d8\u91cf\u4e0e\u56e0\u53d8\u91cf\u4e4b\u95f4\u7684\u4fe1\u606f\u3002<\/p>\n\n\n\n
                                                              \u8bbe\u8ba1\u77e9\u9635\u4e5f\u662f\u5f88\u591a\u673a\u5668\u5b66\u4e60\u7b97\u6cd5\u4e2d\u7684\u91cd\u8981\u6982\u5ff5\uff0c\u4f8b\u5982\u652f\u6301\u5411\u91cf\u673a\u3001\u795e\u7ecf\u7f51\u7edc\u7b49\u3002\u5728\u8fd9\u4e9b\u7b97\u6cd5\u4e2d\uff0c\u8bbe\u8ba1\u77e9\u9635\u901a\u5e38\u7528\u4e8e\u5c06\u8f93\u5165\u6570\u636e\u8868\u793a\u4e3a\u4e00\u4e2a\u5411\u91cf\u6216\u77e9\u9635\u5f62\u5f0f\uff0c\u4ece\u800c\u65b9\u4fbf\u7b97\u6cd5\u7684\u8ba1\u7b97\u548c\u4f18\u5316\u3002<\/p>\n\n\n\n
                                                              \u4ee5\u4e0b\u662f\u4e00\u4e2a\u7b80\u5355\u7684\u8bbe\u8ba1\u77e9\u9635\u793a\u4f8b\uff1a<\/p>\n\n\n\n
                                                              [1\ufffd11\ufffd12\u22ef\ufffd1\ufffd1\ufffd21\ufffd22\u22ef\ufffd2\ufffd\u22ee\u22ee\u22ee\u22f1\u22ee1\ufffd\ufffd1\ufffd\ufffd2\u22ef\ufffd\ufffd\ufffd]\u23a3\u23a1\u200b11\u22ee1\u200bx<\/em>11\u200bx<\/em>21\u200b\u22eex<\/em>n<\/em>1\u200b\u200bx<\/em>12\u200bx<\/em>22\u200b\u22eex<\/em>n<\/em>2\u200b\u200b\u22ef\u22ef\u22f1\u22ef\u200bx<\/em>1p<\/em>\u200bx<\/em>2p<\/em>\u200b\u22eex<\/em>n<\/em>p<\/em>\u200b\u200b\u23a6\u23a4\u200b<\/p>\n\n\n\n
                                                              \u5728\u8be5\u77e9\u9635\u4e2d\uff0c\u6bcf\u4e00\u884c\u8868\u793a\u4e00\u4e2a\u89c2\u6d4b\u503c\uff0c\u7b2c\u4e00\u5217\u4e3a\u622a\u8ddd\u9879\uff0c\u540e\u9762\u7684\u5217\u4e3a\u5404\u4e2a\u7279\u5f81\u3002\u53ef\u4ee5\u770b\u51fa\uff0c\u8bbe\u8ba1\u77e9\u9635\u7684\u6bcf\u4e00\u5217\u90fd\u4ee3\u8868\u4e00\u4e2a\u7279\u5f81\uff0c\u6bcf\u4e00\u884c\u90fd\u4ee3\u8868\u4e00\u4e2a\u89c2\u6d4b\u503c\uff0c\u56e0\u6b64\u53ef\u4ee5\u65b9\u4fbf\u5730\u7528\u4e8e\u5efa\u7acb\u7edf\u8ba1\u6a21\u578b\u6216\u5e94\u7528\u673a\u5668\u5b66\u4e60\u7b97\u6cd5\u3002<\/p>\n\n\n\n\n\n
                                                              \"gun<\/figure>\n\n\n\n
                                                              \u8bf7\u8be6\u7ec6\u89e3\u91ca\u7ec6\u81f4\u5e73\u8861<\/p>\n\n\n\n
                                                              \u7ec6\u81f4\u5e73\u8861\uff08detailed balance\uff09\u662f\u7edf\u8ba1\u7269\u7406\u5b66\u4e2d\u7684\u4e00\u4e2a\u6982\u5ff5\uff0c\u5b83\u662f\u6307\u4e00\u4e2a\u7269\u7406\u7cfb\u7edf\u5728\u8fbe\u5230\u70ed\u5e73\u8861\u65f6\uff0c\u5176\u5728\u72b6\u6001\u7a7a\u95f4\u4e2d\u7684\u4efb\u610f\u4e24\u4e2a\u72b6\u6001 $i$ \u548c $j$ \u4e4b\u95f4\u7684\u8f6c\u79fb\u901f\u7387\u5fc5\u987b\u6ee1\u8db3\uff1a<\/p>\n\n\n\n
                                                              \ufffd\ufffd\ufffd\ufffd\ufffd=\ufffd\ufffd\ufffd\ufffd\ufffdP<\/em>i<\/em>\u200bW<\/em>ij<\/em>\u200b=P<\/em>j<\/em>\u200bW<\/em>ji<\/em>\u200b<\/p>\n\n\n\n
                                                              \u5176\u4e2d $P_i$ \u548c $P_j$ \u5206\u522b\u662f\u72b6\u6001 $i$ \u548c $j$ \u7684\u5e73\u8861\u5206\u5e03\uff0c$W_{ij}$ \u662f\u4ece\u72b6\u6001 $i$ \u8f6c\u79fb\u5230\u72b6\u6001 $j$ \u7684\u8f6c\u79fb\u901f\u7387\uff0c$W_{ji}$ \u662f\u4ece\u72b6\u6001 $j$ \u8f6c\u79fb\u5230\u72b6\u6001 $i$ \u7684\u8f6c\u79fb\u901f\u7387\u3002\u8fd9\u4e2a\u6761\u4ef6\u4fdd\u8bc1\u4e86\u7cfb\u7edf\u5728\u70ed\u5e73\u8861\u72b6\u6001\u4e0b\u7684\u6982\u7387\u5206\u5e03\u4e0d\u4f1a\u53d1\u751f\u6539\u53d8\u3002<\/p>\n\n\n\n
                                                              \u7ec6\u81f4\u5e73\u8861\u7684\u6982\u5ff5\u5728\u9a6c\u5c14\u53ef\u592b\u94fe\u8499\u7279\u5361\u6d1b\uff08Markov chain Monte Carlo\uff0cMCMC\uff09\u7b97\u6cd5\u4e2d\u5f88\u91cd\u8981\uff0c\u56e0\u4e3aMCMC\u7b97\u6cd5\u9700\u8981\u751f\u6210\u4e00\u4e2a\u7b26\u5408\u5e73\u8861\u5206\u5e03\u7684\u6837\u672c\u96c6\u5408\u3002\u5982\u679c\u5728\u72b6\u6001\u8f6c\u79fb\u65f6\u4e0d\u6ee1\u8db3\u7ec6\u81f4\u5e73\u8861\u6761\u4ef6\uff0c\u90a3\u4e48\u751f\u6210\u7684\u6837\u672c\u5c31\u4e0d\u7b26\u5408\u5e73\u8861\u5206\u5e03\uff0c\u5bfc\u81f4MCMC\u7b97\u6cd5\u65e0\u6cd5\u6b63\u786e\u5730\u91c7\u6837\u3002<\/p>\n\n\n\n
                                                              \u7ec6\u81f4\u5e73\u8861\u4e5f\u662f\u5404\u79cd\u52a8\u6001\u6a21\u62df\u65b9\u6cd5\u4e2d\u975e\u5e38\u91cd\u8981\u7684\u6982\u5ff5\uff0c\u4f8b\u5982\u5206\u5b50\u52a8\u529b\u5b66\u6a21\u62df\u548c\u8499\u7279\u5361\u7f57\u6a21\u62df\u3002\u5728\u8fd9\u4e9b\u6a21\u62df\u4e2d\uff0c\u6bcf\u4e00\u6b65\u7684\u72b6\u6001\u8f6c\u79fb\u90fd\u5fc5\u987b\u6ee1\u8db3\u7ec6\u81f4\u5e73\u8861\u6761\u4ef6\uff0c\u4ee5\u786e\u4fdd\u6a21\u62df\u7684\u7ed3\u679c\u662f\u7b26\u5408\u70ed\u529b\u5b66\u5e73\u8861\u7684\u3002<\/p>\n\n\n\n
                                                              \u603b\u4e4b\uff0c\u7ec6\u81f4\u5e73\u8861\u662f\u4e00\u79cd\u975e\u5e38\u57fa\u672c\u7684\u70ed\u529b\u5b66\u5e73\u8861\u6761\u4ef6\uff0c\u5b83\u5728\u5404\u79cd\u7269\u7406\u3001\u5316\u5b66\u3001\u7edf\u8ba1\u5b66\u548c\u8ba1\u7b97\u673a\u79d1\u5b66\u9886\u57df\u4e2d\u90fd\u6709\u5e7f\u6cdb\u7684\u5e94\u7528\u3002<\/p>\n\n\n\n\n\n
                                                              \"gun<\/figure>\n\n\n\n
                                                              \u8bf7\u8be6\u7ec6\u89e3\u91ca\u964d\u7ef4<\/p>\n\n\n\n
                                                              \u964d\u7ef4\u662f\u6307\u901a\u8fc7\u4fdd\u7559\u91cd\u8981\u4fe1\u606f\uff0c\u5c06\u6570\u636e\u4ece\u9ad8\u7ef4\u7a7a\u95f4\u6620\u5c04\u5230\u4f4e\u7ef4\u7a7a\u95f4\u7684\u8fc7\u7a0b\u3002\u5728\u673a\u5668\u5b66\u4e60\u548c\u6570\u636e\u5206\u6790\u4e2d\uff0c\u964d\u7ef4\u901a\u5e38\u7528\u4e8e\u5904\u7406\u9ad8\u7ef4\u6570\u636e\uff0c\u4ee5\u4fbf\u66f4\u597d\u5730\u7406\u89e3\u3001\u53ef\u89c6\u5316\u6216\u7b80\u5316\u6570\u636e\u3002<\/p>\n\n\n\n
                                                              \u964d\u7ef4\u7684\u76ee\u7684\u662f\u5728\u4e0d\u5931\u53bb\u8fc7\u591a\u4fe1\u606f\u7684\u60c5\u51b5\u4e0b\uff0c\u51cf\u5c11\u6570\u636e\u7684\u7ef4\u6570\uff0c\u8fd9\u6709\u5229\u4e8e\u66f4\u597d\u5730\u7406\u89e3\u548c\u5904\u7406\u6570\u636e\u3002\u540c\u65f6\uff0c\u964d\u7ef4\u8fd8\u53ef\u4ee5\u63d0\u9ad8\u673a\u5668\u5b66\u4e60\u7b97\u6cd5\u7684\u6548\u7387\u548c\u6027\u80fd\uff0c\u56e0\u4e3a\u9ad8\u7ef4\u6570\u636e\u901a\u5e38\u4f1a\u5bfc\u81f4\u8fc7\u62df\u5408\u548c\u7ef4\u5ea6\u707e\u96be\u3002<\/p>\n\n\n\n
                                                              \u5e38\u89c1\u7684\u964d\u7ef4\u65b9\u6cd5\u6709\u4e3b\u6210\u5206\u5206\u6790\uff08PCA\uff09\u3001\u7ebf\u6027\u5224\u522b\u5206\u6790\uff08LDA\uff09\u3001\u6d41\u5f62\u5b66\u4e60\u3001\u56e0\u5b50\u5206\u6790\u7b49\u3002<\/p>\n\n\n\n
                                                              \u4e3b\u6210\u5206\u5206\u6790\u662f\u4e00\u79cd\u5e38\u89c1\u7684\u7ebf\u6027\u964d\u7ef4\u65b9\u6cd5\uff0c\u5b83\u901a\u8fc7\u5c06\u6570\u636e\u6295\u5f71\u5230\u6700\u80fd\u8868\u8fbe\u539f\u59cb\u6570\u636e\u65b9\u5dee\u7684\u4f4e\u7ef4\u7a7a\u95f4\u4e2d\u6765\u5b9e\u73b0\u964d\u7ef4\u3002\u5177\u4f53\u800c\u8a00\uff0cPCA\u627e\u5230\u6570\u636e\u7684\u4e3b\u8981\u65b9\u5411\uff08\u5373\u4e3b\u6210\u5206\uff09\uff0c\u5e76\u5c06\u6570\u636e\u6cbf\u7740\u8fd9\u4e9b\u4e3b\u6210\u5206\u8fdb\u884c\u6295\u5f71\uff0c\u4ece\u800c\u5f97\u5230\u964d\u7ef4\u540e\u7684\u6570\u636e\u3002<\/p>\n\n\n\n
                                                              \u6d41\u5f62\u5b66\u4e60\u5219\u662f\u4e00\u79cd\u975e\u7ebf\u6027\u964d\u7ef4\u65b9\u6cd5\uff0c\u5b83\u901a\u8fc7\u5bfb\u627e\u6570\u636e\u5185\u5728\u7684\u6d41\u5f62\u7ed3\u6784\u6765\u5b9e\u73b0\u964d\u7ef4\u3002\u6d41\u5f62\u662f\u6307\u5728\u9ad8\u7ef4\u7a7a\u95f4\u4e2d\u7684\u4f4e\u7ef4\u5b50\u7a7a\u95f4\uff0c\u5176\u4e0a\u7684\u6570\u636e\u4fdd\u6301\u4e86\u4e00\u5b9a\u7684\u5c40\u90e8\u7ed3\u6784\u3002\u6d41\u5f62\u5b66\u4e60\u65b9\u6cd5\u65e8\u5728\u5728\u4fdd\u6301\u6570\u636e\u5c40\u90e8\u7ed3\u6784\u7684\u540c\u65f6\u5c06\u6570\u636e\u6620\u5c04\u5230\u4f4e\u7ef4\u6d41\u5f62\u4e0a\u3002<\/p>\n\n\n\n
                                                              \u964d\u7ef4\u5728\u8bb8\u591a\u9886\u57df\u90fd\u6709\u5e7f\u6cdb\u7684\u5e94\u7528\uff0c\u4f8b\u5982\u56fe\u50cf\u548c\u8bed\u97f3\u5904\u7406\u3001\u81ea\u7136\u8bed\u8a00\u5904\u7406\u3001\u63a8\u8350\u7cfb\u7edf\u3001\u751f\u7269\u4fe1\u606f\u5b66\u7b49\u3002<\/p>\n\n\n\n\n\n
                                                              \"gun<\/figure>\n\n\n\n
                                                              \u8bf7\u8be6\u7ec6\u89e3\u91ca\u6709\u6548\u5bb9\u91cf<\/p>\n\n\n\n
                                                              \u6709\u6548\u5bb9\u91cf\uff08effective capacity\uff09\u662f\u6307\u4e00\u79cd\u901a\u4fe1\u7cfb\u7edf\u7684\u6700\u5927\u53ef\u9760\u4f20\u8f93\u901f\u7387\uff0c\u5b83\u8003\u8651\u4e86\u566a\u58f0\u3001\u5e72\u6270\u548c\u4fe1\u9053\u4e0d\u786e\u5b9a\u6027\u7b49\u56e0\u7d20\u7684\u5f71\u54cd\uff0c\u662f\u7cfb\u7edf\u5728\u67d0\u4e9b\u8bef\u7801\u7387\uff08\u5982 $10^{-6}$\uff09\u4e0b\u7684\u6700\u5927\u4f20\u8f93\u901f\u7387\u3002<\/p>\n\n\n\n
                                                              \u5728\u4fe1\u606f\u8bba\u4e2d\uff0c\u6709\u6548\u5bb9\u91cf\u53ef\u4ee5\u7528\u6765\u63cf\u8ff0\u4fe1\u9053\u7684\u4f20\u8f93\u80fd\u529b\uff0c\u5b83\u662f\u6307\u5728\u4fdd\u8bc1\u4e00\u5b9a\u7684\u9519\u8bef\u7387\u4e0b\uff0c\u4fe1\u9053\u53ef\u4ee5\u4f20\u8f93\u7684\u6700\u5927\u4fe1\u606f\u91cf\u3002\u901a\u5e38\u6765\u8bf4\uff0c\u4fe1\u9053\u7684\u5bb9\u91cf\u662f\u7531\u5176\u5e26\u5bbd\u3001\u4fe1\u566a\u6bd4\u7b49\u56e0\u7d20\u51b3\u5b9a\u7684\uff0c\u4f46\u5728\u5b9e\u9645\u5e94\u7528\u4e2d\uff0c\u4fe1\u9053\u7684\u5b9e\u9645\u5bb9\u91cf\u5f80\u5f80\u6bd4\u7406\u8bba\u503c\u8981\u5c0f\u3002\u8fd9\u662f\u56e0\u4e3a\u5b9e\u9645\u901a\u4fe1\u4e2d\u5b58\u5728\u5404\u79cd\u5e72\u6270\u3001\u566a\u58f0\u7b49\u56e0\u7d20\uff0c\u540c\u65f6\u7531\u4e8e\u4fe1\u9053\u6761\u4ef6\u4e0d\u53ef\u9884\u6d4b\uff0c\u6240\u4ee5\u4fe1\u9053\u7684\u5b9e\u9645\u5bb9\u91cf\u53ef\u80fd\u4f1a\u6709\u5f88\u5927\u7684\u6ce2\u52a8\u3002<\/p>\n\n\n\n
                                                              \u6709\u6548\u5bb9\u91cf\u7684\u6982\u5ff5\u662f\u4e3a\u4e86\u66f4\u52a0\u51c6\u786e\u5730\u63cf\u8ff0\u4fe1\u9053\u7684\u5b9e\u9645\u4f20\u8f93\u80fd\u529b\uff0c\u5b83\u8003\u8651\u4e86\u4fe1\u9053\u7684\u5404\u79cd\u5b9e\u9645\u9650\u5236\u56e0\u7d20\uff0c\u80fd\u591f\u66f4\u597d\u5730\u53cd\u6620\u5b9e\u9645\u5e94\u7528\u4e2d\u7684\u4f20\u8f93\u901f\u7387\u3002\u6709\u6548\u5bb9\u91cf\u7684\u8ba1\u7b97\u6d89\u53ca\u5230\u4fe1\u9053\u7f16\u7801\u3001\u89e3\u7801\u548c\u8c03\u5236\u7b49\u4e00\u7cfb\u5217\u6280\u672f\uff0c\u56e0\u6b64\u662f\u4e00\u4e2a\u76f8\u5bf9\u590d\u6742\u7684\u95ee\u9898\u3002<\/p>\n\n\n\n\n\n
                                                              \"gun<\/figure>\n\n\n\n
                                                              \u8bf7\u8be6\u7ec6\u89e3\u91ca\u7ecf\u9a8c\u5206\u5e03<\/p>\n\n\n\n
                                                              \u7ecf\u9a8c\u5206\u5e03\u662f\u6307\u7531\u4e00\u7ec4\u89c2\u6d4b\u6837\u672c\u5f97\u51fa\u7684\u5206\u5e03\u51fd\u6570\uff0c\u5b83\u662f\u4ece\u6570\u636e\u6837\u672c\u4e2d\u83b7\u5f97\u7684\u6982\u7387\u5206\u5e03\u51fd\u6570\u7684\u8fd1\u4f3c\u503c\u3002\u5728\u5b9e\u8df5\u4e2d\uff0c\u6211\u4eec\u7ecf\u5e38\u9700\u8981\u4ece\u4e00\u4e9b\u6570\u636e\u6837\u672c\u4e2d\u4f30\u8ba1\u51fa\u4e00\u4e2a\u672a\u77e5\u5206\u5e03\u7684\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\u6216\u5206\u5e03\u51fd\u6570\uff0c\u800c\u7ecf\u9a8c\u5206\u5e03\u5c31\u662f\u4e00\u79cd\u5e38\u7528\u7684\u4f30\u8ba1\u65b9\u6cd5\u3002<\/p>\n\n\n\n
                                                              \u5177\u4f53\u5730\uff0c\u5047\u8bbe\u6211\u4eec\u6709\u4e00\u4e2a\u6570\u636e\u6837\u672c $X_1, X_2, \\cdots, X_n$\uff0c\u5b83\u4eec\u6765\u81ea\u67d0\u4e2a\u672a\u77e5\u7684\u6982\u7387\u5206\u5e03 $F$\u3002\u90a3\u4e48\u7ecf\u9a8c\u5206\u5e03\u51fd\u6570 $\\hat{F}$ \u53ef\u4ee5\u5b9a\u4e49\u4e3a\uff1a<\/p>\n\n\n\n
                                                              \ufffd^(\ufffd)=1\ufffd\u2211\ufffd=1\ufffd\ufffd(\u2212\u221e,\ufffd](\ufffd\ufffd)F<\/em>^(x<\/em>)=n<\/em>1\u200bi<\/em>=1\u2211n<\/em>\u200bI<\/em>(\u2212\u221e,x<\/em>]\u200b(X<\/em>i<\/em>\u200b)<\/p>\n\n\n\n
                                                              \u5176\u4e2d $I_{(-\\infty, x]}$ \u662f\u6307\u793a\u51fd\u6570\uff0c\u5373\uff1a<\/p>\n\n\n\n
                                                              \ufffd(\u2212\u221e,\ufffd](\ufffd)={1,\ufffd\u2264\ufffd0,\ufffd>\ufffdI<\/em>(\u2212\u221e,x<\/em>]\u200b(t<\/em>)={1,0,\u200bt<\/em>\u2264x<\/em>t<\/em>>x<\/em>\u200b<\/p>\n\n\n\n
                                                              \u53ef\u4ee5\u770b\u51fa\uff0c\u7ecf\u9a8c\u5206\u5e03\u51fd\u6570\u7684\u5b9a\u4e49\u662f\u5c06 $X_1, X_2, \\cdots, X_n$ \u4e2d\u5c0f\u4e8e\u7b49\u4e8e $x$ \u7684\u6570\u7684\u6bd4\u4f8b\u4f5c\u4e3a $x$ \u7684\u6982\u7387\u4f30\u8ba1\u503c\u3002\u5f53\u6837\u672c\u6570\u91cf $n$ \u8d8a\u5927\u65f6\uff0c\u7ecf\u9a8c\u5206\u5e03\u51fd\u6570\u4f1a\u8d8a\u6765\u8d8a\u63a5\u8fd1\u771f\u5b9e\u7684\u5206\u5e03\u51fd\u6570 $F$\u3002<\/p>\n\n\n\n
                                                              \u7ecf\u9a8c\u5206\u5e03\u5728\u7edf\u8ba1\u5b66\u4e60\u4e2d\u6709\u7740\u5e7f\u6cdb\u7684\u5e94\u7528\u3002\u4f8b\u5982\uff0c\u6211\u4eec\u53ef\u4ee5\u5229\u7528\u7ecf\u9a8c\u5206\u5e03\u6765\u4f30\u8ba1\u4e00\u4e2a\u672a\u77e5\u5206\u5e03\u7684\u671f\u671b\u3001\u65b9\u5dee\u3001\u9ad8\u9636\u77e9\u7b49\u7edf\u8ba1\u91cf\uff1b\u53c8\u6216\u8005\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u7ecf\u9a8c\u5206\u5e03\u4f5c\u4e3a\u5206\u7c7b\u5668\u7684\u8f93\u51fa\u6982\u7387\u5206\u5e03\uff0c\u7528\u4e8e\u5206\u7c7b\u95ee\u9898\u7684\u6c42\u89e3\u3002<\/p>\n\n\n\n\n\n
                                                              \"gun<\/figure>\n\n\n\n
                                                              \u8bf7\u8be6\u7ec6\u89e3\u91ca\u8303\u6570<\/p>\n\n\n\n
                                                              \u5728\u6570\u5b66\u4e2d\uff0c\u8303\u6570\u662f\u4e00\u4e2a\u51fd\u6570\uff0c\u5b83\u5c06\u4e00\u4e2a\u5411\u91cf\u7a7a\u95f4\u4e2d\u7684\u5411\u91cf\u6620\u5c04\u5230\u975e\u8d1f\u5b9e\u6570\uff0c\u8868\u793a\u5411\u91cf\u7684\u5927\u5c0f\u6216\u957f\u5ea6\u3002\u8303\u6570\u5177\u6709\u4ee5\u4e0b\u6027\u8d28\uff1a<\/p>\n\n\n\n
                                                                \n
                                                              1. \u975e\u8d1f\u6027\uff1a\u5bf9\u4e8e\u4efb\u610f\u5411\u91cf $\\mathbf{x}$\uff0c\u8303\u6570 $|\\mathbf{x}|$ \u5927\u4e8e\u7b49\u4e8e\u96f6\uff0c\u4e14\u7b49\u4e8e\u96f6\u5f53\u4e14\u4ec5\u5f53\u5411\u91cf $\\mathbf{x}$ \u662f\u96f6\u5411\u91cf\u3002<\/li>\n\n\n\n
                                                              2. \u9f50\u6b21\u6027\uff1a\u5bf9\u4e8e\u4efb\u610f\u5411\u91cf $\\mathbf{x}$ \u548c\u4efb\u610f\u5b9e\u6570 $a$\uff0c\u6709 $|a\\mathbf{x}|=|a||\\mathbf{x}|$\u3002<\/li>\n\n\n\n
                                                              3. \u4e09\u89d2\u4e0d\u7b49\u5f0f\uff1a\u5bf9\u4e8e\u4efb\u610f\u5411\u91cf $\\mathbf{x}$ \u548c $\\mathbf{y}$\uff0c\u6709 $|\\mathbf{x}+\\mathbf{y}| \\leq |\\mathbf{x}|+|\\mathbf{y}|$\u3002<\/li>\n<\/ol>\n\n\n\n
                                                                \u5e38\u7528\u7684\u8303\u6570\u6709\u4ee5\u4e0b\u51e0\u79cd\uff1a<\/p>\n\n\n\n
                                                                  \n
                                                                1. $L^1$ \u8303\u6570\uff1a\u4e5f\u79f0\u66fc\u54c8\u987f\u8ddd\u79bb\uff0c\u5b9a\u4e49\u4e3a\u5411\u91cf\u5143\u7d20\u7edd\u5bf9\u503c\u4e4b\u548c\uff1a$|\\mathbf{x}|_1 = \\sum_i |x_i|$\u3002<\/li>\n\n\n\n
                                                                2. $L^2$ \u8303\u6570\uff1a\u4e5f\u79f0\u6b27\u51e0\u91cc\u5f97\u8ddd\u79bb\uff0c\u5b9a\u4e49\u4e3a\u5411\u91cf\u5143\u7d20\u5e73\u65b9\u548c\u7684\u5e73\u65b9\u6839\uff1a$|\\mathbf{x}|_2 = \\sqrt{\\sum_i x_i^2}$\u3002<\/li>\n\n\n\n
                                                                3. $L^\\infty$ \u8303\u6570\uff1a\u4e5f\u79f0\u5207\u6bd4\u96ea\u592b\u8ddd\u79bb\uff0c\u5b9a\u4e49\u4e3a\u5411\u91cf\u5143\u7d20\u7684\u6700\u5927\u503c\uff1a$|\\mathbf{x}|_\\infty = \\max_i |x_i|$\u3002<\/li>\n<\/ol>\n\n\n\n
                                                                  \u8303\u6570\u5e38\u5e38\u88ab\u7528\u4e8e\u8861\u91cf\u4e24\u4e2a\u5411\u91cf\u4e4b\u95f4\u7684\u8ddd\u79bb\uff0c\u4e5f\u53ef\u4ee5\u7528\u4e8e\u6b63\u5219\u5316\u548c\u4f18\u5316\u95ee\u9898\u3002\u5728\u673a\u5668\u5b66\u4e60\u4e2d\uff0c\u5e38\u5e38\u4f7f\u7528 $L^1$ \u548c $L^2$ \u8303\u6570\u6765\u7ea6\u675f\u6a21\u578b\u53c2\u6570\uff0c\u4f8b\u5982 $L^1$ \u6b63\u5219\u5316\u548c $L^2$ \u6b63\u5219\u5316\u3002<\/p>\n\n\n\n\n\n
                                                                  \"gun<\/figure>\n\n\n\n
                                                                  \u8bf7\u8be6\u7ec6\u89e3\u91ca\u53d8\u5dee\u56e0\u7d20<\/p>\n\n\n\n
                                                                  \u53d8\u5dee\u56e0\u7d20\uff08vanishing\/exploding gradient\uff09\u662f\u6307\u5728\u795e\u7ecf\u7f51\u7edc\u4e2d\uff0c\u968f\u7740\u53cd\u5411\u4f20\u64ad\u7b97\u6cd5\u7684\u6df1\u5165\uff0c\u68af\u5ea6\u9010\u6e10\u53d8\u5f97\u5f88\u5c0f\uff08vanishing\uff09\u6216\u5f88\u5927\uff08exploding\uff09\u7684\u73b0\u8c61\u3002<\/p>\n\n\n\n
                                                                  \u5728\u795e\u7ecf\u7f51\u7edc\u7684\u53cd\u5411\u4f20\u64ad\u8fc7\u7a0b\u4e2d\uff0c\u6bcf\u4e00\u5c42\u90fd\u9700\u8981\u8ba1\u7b97\u76f8\u5e94\u7684\u68af\u5ea6\uff0c\u7136\u540e\u5c06\u8fd9\u4e9b\u68af\u5ea6\u4f20\u9012\u7ed9\u524d\u4e00\u5c42\uff0c\u76f4\u5230\u4f20\u9012\u5230\u7b2c\u4e00\u5c42\u3002\u7531\u4e8e\u94fe\u5f0f\u6cd5\u5219\u7684\u539f\u56e0\uff0c\u53cd\u5411\u4f20\u64ad\u7b97\u6cd5\u4e2d\u7684\u68af\u5ea6\u662f\u7531\u4e0a\u4e00\u5c42\u7684\u68af\u5ea6\u548c\u5f53\u524d\u5c42\u7684\u6fc0\u6d3b\u51fd\u6570\u5bfc\u6570\u7684\u4e58\u79ef\u6240\u7ec4\u6210\u7684\u3002\u5f53\u7f51\u7edc\u6df1\u5ea6\u589e\u52a0\u65f6\uff0c\u7531\u4e8e\u8fd9\u4e9b\u68af\u5ea6\u7684\u4e58\u79ef\uff0c\u68af\u5ea6\u7684\u5927\u5c0f\u53ef\u80fd\u4f1a\u6307\u6570\u7ea7\u5730\u589e\u52a0\u6216\u51cf\u5c11\uff0c\u8fd9\u53ef\u80fd\u5bfc\u81f4\u68af\u5ea6\u7206\u70b8\u6216\u68af\u5ea6\u6d88\u5931\u7684\u95ee\u9898\u3002<\/p>\n\n\n\n
                                                                  \u68af\u5ea6\u7206\u70b8\u7684\u95ee\u9898\u53ef\u4ee5\u901a\u8fc7\u68af\u5ea6\u88c1\u526a\u6765\u7f13\u89e3\uff0c\u800c\u68af\u5ea6\u6d88\u5931\u7684\u95ee\u9898\u5219\u6bd4\u8f83\u96be\u4ee5\u89e3\u51b3\u3002\u4e3a\u4e86\u89e3\u51b3\u68af\u5ea6\u6d88\u5931\u7684\u95ee\u9898\uff0c\u51fa\u73b0\u4e86\u4e00\u4e9b\u53d8\u4f53\u795e\u7ecf\u7f51\u7edc\uff0c\u5982LSTM\u548cGRU\uff0c\u5b83\u4eec\u80fd\u591f\u66f4\u597d\u5730\u5904\u7406\u957f\u671f\u4f9d\u8d56\u5173\u7cfb\uff0c\u540c\u65f6\u4e5f\u51cf\u8f7b\u4e86\u68af\u5ea6\u6d88\u5931\u7684\u95ee\u9898\u3002<\/p>\n\n\n\n
                                                                  \u603b\u4e4b\uff0c\u53d8\u5dee\u56e0\u7d20\u662f\u795e\u7ecf\u7f51\u7edc\u4e2d\u4e00\u4e2a\u91cd\u8981\u7684\u95ee\u9898\uff0c\u9700\u8981\u5728\u8bbe\u8ba1\u7f51\u7edc\u7ed3\u6784\u3001\u521d\u59cb\u5316\u53c2\u6570\u548c\u8bad\u7ec3\u8fc7\u7a0b\u4e2d\u8fdb\u884c\u6ce8\u610f\u548c\u8c03\u6574\u3002<\/p>\n\n\n\n\n\n
                                                                  \"gun<\/figure>\n\n\n\n
                                                                  \u8bf7\u8be6\u7ec6\u89e3\u91ca\u7cbe\u8c03<\/p>\n\n\n\n
                                                                  \u7cbe\u8c03\uff08fine-tuning\uff09\u662f\u6307\u5728\u4e00\u4e2a\u5df2\u7ecf\u9884\u8bad\u7ec3\u597d\u7684\u6a21\u578b\u7684\u57fa\u7840\u4e0a\uff0c\u901a\u8fc7\u5bf9\u4e00\u5c0f\u90e8\u5206\u53c2\u6570\u8fdb\u884c\u5fae\u8c03\u6765\u9002\u5e94\u7279\u5b9a\u7684\u4efb\u52a1\u3002\u5728\u673a\u5668\u5b66\u4e60\u9886\u57df\uff0c\u9884\u8bad\u7ec3\u6a21\u578b\u662f\u6307\u5728\u5927\u91cf\u6570\u636e\u4e0a\u8bad\u7ec3\u51fa\u6765\u7684\u901a\u7528\u7684\u6a21\u578b\uff0c\u800c\u7cbe\u8c03\u5219\u662f\u6307\u5728\u5c11\u91cf\u6570\u636e\u4e0a\u5bf9\u8be5\u6a21\u578b\u8fdb\u884c\u5fae\u8c03\u4ee5\u9002\u5e94\u7279\u5b9a\u4efb\u52a1\u7684\u8fc7\u7a0b\u3002<\/p>\n\n\n\n
                                                                  \u901a\u5e38\u60c5\u51b5\u4e0b\uff0c\u9884\u8bad\u7ec3\u6a21\u578b\u662f\u901a\u8fc7\u5927\u89c4\u6a21\u7684\u6570\u636e\u96c6\u8fdb\u884c\u8bad\u7ec3\u5f97\u5230\u7684\uff0c\u4f8b\u5982\u5728\u81ea\u7136\u8bed\u8a00\u5904\u7406\u9886\u57df\u4e2d\uff0c\u53ef\u4ee5\u4f7f\u7528\u6d77\u91cf\u7684\u6587\u672c\u6570\u636e\u6765\u8bad\u7ec3\u8bed\u8a00\u6a21\u578b\u3002\u7531\u4e8e\u8bad\u7ec3\u6570\u636e\u91cf\u5927\uff0c\u9884\u8bad\u7ec3\u6a21\u578b\u901a\u5e38\u80fd\u591f\u63d0\u53d6\u5230\u6570\u636e\u4e2d\u7684\u901a\u7528\u7279\u5f81\uff0c\u5177\u6709\u8f83\u597d\u7684\u6cdb\u5316\u80fd\u529b\u3002\u5728\u5b9e\u9645\u5e94\u7528\u4e2d\uff0c\u5f88\u5c11\u4f1a\u6709\u5145\u8db3\u7684\u6570\u636e\u7528\u4e8e\u9488\u5bf9\u7279\u5b9a\u4efb\u52a1\u8fdb\u884c\u8bad\u7ec3\uff0c\u6b64\u65f6\u53ef\u4ee5\u901a\u8fc7\u7cbe\u8c03\u6765\u5229\u7528\u5df2\u6709\u7684\u9884\u8bad\u7ec3\u6a21\u578b\u3002<\/p>\n\n\n\n
                                                                  \u7cbe\u8c03\u7684\u5177\u4f53\u5b9e\u73b0\u901a\u5e38\u662f\u5728\u9884\u8bad\u7ec3\u6a21\u578b\u7684\u57fa\u7840\u4e0a\u6dfb\u52a0\u4e00\u4e9b\u65b0\u7684\u7f51\u7edc\u5c42\uff0c\u7136\u540e\u518d\u4f7f\u7528\u7279\u5b9a\u7684\u6570\u636e\u96c6\u5bf9\u65b0\u6dfb\u52a0\u7684\u7f51\u7edc\u5c42\u8fdb\u884c\u8bad\u7ec3\uff0c\u4f7f\u5f97\u6574\u4e2a\u6a21\u578b\u80fd\u591f\u9002\u5e94\u7279\u5b9a\u7684\u4efb\u52a1\u3002\u5728\u8bad\u7ec3\u8fc7\u7a0b\u4e2d\uff0c\u53ef\u4ee5\u6839\u636e\u9700\u8981\u9009\u62e9\u662f\u5426\u51bb\u7ed3\u9884\u8bad\u7ec3\u6a21\u578b\u7684\u6743\u91cd\uff0c\u4ee5\u4fbf\u66f4\u597d\u5730\u8fdb\u884c\u5fae\u8c03\u3002<\/p>\n\n\n\n
                                                                  \u7cbe\u8c03\u7684\u4f18\u70b9\u5728\u4e8e\u5b83\u80fd\u591f\u663e\u8457\u63d0\u9ad8\u6a21\u578b\u7684\u6027\u80fd\uff0c\u540c\u65f6\u5728\u6570\u636e\u91cf\u8f83\u5c0f\u7684\u60c5\u51b5\u4e0b\u4e5f\u80fd\u591f\u53d6\u5f97\u4e0d\u9519\u7684\u6548\u679c\u3002\u7f3a\u70b9\u5728\u4e8e\u9700\u8981\u5728\u7279\u5b9a\u4efb\u52a1\u4e0a\u8fdb\u884c\u8bad\u7ec3\uff0c\u800c\u8fd9\u901a\u5e38\u9700\u8981\u82b1\u8d39\u5927\u91cf\u7684\u65f6\u95f4\u548c\u8ba1\u7b97\u8d44\u6e90\u3002<\/p>\n\n\n\n\n\n
                                                                  \"gun<\/figure>\n\n\n\n
                                                                  \u8bf7\u8be6\u7ec6\u89e3\u91ca\u95e8\u63a7<\/p>\n\n\n\n
                                                                  \u95e8\u63a7\u662f\u6307\u5728\u6df1\u5ea6\u5b66\u4e60\u4e2d\u7528\u4e8e\u63a7\u5236\u795e\u7ecf\u7f51\u7edc\u4e2d\u4fe1\u606f\u6d41\u52a8\u7684\u4e00\u79cd\u673a\u5236\u3002\u5b83\u53ef\u4ee5\u5728\u8f93\u5165\u6570\u636e\u548c\u7f51\u7edc\u4e2d\u95f4\u7684\u4fe1\u606f\u4f20\u9012\u8fc7\u7a0b\u4e2d\u63a7\u5236\u67d0\u4e9b\u4fe1\u606f\u662f\u5426\u4f20\u9012\u3002\u8fd9\u79cd\u673a\u5236\u53ef\u4ee5\u4f7f\u5f97\u7f51\u7edc\u66f4\u52a0\u7075\u6d3b\u3001\u9002\u5e94\u6027\u66f4\u5f3a\uff0c\u540c\u65f6\u4e5f\u53ef\u4ee5\u6709\u6548\u5730\u89e3\u51b3\u4e00\u4e9b\u5e38\u89c1\u7684\u95ee\u9898\uff0c\u5982\u957f\u671f\u4f9d\u8d56\u95ee\u9898\u548c\u8fc7\u62df\u5408\u95ee\u9898\u3002<\/p>\n\n\n\n
                                                                  \u5728\u6df1\u5ea6\u5b66\u4e60\u4e2d\uff0c\u95e8\u63a7\u673a\u5236\u901a\u5e38\u662f\u901a\u8fc7\u4f7f\u7528\u95e8\u63a7\u5355\u5143\u6765\u5b9e\u73b0\u7684\u3002\u5e38\u89c1\u7684\u95e8\u63a7\u5355\u5143\u5305\u62ec\uff1asigmoid\u95e8\u3001ReLU\u95e8\u3001tanh\u95e8\u7b49\u3002<\/p>\n\n\n\n
                                                                  \u5176\u4e2d\uff0csigmoid\u95e8\u53ef\u4ee5\u7528\u4e8e\u63a7\u5236\u4fe1\u606f\u7684\u4f20\u9012\u6982\u7387\uff0c\u5b83\u7684\u8f93\u51fa\u503c\u8303\u56f4\u57280\u52301\u4e4b\u95f4\uff0c\u53ef\u4ee5\u8868\u793a\u4fe1\u606f\u662f\u5426\u901a\u8fc7\u7684\u6982\u7387\uff1bReLU\u95e8\u53ef\u4ee5\u7528\u4e8e\u63a7\u5236\u4fe1\u606f\u662f\u5426\u901a\u8fc7\uff0c\u5f53\u8f93\u5165\u4e3a\u8d1f\u6570\u65f6\uff0cReLU\u95e8\u8f93\u51fa0\uff0c\u8868\u793a\u4fe1\u606f\u4e0d\u901a\u8fc7\uff1btanh\u95e8\u53ef\u4ee5\u7528\u4e8e\u63a7\u5236\u4fe1\u606f\u7684\u5f3a\u5ea6\uff0c\u5f53\u8f93\u5165\u503c\u8f83\u5927\u65f6\uff0ctanh\u95e8\u8f93\u51fa\u63a5\u8fd11\u7684\u503c\uff0c\u8868\u793a\u4fe1\u606f\u901a\u8fc7\u8f83\u5f3a\u3002<\/p>\n\n\n\n
                                                                  \u95e8\u63a7\u673a\u5236\u88ab\u5e7f\u6cdb\u5e94\u7528\u4e8e\u6df1\u5ea6\u5b66\u4e60\u4e2d\u7684\u5faa\u73af\u795e\u7ecf\u7f51\u7edc\uff08RNN\uff09\u548c\u957f\u77ed\u65f6\u8bb0\u5fc6\u7f51\u7edc\uff08LSTM\uff09\uff0c\u5b83\u4eec\u901a\u8fc7\u4f7f\u7528\u95e8\u63a7\u5355\u5143\u6765\u89e3\u51b3\u957f\u671f\u4f9d\u8d56\u95ee\u9898\uff0c\u4ece\u800c\u5728\u81ea\u7136\u8bed\u8a00\u5904\u7406\u548c\u8bed\u97f3\u8bc6\u522b\u7b49\u4efb\u52a1\u4e2d\u53d6\u5f97\u4e86\u91cd\u8981\u7684\u8fdb\u5c55\u3002<\/p>\n\n\n\n\n\n
                                                                  \"gun<\/figure>\n\n\n\n
                                                                  \u8bf7\u8be6\u7ec6\u89e3\u91ca\u4f3d\u9a6c\u5206\u5e03<\/p>\n\n\n\n
                                                                  \u4f3d\u9a6c\u5206\u5e03\u662f\u4e00\u79cd\u5e38\u89c1\u7684\u6982\u7387\u5206\u5e03\uff0c\u901a\u5e38\u7528\u4e8e\u6a21\u62df\u968f\u673a\u4e8b\u4ef6\u7684\u6301\u7eed\u65f6\u95f4\uff0c\u6bd4\u5982\u4e00\u4e2a\u4eba\u5728\u67d0\u4e2a\u6d3b\u52a8\u4e2d\u505c\u7559\u7684\u65f6\u95f4\u6216\u8005\u4e00\u4ef6\u673a\u5668\u7684\u5bff\u547d\u7b49\u3002\u5b83\u5728\u7edf\u8ba1\u5b66\u4e2d\u5e7f\u6cdb\u5e94\u7528\uff0c\u4f8b\u5982\u5728\u91d1\u878d\u3001\u533b\u5b66\u3001\u5de5\u7a0b\u7b49\u9886\u57df\u3002<\/p>\n\n\n\n
                                                                  \u4f3d\u9a6c\u5206\u5e03\u7684\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\u4e3a\uff1a<\/p>\n\n\n\n
                                                                  \ufffd(\ufffd;\ufffd,\ufffd)=\ufffd\ufffd\ufffd\ufffd\u22121\ufffd\u2212\ufffd\ufffd\u0393(\ufffd)f<\/em>(x<\/em>;\u03b1<\/em>,\u03b2<\/em>)=\u0393(\u03b1<\/em>)\u03b2<\/em>\u03b1<\/em>x<\/em>\u03b1<\/em>\u22121e<\/em>\u2212\u03b2<\/em>x<\/em>\u200b<\/p>\n\n\n\n
                                                                  \u5176\u4e2d\uff0c$\\alpha$ \u548c $\\beta$ \u662f\u5206\u5e03\u7684\u53c2\u6570\uff0c$\\Gamma$ \u662f\u4f3d\u9a6c\u51fd\u6570\uff0c\u5b9a\u4e49\u4e3a\uff1a<\/p>\n\n\n\n
                                                                  \u0393(\ufffd)=\u222b0\u221e\ufffd\ufffd\u22121\ufffd\u2212\ufffd\ufffd\ufffd\u0393(\u03b1<\/em>)=\u222b0\u221e\u200bx<\/em>\u03b1<\/em>\u22121e<\/em>\u2212x<\/em>d<\/em>x<\/em><\/p>\n\n\n\n
                                                                  \u4f3d\u9a6c\u5206\u5e03\u7684\u671f\u671b\u548c\u65b9\u5dee\u5206\u522b\u4e3a\uff1a<\/p>\n\n\n\n
                                                                  \ufffd(\ufffd)=\ufffd\ufffdE<\/em>(X<\/em>)=\u03b2<\/em>\u03b1<\/em>\u200b<\/p>\n\n\n\n
                                                                  \ufffd\ufffd\ufffd(\ufffd)=\ufffd\ufffd2Va<\/em>r<\/em>(X<\/em>)=\u03b2<\/em>2\u03b1<\/em>\u200b<\/p>\n\n\n\n
                                                                  \u4f3d\u9a6c\u5206\u5e03\u8fd8\u6709\u4e00\u4e2a\u91cd\u8981\u7684\u6027\u8d28\uff0c\u5373\u5b83\u662f\u6307\u6570\u5206\u5e03\u548c\u5361\u65b9\u5206\u5e03\u7684\u5171\u8f6d\u5148\u9a8c\u5206\u5e03\uff0c\u56e0\u6b64\u5728\u8d1d\u53f6\u65af\u7edf\u8ba1\u5b66\u4e2d\u6709\u5e7f\u6cdb\u7684\u5e94\u7528\u3002<\/p>\n\n\n\n
                                                                  \u5728Python\u4e2d\uff0c\u53ef\u4ee5\u4f7f\u7528SciPy\u5e93\u6765\u751f\u6210\u4f3d\u9a6c\u5206\u5e03\u968f\u673a\u53d8\u91cf\uff0c\u5e76\u8fdb\u884c\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\u7684\u7ed8\u5236\u548c\u53c2\u6570\u62df\u5408\u7b49\u64cd\u4f5c\uff0c\u793a\u4f8b\u4ee3\u7801\u5982\u4e0b\uff1a<\/p>\n\n\n\n
                                                                  pythonCopy codeimport numpy as np\nimport matplotlib.pyplot as plt\nfrom scipy.stats import gamma\n\n# \u751f\u6210\u968f\u673a\u53d8\u91cf\nalpha = 2\nbeta = 1\nX = gamma.rvs(a=alpha, scale=1\/beta, size=1000)\n\n# \u7ed8\u5236\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\nx = np.linspace(0, 10, 100)\ny = gamma.pdf(x, a=alpha, scale=1\/beta)\nplt.plot(x, y)\n\n# \u53c2\u6570\u62df\u5408\nparams = gamma.fit(X)\nprint(\"\u62df\u5408\u7684\u53c2\u6570\uff1a\", params)\n\nplt.show()\n<\/code><\/pre>\n\n\n\n
                                                                  \u8be5\u4ee3\u7801\u751f\u6210\u4e00\u4e2a\u53c2\u6570\u4e3a $\\alpha=2$\uff0c$\\beta=1$ \u7684\u4f3d\u9a6c\u5206\u5e03\u968f\u673a\u53d8\u91cf\uff0c\u7ed8\u5236\u5176\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\uff0c\u5e76\u4f7f\u7528 SciPy \u5e93\u8fdb\u884c\u53c2\u6570\u62df\u5408\u3002\u5728\u5b9e\u9645\u5e94\u7528\u4e2d\uff0c\u53ef\u4ee5\u6839\u636e\u6570\u636e\u7684\u5206\u5e03\u60c5\u51b5\u9009\u62e9\u5408\u9002\u7684\u5206\u5e03\u7c7b\u578b\uff0c\u5e76\u6839\u636e\u6570\u636e\u62df\u5408\u5206\u5e03\u7684\u53c2\u6570\u3002<\/p>\n\n\n\n\n\n
                                                                  \"gun<\/figure>\n\n\n\n
                                                                  \u8bf7\u8be6\u7ec6\u89e3\u91ca\u9ad8\u65af\u6838<\/p>\n\n\n\n
                                                                  \u9ad8\u65af\u6838\u662f\u4e00\u79cd\u5e38\u7528\u7684\u6838\u51fd\u6570\uff0c\u5b83\u53ef\u4ee5\u5c06\u4f4e\u7ef4\u7684\u6570\u636e\u6620\u5c04\u5230\u9ad8\u7ef4\u7684\u7279\u5f81\u7a7a\u95f4\uff0c\u5e76\u5728\u8be5\u7a7a\u95f4\u4e2d\u8fdb\u884c\u975e\u7ebf\u6027\u5206\u7c7b\u6216\u56de\u5f52\u3002\u9ad8\u65af\u6838\u51fd\u6570\u5728\u673a\u5668\u5b66\u4e60\u4e2d\u88ab\u5e7f\u6cdb\u5e94\u7528\u4e8e\u652f\u6301\u5411\u91cf\u673a\u3001\u6838\u5cad\u56de\u5f52\u548c\u9ad8\u65af\u8fc7\u7a0b\u7b49\u7b97\u6cd5\u4e2d\u3002<\/p>\n\n\n\n
                                                                  \u9ad8\u65af\u6838\u7684\u6570\u5b66\u8868\u8fbe\u5f0f\u4e3a\uff1a<\/p>\n\n\n\n
                                                                  \ufffd(\ufffd\ufffd,\ufffd\ufffd)=exp\u2061(\u2212\u2225\ufffd\ufffd\u2212\ufffd\ufffd\u222522\ufffd2)k<\/em>(x<\/em>i<\/em>\u200b,x<\/em>j<\/em>\u200b)=exp(\u22122\u03c3<\/em>2\u2225x<\/em>i<\/em>\u200b\u2212x<\/em>j<\/em>\u200b\u22252\u200b)<\/p>\n\n\n\n
                                                                  \u5176\u4e2d\uff0c$x_i$\u548c$x_j$\u662f\u8f93\u5165\u6837\u672c\u7684\u7279\u5f81\u5411\u91cf\uff0c$\\sigma$\u662f\u9ad8\u65af\u6838\u51fd\u6570\u7684\u5e26\u5bbd\u53c2\u6570\uff0c\u7528\u4e8e\u63a7\u5236\u6837\u672c\u70b9\u4e4b\u95f4\u7684\u76f8\u4f3c\u5ea6\u3002\u5f53\u6837\u672c\u70b9\u4e4b\u95f4\u7684\u8ddd\u79bb\u5c0f\u4e8e\u5e26\u5bbd\u53c2\u6570$\\sigma$\u65f6\uff0c\u5b83\u4eec\u4e4b\u95f4\u7684\u76f8\u4f3c\u5ea6\u5c31\u5f88\u9ad8\uff0c\u53cd\u4e4b\u5219\u5f88\u4f4e\u3002<\/p>\n\n\n\n
                                                                  \u9ad8\u65af\u6838\u51fd\u6570\u53ef\u4ee5\u5c06\u8f93\u5165\u6837\u672c\u4ece\u539f\u59cb\u7684\u4f4e\u7ef4\u7a7a\u95f4\u6620\u5c04\u5230\u65e0\u7a77\u7ef4\u7684\u7279\u5f81\u7a7a\u95f4\u4e2d\uff0c\u901a\u8fc7\u8ba1\u7b97\u4e24\u4e2a\u6837\u672c\u5728\u8be5\u7a7a\u95f4\u4e2d\u7684\u5185\u79ef\u6765\u5ea6\u91cf\u5b83\u4eec\u4e4b\u95f4\u7684\u76f8\u4f3c\u5ea6\u3002\u5177\u4f53\u6765\u8bf4\uff0c\u5047\u8bbe\u5c06\u8f93\u5165\u6837\u672c\u6620\u5c04\u5230\u7279\u5f81\u7a7a\u95f4\u4e2d\u7684\u51fd\u6570\u4e3a$\\phi(x)$\uff0c\u5219\u4e24\u4e2a\u6837\u672c$x_i$\u548c$x_j$\u5728\u7279\u5f81\u7a7a\u95f4\u4e2d\u7684\u5185\u79ef\u53ef\u4ee5\u8868\u793a\u4e3a\uff1a<\/p>\n\n\n\n
                                                                  \ufffd(\ufffd\ufffd)\u22c5\ufffd(\ufffd\ufffd)=exp\u2061(\u2212\u2225\ufffd\ufffd\u2212\ufffd\ufffd\u222522\ufffd2)\u03d5<\/em>(x<\/em>i<\/em>\u200b)\u22c5\u03d5<\/em>(x<\/em>j<\/em>\u200b)=exp(\u22122\u03c3<\/em>2\u2225x<\/em>i<\/em>\u200b\u2212x<\/em>j<\/em>\u200b\u22252\u200b)<\/p>\n\n\n\n
                                                                  \u7531\u4e8e\u7279\u5f81\u7a7a\u95f4\u662f\u65e0\u7a77\u7ef4\u7684\uff0c\u56e0\u6b64\u65e0\u6cd5\u76f4\u63a5\u8ba1\u7b97\u6837\u672c\u5728\u7279\u5f81\u7a7a\u95f4\u4e2d\u7684\u5185\u79ef\u3002\u4e0d\u8fc7\uff0c\u901a\u8fc7\u6838\u51fd\u6570\u7684\u5b9a\u4e49\uff0c\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u6838\u51fd\u6570\u7684\u5f62\u5f0f\u6765\u8ba1\u7b97\u4e24\u4e2a\u6837\u672c\u5728\u7279\u5f81\u7a7a\u95f4\u4e2d\u7684\u5185\u79ef\uff0c\u4ece\u800c\u907f\u514d\u4e86\u76f4\u63a5\u8ba1\u7b97\u9ad8\u7ef4\u5411\u91cf\u5185\u79ef\u7684\u8ba1\u7b97\u590d\u6742\u5ea6\u3002<\/p>\n\n\n\n\n\n
                                                                  \"gun<\/figure>\n\n\n\n
                                                                  \u8bf7\u8be6\u7ec6\u89e3\u91ca\u9009\u901a\u5668<\/p>\n\n\n\n
                                                                  \u9009\u901a\u5668\uff0c\u4e5f\u79f0\u4e3a\u95e8\u63a7\u5668\uff08gate\uff09\uff0c\u662f\u6307\u5728\u795e\u7ecf\u7f51\u7edc\u4e2d\u7528\u4e8e\u63a7\u5236\u4fe1\u606f\u6d41\u91cf\u7684\u4e00\u79cd\u673a\u5236\u3002\u5728\u6df1\u5ea6\u5b66\u4e60\u4e2d\uff0c\u901a\u5e38\u91c7\u7528\u95e8\u63a7\u5faa\u73af\u5355\u5143\uff08GRU\uff09\u548c\u957f\u77ed\u65f6\u8bb0\u5fc6\u7f51\u7edc\uff08LSTM\uff09\u7b49\u6a21\u578b\u6765\u5b9e\u73b0\u95e8\u63a7\u673a\u5236\u3002<\/p>\n\n\n\n
                                                                  \u95e8\u63a7\u673a\u5236\u7684\u4f5c\u7528\u662f\u5141\u8bb8\u7f51\u7edc\u9009\u62e9\u6027\u5730\u63a7\u5236\u8f93\u5165\u4fe1\u606f\u7684\u6d41\u52a8\uff0c\u4ece\u800c\u5b9e\u73b0\u5bf9\u4e0d\u540c\u4fe1\u606f\u7684\u52a0\u6743\u5904\u7406\u3002\u5728LSTM\u4e2d\uff0c\u5305\u542b\u8f93\u5165\u95e8\u3001\u9057\u5fd8\u95e8\u548c\u8f93\u51fa\u95e8\u4e09\u79cd\u95e8\u63a7\u673a\u5236\uff0c\u8fd9\u4e9b\u95e8\u63a7\u5236\u4e86\u4fe1\u606f\u5728LSTM\u4e2d\u7684\u6d41\u52a8\u65b9\u5f0f\uff0c\u4f7f\u5f97\u6a21\u578b\u53ef\u4ee5\u6709\u6548\u5730\u5904\u7406\u5e8f\u5217\u6570\u636e\u3002\u5728GRU\u4e2d\uff0c\u5305\u542b\u91cd\u7f6e\u95e8\u548c\u66f4\u65b0\u95e8\u4e24\u79cd\u95e8\u63a7\u673a\u5236\uff0c\u91cd\u7f6e\u95e8\u7528\u4e8e\u63a7\u5236\u54ea\u4e9b\u5386\u53f2\u4fe1\u606f\u9700\u8981\u88ab\u5ffd\u7565\uff0c\u800c\u66f4\u65b0\u95e8\u5219\u63a7\u5236\u54ea\u4e9b\u4fe1\u606f\u9700\u8981\u88ab\u4fdd\u7559\u3002<\/p>\n\n\n\n
                                                                  \u9009\u901a\u5668\u5728\u81ea\u7136\u8bed\u8a00\u5904\u7406\u3001\u8bed\u97f3\u8bc6\u522b\u548c\u56fe\u50cf\u5904\u7406\u7b49\u9886\u57df\u4e2d\u5f97\u5230\u5e7f\u6cdb\u5e94\u7528\u3002\u4f8b\u5982\uff0c\u5728\u81ea\u7136\u8bed\u8a00\u5904\u7406\u4e2d\uff0cLSTM\u548cGRU\u7b49\u6a21\u578b\u53ef\u4ee5\u7528\u4e8e\u5b9e\u73b0\u6587\u672c\u751f\u6210\u3001\u60c5\u611f\u5206\u6790\u548c\u673a\u5668\u7ffb\u8bd1\u7b49\u4efb\u52a1\u3002\u5728\u56fe\u50cf\u5904\u7406\u4e2d\uff0c\u95e8\u63a7\u5377\u79ef\u7f51\u7edc\uff08GCN\uff09\u7b49\u6a21\u578b\u53ef\u4ee5\u7528\u4e8e\u5b9e\u73b0\u56fe\u50cf\u5206\u5272\u548c\u56fe\u50cf\u751f\u6210\u7b49\u4efb\u52a1\u3002<\/p>\n\n\n\n\n\n
                                                                  \"gun<\/figure>\n\n\n\n
                                                                  \u8bf7\u8be6\u7ec6\u89e3\u91ca\u6cdb\u5316<\/p>\n\n\n\n
                                                                  \u6cdb\u5316\u662f\u6307\u673a\u5668\u5b66\u4e60\u7b97\u6cd5\u5728\u8bad\u7ec3\u6570\u636e\u96c6\u4e0a\u5b66\u4e60\u5230\u7684\u77e5\u8bc6\u53ef\u4ee5\u88ab\u5e94\u7528\u4e8e\u65b0\u7684\u6570\u636e\u96c6\uff0c\u800c\u4e0d\u4ec5\u4ec5\u662f\u5728\u8bad\u7ec3\u96c6\u4e0a\u5f97\u5230\u597d\u7684\u7ed3\u679c\u3002\u7b80\u5355\u6765\u8bf4\uff0c\u6cdb\u5316\u80fd\u529b\u597d\u7684\u7b97\u6cd5\u5728\u9762\u5bf9\u65b0\u7684\u6570\u636e\u65f6\uff0c\u53ef\u4ee5\u505a\u51fa\u6b63\u786e\u7684\u9884\u6d4b\u6216\u5206\u7c7b\u3002<\/p>\n\n\n\n
                                                                  \u5728\u673a\u5668\u5b66\u4e60\u4e2d\uff0c\u6211\u4eec\u901a\u5e38\u5c06\u6570\u636e\u96c6\u5212\u5206\u4e3a\u8bad\u7ec3\u96c6\u548c\u6d4b\u8bd5\u96c6\uff0c\u8bad\u7ec3\u96c6\u7528\u4e8e\u8bad\u7ec3\u6a21\u578b\uff0c\u6d4b\u8bd5\u96c6\u7528\u4e8e\u8bc4\u4f30\u6a21\u578b\u7684\u6cdb\u5316\u80fd\u529b\u3002\u5982\u679c\u6a21\u578b\u5728\u6d4b\u8bd5\u96c6\u4e0a\u8868\u73b0\u826f\u597d\uff0c\u5219\u8bf4\u660e\u5b83\u5177\u6709\u826f\u597d\u7684\u6cdb\u5316\u80fd\u529b\u3002<\/p>\n\n\n\n
                                                                  \u5b9e\u73b0\u9ad8\u6cdb\u5316\u80fd\u529b\u7684\u673a\u5668\u5b66\u4e60\u7b97\u6cd5\u9700\u8981\u9075\u5faa\u4ee5\u4e0b\u51e0\u4e2a\u539f\u5219\uff1a<\/p>\n\n\n\n
                                                                    \n
                                                                  1. \u6a21\u578b\u7684\u590d\u6742\u5ea6\u5e94\u8be5\u4e0e\u6570\u636e\u96c6\u7684\u590d\u6742\u5ea6\u76f8\u5339\u914d\uff0c\u907f\u514d\u8fc7\u62df\u5408\u6216\u6b20\u62df\u5408\u3002<\/li>\n\n\n\n
                                                                  2. \u6570\u636e\u96c6\u5e94\u8be5\u8db3\u591f\u5927\uff0c\u80fd\u591f\u4ee3\u8868\u95ee\u9898\u7a7a\u95f4\u7684\u5404\u79cd\u60c5\u51b5\u3002<\/li>\n\n\n\n
                                                                  3. \u6570\u636e\u96c6\u5e94\u8be5\u968f\u673a\u62bd\u6837\uff0c\u907f\u514d\u6a21\u578b\u8fc7\u62df\u5408\u7279\u5b9a\u7684\u6570\u636e\u3002<\/li>\n\n\n\n
                                                                  4. \u4f7f\u7528\u6b63\u5219\u5316\u65b9\u6cd5\u6765\u907f\u514d\u8fc7\u62df\u5408\u3002<\/li>\n\n\n\n
                                                                  5. \u5bf9\u6570\u636e\u8fdb\u884c\u9884\u5904\u7406\uff0c\u5982\u6807\u51c6\u5316\u3001\u5f52\u4e00\u5316\u7b49\u3002<\/li>\n\n\n\n
                                                                  6. \u4f7f\u7528\u4ea4\u53c9\u9a8c\u8bc1\u65b9\u6cd5\u6765\u8bc4\u4f30\u6a21\u578b\u7684\u6cdb\u5316\u80fd\u529b\u3002<\/li>\n<\/ol>\n\n\n\n
                                                                    \u4ee5\u4e0a\u539f\u5219\u90fd\u662f\u4e3a\u4e86\u8ba9\u673a\u5668\u5b66\u4e60\u7b97\u6cd5\u5728\u9762\u5bf9\u65b0\u6570\u636e\u65f6\uff0c\u80fd\u591f\u6709\u66f4\u597d\u7684\u9002\u5e94\u6027\u548c\u6cdb\u5316\u80fd\u529b\u3002<\/p>\n\n\n\n\n\n
                                                                    \"gun<\/figure>\n\n\n\n
                                                                    \u8bf7\u8be6\u7ec6\u89e3\u91ca\u5e7f\u4e49\u51fd\u6570<\/p>\n\n\n\n
                                                                    \u5e7f\u4e49\u51fd\u6570\u662f\u4e00\u7c7b\u5728\u6570\u5b66\u3001\u7269\u7406\u3001\u5de5\u7a0b\u7b49\u9886\u57df\u4e2d\u975e\u5e38\u5e38\u89c1\u7684\u7279\u6b8a\u51fd\u6570\uff0c\u4e5f\u88ab\u79f0\u4e3a\u5206\u5e03\u51fd\u6570\u3002\u5e7f\u4e49\u51fd\u6570\u7684\u5b9a\u4e49\u662f\u5bf9\u5b9e\u51fd\u6570\u7684\u63a8\u5e7f\uff0c\u5b83\u53ef\u4ee5\u770b\u4f5c\u662f\u5bf9\u51fd\u6570\u7684\u67d0\u79cd\u6cdb\u5316\uff0c\u6709\u65f6\u4e5f\u53ef\u4ee5\u770b\u4f5c\u662f\u201c\u5e26\u6743\u51fd\u6570\u201d\u7684\u6982\u5ff5\u3002<\/p>\n\n\n\n
                                                                    \u901a\u5e38\u60c5\u51b5\u4e0b\uff0c\u51fd\u6570\u662f\u5728\u70b9\u4e0a\u5b9a\u4e49\u7684\uff0c\u800c\u5e7f\u4e49\u51fd\u6570\u4e0d\u4e00\u5b9a\u662f\u70b9\u51fd\u6570\uff0c\u800c\u662f\u5b9a\u4e49\u5728\u51fd\u6570\u7c7b\u4e0a\u7684\u51fd\u6570\u3002\u56e0\u6b64\uff0c\u5e7f\u4e49\u51fd\u6570\u4e0e\u70b9\u51fd\u6570\u7684\u533a\u522b\u5728\u4e8e\uff0c\u5b83\u7684\u5b9a\u4e49\u57df\u4e0d\u662f\u6570\u8f74\u4e0a\u7684\u70b9\uff0c\u800c\u662f\u5305\u62ec\u6570\u8f74\u4e0a\u7684\u6240\u6709\u8fde\u7eed\u51fd\u6570\u6216\u8005\u5927\u90e8\u5206\u51fd\u6570\u7684\u96c6\u5408\u3002<\/p>\n\n\n\n
                                                                    \u5e7f\u4e49\u51fd\u6570\u7684\u4e00\u4e2a\u91cd\u8981\u6027\u8d28\u662f\u5b83\u4eec\u53ef\u4ee5\u7528\u4e8e\u63cf\u8ff0\u67d0\u4e9b\u7269\u7406\u73b0\u8c61\u4e2d\u7684\u5f31\u89e3\u6216\u8005\u4e0d\u8fde\u7eed\u89e3\uff0c\u4f8b\u5982\u5728\u70ed\u4f20\u5bfc\u65b9\u7a0b\u3001\u7535\u78c1\u573a\u65b9\u7a0b\u3001\u91cf\u5b50\u529b\u5b66\u6ce2\u52a8\u65b9\u7a0b\u7b49\u7684\u6c42\u89e3\u4e2d\uff0c\u5e7f\u4e49\u51fd\u6570\u7ecf\u5e38\u88ab\u7528\u4e8e\u63cf\u8ff0\u4e0d\u8fde\u7eed\u6761\u4ef6\u7684\u8fb9\u754c\u6761\u4ef6\u6216\u8005\u521d\u503c\u6761\u4ef6\u3002<\/p>\n\n\n\n
                                                                    \u5e38\u89c1\u7684\u5e7f\u4e49\u51fd\u6570\u5305\u62ec\u72c4\u62c9\u514b\u03b4\u51fd\u6570\uff08Dirac delta function\uff09\u3001\u6d77\u68ee\u4f2fc\u51fd\u6570\uff08Heaviside step function\uff09\u3001\u62c9\u666e\u62c9\u65af\u51fd\u6570\uff08Laplace function\uff09\u7b49\uff0c\u8fd9\u4e9b\u5e7f\u4e49\u51fd\u6570\u5728\u7269\u7406\u5b66\u3001\u5de5\u7a0b\u5b66\u3001\u6570\u5b66\u548c\u8ba1\u7b97\u673a\u79d1\u5b66\u7b49\u9886\u57df\u90fd\u6709\u5e7f\u6cdb\u5e94\u7528\u3002<\/p>\n\n\n\n
                                                                    \u5e7f\u4e49\u51fd\u6570\u4e5f\u53ef\u4ee5\u901a\u8fc7\u4e00\u4e9b\u64cd\u4f5c\u6765\u5f97\u5230\u65b0\u7684\u5e7f\u4e49\u51fd\u6570\uff0c\u4f8b\u5982\u52a0\u3001\u4e58\u3001\u5bfc\u3001\u5377\u79ef\u3001\u5085\u91cc\u53f6\u53d8\u6362\u7b49\u64cd\u4f5c\u3002\u540c\u65f6\uff0c\u5e7f\u4e49\u51fd\u6570\u8fd8\u6709\u4e00\u4e9b\u91cd\u8981\u7684\u6027\u8d28\uff0c\u4f8b\u5982\u7ebf\u6027\u6027\u3001\u53ef\u5fae\u6027\u3001\u53ef\u79ef\u6027\u7b49\u7b49\uff0c\u8fd9\u4e9b\u6027\u8d28\u4f7f\u5f97\u5e7f\u4e49\u51fd\u6570\u5728\u5404\u4e2a\u9886\u57df\u4e2d\u90fd\u6709\u7740\u5e7f\u6cdb\u7684\u5e94\u7528\u3002<\/p>\n\n\n\n\n\n
                                                                    \"gun<\/figure>\n\n\n\n
                                                                    \u8bf7\u8be6\u7ec6\u89e3\u91ca\u751f\u6210\u6a21\u578b<\/p>\n\n\n\n
                                                                    \u751f\u6210\u6a21\u578b\u662f\u4e00\u7c7b\u53ef\u4ee5\u5b66\u4e60\u4e00\u4e2a\u6570\u636e\u96c6\u7684\u6982\u7387\u5206\u5e03\u5e76\u4e14\u80fd\u591f\u5229\u7528\u8be5\u5206\u5e03\u6765\u751f\u6210\u65b0\u7684\u6837\u672c\u6570\u636e\u7684\u6a21\u578b\u3002\u5728\u673a\u5668\u5b66\u4e60\u4e2d\uff0c\u901a\u5e38\u4f7f\u7528\u751f\u6210\u6a21\u578b\u6765\u5b66\u4e60\u6837\u672c\u6570\u636e\u96c6\u4e2d\u7684\u6570\u636e\u5206\u5e03\uff0c\u4ee5\u4fbf\u751f\u6210\u4e0e\u539f\u59cb\u6570\u636e\u96c6\u7c7b\u4f3c\u7684\u65b0\u6570\u636e\u3002\u8fd9\u4e9b\u6a21\u578b\u901a\u5e38\u53ef\u4ee5\u7528\u6765\u751f\u6210\u56fe\u50cf\u3001\u6587\u672c\u548c\u97f3\u9891\u7b49\u5185\u5bb9\u3002<\/p>\n\n\n\n
                                                                    \u751f\u6210\u6a21\u578b\u4e0e\u5224\u522b\u6a21\u578b\u4e0d\u540c\u3002\u5224\u522b\u6a21\u578b\u662f\u4e00\u79cd\u76f4\u63a5\u5b66\u4e60\u8f93\u5165\u548c\u8f93\u51fa\u4e4b\u95f4\u7684\u5173\u7cfb\uff0c\u5373\u8f93\u5165\u6620\u5c04\u5230\u8f93\u51fa\u7684\u51fd\u6570\u3002\u4e0e\u6b64\u76f8\u6bd4\uff0c\u751f\u6210\u6a21\u578b\u662f\u4e00\u79cd\u5b66\u4e60\u8f93\u5165\u548c\u8f93\u51fa\u4e4b\u95f4\u8054\u5408\u6982\u7387\u5206\u5e03\u7684\u6a21\u578b\uff0c\u53ef\u4ee5\u5bf9\u65b0\u6570\u636e\u8fdb\u884c\u91c7\u6837\u3002<\/p>\n\n\n\n
                                                                    \u751f\u6210\u6a21\u578b\u53ef\u4ee5\u5206\u4e3a\u6982\u7387\u5bc6\u5ea6\u4f30\u8ba1\u6a21\u578b\u548c\u751f\u6210\u5bf9\u6297\u7f51\u7edc (GAN) \u6a21\u578b\u4e24\u7c7b\u3002\u6982\u7387\u5bc6\u5ea6\u4f30\u8ba1\u6a21\u578b\u57fa\u4e8e\u7ed9\u5b9a\u7684\u6570\u636e\u96c6\u5b66\u4e60\u6982\u7387\u5206\u5e03\uff0c\u4f8b\u5982\u9ad8\u65af\u6df7\u5408\u6a21\u578b\u3001\u53d8\u5206\u81ea\u7f16\u7801\u5668\u548c\u81ea\u56de\u5f52\u6a21\u578b\u3002\u751f\u6210\u5bf9\u6297\u7f51\u7edc\u6a21\u578b\u901a\u8fc7\u540c\u65f6\u8bad\u7ec3\u4e00\u4e2a\u751f\u6210\u5668\u548c\u4e00\u4e2a\u5224\u522b\u5668\uff0c\u4f7f\u5f97\u751f\u6210\u5668\u53ef\u4ee5\u751f\u6210\u770b\u8d77\u6765\u50cf\u771f\u5b9e\u6570\u636e\u7684\u65b0\u6570\u636e\uff0c\u800c\u5224\u522b\u5668\u5219\u5c1d\u8bd5\u5c06\u751f\u6210\u7684\u6570\u636e\u4e0e\u771f\u5b9e\u6570\u636e\u533a\u5206\u5f00\u6765\u3002<\/p>\n\n\n\n
                                                                    \u751f\u6210\u6a21\u578b\u7684\u4f18\u70b9\u5728\u4e8e\u53ef\u4ee5\u751f\u6210\u770b\u8d77\u6765\u771f\u5b9e\u7684\u65b0\u6570\u636e\uff0c\u8fd9\u5728\u6570\u636e\u589e\u5f3a\u3001\u751f\u6210\u6587\u672c\u3001\u56fe\u50cf\u548c\u97f3\u9891\u7b49\u4efb\u52a1\u4e2d\u975e\u5e38\u6709\u7528\u3002\u6b64\u5916\uff0c\u751f\u6210\u6a21\u578b\u4e5f\u53ef\u4ee5\u7528\u4e8e\u7f3a\u5931\u6570\u636e\u7684\u63d2\u503c\u3001\u5f02\u5e38\u68c0\u6d4b\u548c\u56fe\u50cf\u4fee\u590d\u7b49\u4efb\u52a1\u3002<\/p>\n\n\n\n\n\n
                                                                    \"gun<\/figure>\n\n\n\n
                                                                    \u8bf7\u8be6\u7ec6\u89e3\u91ca\u68af\u5ea6\u622a\u65ad<\/p>\n\n\n\n
                                                                    \u68af\u5ea6\u622a\u65ad\uff08Gradient Clipping\uff09\u662f\u4e00\u79cd\u7528\u4e8e\u7f13\u89e3\u68af\u5ea6\u7206\u70b8\uff08Gradient Exploding\uff09\u95ee\u9898\u7684\u6280\u672f\u3002\u5728\u6df1\u5ea6\u795e\u7ecf\u7f51\u7edc\u7684\u8bad\u7ec3\u8fc7\u7a0b\u4e2d\uff0c\u68af\u5ea6\u53ef\u80fd\u4f1a\u53d8\u5f97\u975e\u5e38\u5927\uff0c\u5bfc\u81f4\u6743\u91cd\u7684\u66f4\u65b0\u8fc7\u4e8e\u5267\u70c8\uff0c\u6a21\u578b\u65e0\u6cd5\u6536\u655b\u6216\u4e0d\u7a33\u5b9a\u3002\u68af\u5ea6\u622a\u65ad\u6280\u672f\u901a\u8fc7\u5c06\u68af\u5ea6\u9650\u5236\u5728\u4e00\u5b9a\u8303\u56f4\u5185\uff0c\u9632\u6b62\u68af\u5ea6\u8fc7\u5927\uff0c\u4f7f\u5f97\u6743\u91cd\u66f4\u65b0\u8fc7\u7a0b\u66f4\u52a0\u5e73\u7f13\u7a33\u5b9a\u3002<\/p>\n\n\n\n
                                                                    \u5177\u4f53\u6765\u8bf4\uff0c\u68af\u5ea6\u622a\u65ad\u6280\u672f\u5c06\u68af\u5ea6\u88c1\u526a\u4e3a\u4e00\u4e2a\u6700\u5927\u503c\uff0c\u5982\u679c\u68af\u5ea6\u7684\u8303\u6570\u8d85\u8fc7\u4e86\u8fd9\u4e2a\u6700\u5927\u503c\uff0c\u5219\u5c06\u5176\u7f29\u653e\u5230\u6700\u5927\u503c\u7684\u8303\u56f4\u5185\u3002\u8fd9\u4e2a\u6700\u5927\u503c\u53ef\u4ee5\u4f5c\u4e3a\u4e00\u4e2a\u8d85\u53c2\u6570\u8fdb\u884c\u8c03\u6574\uff0c\u901a\u5e38\u8bbe\u7f6e\u4e3a\u4e00\u4e2a\u8f83\u5c0f\u7684\u6570\u503c\uff0c\u6bd4\u59821.0\u62165.0\u3002<\/p>\n\n\n\n
                                                                    \u68af\u5ea6\u622a\u65ad\u53ef\u4ee5\u5e94\u7528\u4e8e\u4efb\u4f55\u4f7f\u7528\u68af\u5ea6\u4e0b\u964d\u6216\u5176\u53d8\u79cd\u7684\u4f18\u5316\u7b97\u6cd5\uff0c\u5982\u968f\u673a\u68af\u5ea6\u4e0b\u964d\uff08SGD\uff09\u3001\u52a8\u91cf\u68af\u5ea6\u4e0b\u964d\uff08Momentum\uff09\u3001\u81ea\u9002\u5e94\u68af\u5ea6\u7b97\u6cd5\uff08Adam\u3001Adagrad\u3001Adadelta\u7b49\uff09\u7b49\u3002<\/p>\n\n\n\n
                                                                    \u4ee5\u4e0b\u662f\u68af\u5ea6\u622a\u65ad\u7684\u4ee3\u7801\u793a\u4f8b\uff1a<\/p>\n\n\n\n
                                                                    pythonCopy codeimport torch.nn as nn\nimport torch.optim as optim\n\n# \u5b9a\u4e49\u6a21\u578b\nclass MyModel(nn.Module):\n    def __init__(self):\n        super(MyModel, self).__init__()\n        self.linear = nn.Linear(10, 1)\n\n    def forward(self, x):\n        return self.linear(x)\n\nmodel = MyModel()\n\n# \u5b9a\u4e49\u635f\u5931\u51fd\u6570\u548c\u4f18\u5316\u5668\ncriterion = nn.MSELoss()\noptimizer = optim.SGD(model.parameters(), lr=0.01)\n\n# \u5b9a\u4e49\u68af\u5ea6\u622a\u65ad\u7684\u6700\u5927\u503c\nclip_value = 1.0\n\n# \u8bad\u7ec3\u8fc7\u7a0b\u4e2d\u8fdb\u884c\u68af\u5ea6\u622a\u65ad\nfor input, target in data_loader:\n    # \u524d\u5411\u4f20\u64ad\n    output = model(input)\n    # \u8ba1\u7b97\u635f\u5931\n    loss = criterion(output, target)\n    # \u53cd\u5411\u4f20\u64ad\n    optimizer.zero_grad()\n    loss.backward()\n    # \u5bf9\u68af\u5ea6\u8fdb\u884c\u622a\u65ad\n    nn.utils.clip_grad_norm_(model.parameters(), clip_value)\n    # \u66f4\u65b0\u53c2\u6570\n    optimizer.step()\n<\/code><\/pre>\n\n\n\n
                                                                    \u5728\u4e0a\u9762\u7684\u4ee3\u7801\u4e2d\uff0cnn.utils.clip_grad_norm_()<\/code>\u51fd\u6570\u5bf9\u6a21\u578b\u7684\u68af\u5ea6\u8fdb\u884c\u622a\u65ad\uff0c\u5e76\u5c06\u88c1\u526a\u540e\u7684\u68af\u5ea6\u7528\u4e8e\u66f4\u65b0\u6a21\u578b\u53c2\u6570\u3002<\/p>\n\n\n\n\n\n
                                                                    \"gun<\/figure>\n\n\n\n
                                                                    \u8bf7\u8be6\u7ec6\u89e3\u91ca\u7f51\u683c\u641c\u7d22<\/p>\n\n\n\n
                                                                    \u7f51\u683c\u641c\u7d22\u662f\u4e00\u79cd\u8d85\u53c2\u6570\u4f18\u5316\u65b9\u6cd5\uff0c\u5b83\u53ef\u4ee5\u5728\u7ed9\u5b9a\u7684\u8d85\u53c2\u6570\u7a7a\u95f4\u4e2d\u7a77\u4e3e\u6bcf\u79cd\u53ef\u80fd\u7684\u8d85\u53c2\u6570\u7ec4\u5408\uff0c\u5e76\u9009\u62e9\u8868\u73b0\u6700\u597d\u7684\u8d85\u53c2\u6570\u7ec4\u5408\u4f5c\u4e3a\u6a21\u578b\u7684\u6700\u7ec8\u8d85\u53c2\u6570\u8bbe\u7f6e\u3002<\/p>\n\n\n\n
                                                                    \u5177\u4f53\u6765\u8bf4\uff0c\u7f51\u683c\u641c\u7d22\u4f1a\u5148\u5b9a\u4e49\u4e00\u7ec4\u8d85\u53c2\u6570\u7a7a\u95f4\uff0c\u901a\u5e38\u5305\u62ec\u6a21\u578b\u7684\u5404\u4e2a\u8d85\u53c2\u6570\u53ca\u5176\u5bf9\u5e94\u7684\u53d6\u503c\u8303\u56f4\uff0c\u7136\u540e\u7a77\u4e3e\u6240\u6709\u53ef\u80fd\u7684\u8d85\u53c2\u6570\u7ec4\u5408\uff0c\u5bf9\u4e8e\u6bcf\u4e2a\u7ec4\u5408\uff0c\u5229\u7528\u4ea4\u53c9\u9a8c\u8bc1\u7b49\u65b9\u6cd5\u5728\u8bad\u7ec3\u96c6\u4e0a\u8bad\u7ec3\u6a21\u578b\u5e76\u5728\u9a8c\u8bc1\u96c6\u4e0a\u8fdb\u884c\u8bc4\u4f30\uff0c\u6700\u540e\u9009\u62e9\u8868\u73b0\u6700\u597d\u7684\u8d85\u53c2\u6570\u7ec4\u5408\u4f5c\u4e3a\u6a21\u578b\u7684\u6700\u7ec8\u8d85\u53c2\u6570\u8bbe\u7f6e\u3002<\/p>\n\n\n\n
                                                                    \u7f51\u683c\u641c\u7d22\u7684\u4f18\u70b9\u662f\u7b80\u5355\u6613\u7528\uff0c\u4e0d\u9700\u8981\u5bf9\u6a21\u578b\u8fdb\u884c\u592a\u591a\u7684\u5047\u8bbe\u6216\u8005\u5148\u9a8c\u77e5\u8bc6\uff0c\u540c\u65f6\u4e5f\u9002\u7528\u4e8e\u5927\u591a\u6570\u7684\u6a21\u578b\u548c\u6570\u636e\u96c6\u3002\u4f46\u662f\u7f3a\u70b9\u662f\u9700\u8981\u8017\u8d39\u5927\u91cf\u7684\u8ba1\u7b97\u8d44\u6e90\u548c\u65f6\u95f4\uff0c\u5c24\u5176\u662f\u5f53\u8d85\u53c2\u6570\u7a7a\u95f4\u6bd4\u8f83\u5927\u65f6\u3002\u6b64\u5916\uff0c\u7f51\u683c\u641c\u7d22\u7684\u641c\u7d22\u7a7a\u95f4\u5f88\u5bb9\u6613\u53d7\u5230\u8d85\u53c2\u6570\u6570\u91cf\u7684\u5f71\u54cd\uff0c\u5f53\u8d85\u53c2\u6570\u6570\u91cf\u589e\u591a\u65f6\uff0c\u641c\u7d22\u7a7a\u95f4\u4e5f\u4f1a\u5448\u6307\u6570\u7ea7\u522b\u589e\u957f\uff0c\u4ece\u800c\u5bfc\u81f4\u8ba1\u7b97\u65f6\u95f4\u6210\u500d\u589e\u52a0\u3002<\/p>\n\n\n\n
                                                                    \u4e0b\u9762\u662f\u4e00\u4e2a\u4f7f\u7528\u7f51\u683c\u641c\u7d22\u8fdb\u884c\u8d85\u53c2\u6570\u4f18\u5316\u7684\u793a\u4f8b\u4ee3\u7801\uff1a<\/p>\n\n\n\n
                                                                    pythonCopy codefrom sklearn.model_selection import GridSearchCV\nfrom sklearn.svm import SVC\nfrom sklearn.datasets import load_digits\n\n# \u52a0\u8f7d\u624b\u5199\u6570\u5b57\u6570\u636e\u96c6\ndigits = load_digits()\n\n# \u5b9a\u4e49SVM\u6a21\u578b\u548c\u8d85\u53c2\u6570\u7a7a\u95f4\nsvm_model = SVC()\nparam_grid = {'C': [1, 10, 100], 'kernel': ['linear', 'rbf'], 'gamma': [0.001, 0.01, 0.1]}\n\n# \u7f51\u683c\u641c\u7d22\ngrid_search = GridSearchCV(svm_model, param_grid, cv=5)\ngrid_search.fit(digits.data, digits.target)\n\n# \u8f93\u51fa\u6700\u4f73\u8d85\u53c2\u6570\u7ec4\u5408\u548c\u5bf9\u5e94\u7684\u51c6\u786e\u7387\nprint(grid_search.best_params_)\nprint(grid_search.best_score_)\n<\/code><\/pre>\n\n\n\n
                                                                    \u5728\u4e0a\u9762\u7684\u793a\u4f8b\u4ee3\u7801\u4e2d\uff0c\u6211\u4eec\u4f7f\u7528Scikit-learn\u5e93\u4e2d\u7684GridSearchCV\u7c7b\u5b9e\u73b0\u7f51\u683c\u641c\u7d22\u3002\u9996\u5148\uff0c\u6211\u4eec\u52a0\u8f7d\u4e86Scikit-learn\u5e93\u4e2d\u81ea\u5e26\u7684\u624b\u5199\u6570\u5b57\u6570\u636e\u96c6\uff0c\u7136\u540e\u5b9a\u4e49\u4e86\u4e00\u4e2aSVM\u6a21\u578b\u548c\u8d85\u53c2\u6570\u7a7a\u95f4\u3002\u63a5\u7740\uff0c\u6211\u4eec\u521b\u5efa\u4e86\u4e00\u4e2aGridSearchCV\u5bf9\u8c61\uff0c\u5e76\u5c06SVM\u6a21\u578b\u548c\u8d85\u53c2\u6570\u7a7a\u95f4\u4f5c\u4e3a\u53c2\u6570\u4f20\u5165\uff0c\u540c\u65f6\u6307\u5b9a\u4e86\u4ea4\u53c9\u9a8c\u8bc1\u7684\u6298\u6570\u4e3a5\u3002\u6700\u540e\uff0c\u6211\u4eec\u8c03\u7528GridSearchCV\u5bf9\u8c61\u7684fit\u65b9\u6cd5\u8fdb\u884c\u7f51\u683c\u641c\u7d22\uff0c\u5e76\u8f93\u51fa\u4e86\u6700\u4f73\u8d85\u53c2\u6570\u7ec4\u5408\u548c\u5bf9\u5e94\u7684\u51c6\u786e\u7387\u3002<\/p>\n\n\n\n\n\n
                                                                    \"gun<\/figure>\n\n\n\n
                                                                    \u8bf7\u8be6\u7ec6\u89e3\u91ca\u6c49\u660e\u8ddd\u79bb<\/p>\n\n\n\n
                                                                    \u6c49\u660e\u8ddd\u79bb\uff08Hamming distance\uff09\u662f\u6307\u5728\u4fe1\u606f\u79d1\u5b66\u4e2d\uff0c\u4e24\u4e2a\u7b49\u957f\u5b57\u7b26\u4e32\u5728\u5bf9\u5e94\u4f4d\u7f6e\u4e0a\u4e0d\u540c\u5b57\u7b26\u7684\u6570\u76ee\u3002\u4e5f\u5c31\u662f\u8bf4\uff0c\u5b83\u662f\u5c06\u4e00\u4e2a\u5b57\u7b26\u4e32\u53d8\u6362\u6210\u53e6\u5916\u4e00\u4e2a\u5b57\u7b26\u4e32\u6240\u9700\u8981\u66ff\u6362\u7684\u5b57\u7b26\u4e2a\u6570\u3002<\/p>\n\n\n\n
                                                                    \u6c49\u660e\u8ddd\u79bb\u7ecf\u5e38\u7528\u4e8e\u7f16\u7801\u7406\u8bba\u3001\u4fe1\u606f\u5904\u7406\u4ee5\u53ca\u9519\u8bef\u68c0\u6d4b\u548c\u7ea0\u6b63\u7b49\u9886\u57df\u3002\u5728\u8ba1\u7b97\u673a\u79d1\u5b66\u4e2d\uff0c\u6c49\u660e\u8ddd\u79bb\u7528\u4e8e\u5ea6\u91cf\u4e24\u4e2a\u4e8c\u8fdb\u5236\u4e32\u4e4b\u95f4\u7684\u8ddd\u79bb\uff0c\u901a\u5e38\u7528\u4e8e\u6570\u636e\u4f20\u8f93\u9519\u8bef\u63a7\u5236\u4e2d\u7684\u7ea0\u9519\u7801\u7684\u8bbe\u8ba1\u3002<\/p>\n\n\n\n
                                                                    \u4f8b\u5982\uff0c\u5bf9\u4e8e\u4e24\u4e2a\u7b49\u957f\u7684\u5b57\u7b26\u4e32\uff0c1011101\u548c1001001\uff0c\u5b83\u4eec\u7684\u6c49\u660e\u8ddd\u79bb\u662f2\uff0c\u56e0\u4e3a\u5b83\u4eec\u5728\u7b2c2\u4e2a\u548c\u7b2c5\u4e2a\u4f4d\u7f6e\u4e0a\u7684\u5b57\u7b26\u4e0d\u540c\u3002<\/p>\n\n\n\n
                                                                    \u6c49\u660e\u8ddd\u79bb\u7684\u8ba1\u7b97\u53ef\u4ee5\u901a\u8fc7\u6bd4\u8f83\u4e24\u4e2a\u5b57\u7b26\u4e32\u5728\u76f8\u5e94\u4f4d\u7f6e\u4e0a\u7684\u5b57\u7b26\u662f\u5426\u76f8\u540c\u6765\u5b9e\u73b0\u3002\u5bf9\u4e8e\u957f\u5ea6\u4e3a $n$ \u7684\u5b57\u7b26\u4e32\uff0c\u5b83\u4eec\u4e4b\u95f4\u7684\u6c49\u660e\u8ddd\u79bb\u53ef\u4ee5\u7528\u4ee5\u4e0b\u516c\u5f0f\u8ba1\u7b97\uff1a<\/p>\n\n\n\n
                                                                    \ufffd\ufffd(\ufffd,\ufffd)=\u2211\ufffd=1\ufffd1{\ufffd\ufffd\u2260\ufffd\ufffd}d<\/em>H<\/em>\u200b(x<\/em>,y<\/em>)=\u2211i<\/em>=1n<\/em>\u200b1{x<\/em>i<\/em>\u200b\ue020=y<\/em>i<\/em>\u200b}\u200b<\/p>\n\n\n\n
                                                                    \u5176\u4e2d $x_i$ \u548c $y_i$ \u5206\u522b\u662f\u5b57\u7b26\u4e32 $x$ \u548c $y$ \u7684\u7b2c $i$ \u4e2a\u5b57\u7b26\uff0c$\\mathbb{1}$ \u662f\u6307\u793a\u51fd\u6570\uff0c\u5982\u679c\u6761\u4ef6\u6210\u7acb\u5219\u8fd4\u56de1\uff0c\u5426\u5219\u8fd4\u56de0\u3002<\/p>\n\n\n\n\n\n
                                                                    \"gun<\/figure>\n\n\n\n
                                                                    \u8bf7\u8be6\u7ec6\u89e3\u91ca\u5f02\u65b9\u5dee<\/p>\n\n\n\n
                                                                    \u5f02\u65b9\u5dee\uff08Heteroscedasticity\uff09\u662f\u6307\u968f\u673a\u53d8\u91cf\u7684\u65b9\u5dee\u4e0d\u662f\u6052\u5b9a\u7684\uff0c\u800c\u662f\u968f\u7740\u81ea\u53d8\u91cf\u6216\u5176\u4ed6\u56e0\u7d20\u7684\u53d8\u5316\u800c\u53d8\u5316\u7684\u73b0\u8c61\u3002\u5728\u56de\u5f52\u5206\u6790\u4e2d\uff0c\u5f02\u65b9\u5dee\u4f1a\u5bfc\u81f4\u6807\u51c6\u8bef\u5dee\u4f30\u8ba1\u504f\u5dee\uff0c\u8fdb\u800c\u5f71\u54cd\u7cfb\u6570\u7684\u663e\u8457\u6027\u68c0\u9a8c\u548c\u7f6e\u4fe1\u533a\u95f4\u7684\u4f30\u8ba1\u3002\u56e0\u6b64\uff0c\u6b63\u786e\u8bc6\u522b\u5e76\u5904\u7406\u5f02\u65b9\u5dee\u662f\u56de\u5f52\u5206\u6790\u4e2d\u7684\u91cd\u8981\u95ee\u9898\u3002<\/p>\n\n\n\n
                                                                    \u5728\u5b9e\u9645\u5e94\u7528\u4e2d\uff0c\u5e38\u5e38\u901a\u8fc7\u68c0\u9a8c\u6b8b\u5dee\u7684\u65b9\u5dee\u662f\u5426\u4e0e\u9884\u6d4b\u503c\u76f8\u5173\u6765\u68c0\u9a8c\u5f02\u65b9\u5dee\u3002\u5982\u679c\u68c0\u9a8c\u7ed3\u679c\u53d1\u73b0\u5b58\u5728\u5f02\u65b9\u5dee\uff0c\u53ef\u4ee5\u91c7\u7528\u4ee5\u4e0b\u65b9\u6cd5\u8fdb\u884c\u5904\u7406\uff1a<\/p>\n\n\n\n
                                                                    1.\u5f02\u65b9\u5dee\u7a33\u5065\u6807\u51c6\u8bef\u5dee\u4f30\u8ba1\uff1a\u901a\u8fc7\u4f7f\u7528\u5f02\u65b9\u5dee\u7a33\u5065\u6807\u51c6\u8bef\u5dee\u6765\u4ee3\u66ff\u666e\u901a\u7684\u6807\u51c6\u8bef\u5dee\uff0c\u53ef\u4ee5\u4f7f\u5f97\u56de\u5f52\u7cfb\u6570\u7684\u4f30\u8ba1\u66f4\u52a0\u51c6\u786e\u3002\u5f02\u65b9\u5dee\u7a33\u5065\u6807\u51c6\u8bef\u5dee\u4f30\u8ba1\u901a\u5e38\u4f7f\u7528\u5e7f\u4e49\u6700\u5c0f\u4e8c\u4e58\u6cd5\uff08GLS\uff09\u6765\u5b9e\u73b0\u3002<\/p>\n\n\n\n
                                                                    2.\u6570\u636e\u53d8\u6362\uff1a\u901a\u8fc7\u5bf9\u81ea\u53d8\u91cf\u6216\u56e0\u53d8\u91cf\u8fdb\u884c\u53d8\u6362\uff0c\u53ef\u4ee5\u4f7f\u5f97\u6570\u636e\u7684\u65b9\u5dee\u53d8\u5f97\u66f4\u52a0\u7a33\u5b9a\uff0c\u4ece\u800c\u907f\u514d\u5f02\u65b9\u5dee\u7684\u5f71\u54cd\u3002\u5e38\u89c1\u7684\u6570\u636e\u53d8\u6362\u65b9\u6cd5\u5305\u62ec\u5bf9\u6570\u53d8\u6362\u3001\u5e73\u65b9\u6839\u53d8\u6362\u7b49\u3002<\/p>\n\n\n\n
                                                                    3.\u4f7f\u7528\u5f02\u65b9\u5dee\u6a21\u578b\uff1a\u5728\u67d0\u4e9b\u60c5\u51b5\u4e0b\uff0c\u53ef\u4ee5\u4f7f\u7528\u5177\u6709\u5f02\u65b9\u5dee\u7ed3\u6784\u7684\u6a21\u578b\u6765\u5904\u7406\u5f02\u65b9\u5dee\u3002\u4f8b\u5982\uff0c\u53ef\u4ee5\u4f7f\u7528\u5e7f\u4e49\u7ebf\u6027\u6a21\u578b\uff08GLM\uff09\u548c\u5e7f\u4e49\u52a0\u6027\u6a21\u578b\uff08GAM\uff09\u7b49\u3002<\/p>\n\n\n\n
                                                                    \u5904\u7406\u5f02\u65b9\u5dee\u7684\u65b9\u6cd5\u9700\u8981\u6839\u636e\u5177\u4f53\u60c5\u51b5\u800c\u5b9a\uff0c\u9009\u62e9\u5408\u9002\u7684\u65b9\u6cd5\u53ef\u4ee5\u4f7f\u5f97\u56de\u5f52\u5206\u6790\u7684\u7ed3\u679c\u66f4\u52a0\u51c6\u786e\u548c\u53ef\u9760\u3002<\/p>\n\n\n\n\n\n
                                                                    \"gun<\/figure>\n\n\n\n
                                                                    \u8bf7\u8be6\u7ec6\u89e3\u91ca\u8d85\u53c2\u6570<\/p>\n\n\n\n
                                                                    \u8d85\u53c2\u6570\uff08hyperparameters\uff09\u662f\u673a\u5668\u5b66\u4e60\u6a21\u578b\u4e2d\u9700\u8981\u624b\u52a8\u8bbe\u7f6e\u7684\u53c2\u6570\uff0c\u4e0d\u540c\u4e8e\u6a21\u578b\u4e2d\u7684\u6743\u91cd\uff08weights\uff09\uff0c\u8d85\u53c2\u6570\u51b3\u5b9a\u4e86\u6a21\u578b\u7684\u57fa\u672c\u7ed3\u6784\u548c\u5b66\u4e60\u8fc7\u7a0b\u7684\u4e00\u4e9b\u5173\u952e\u6027\u8d28\u3002\u901a\u5e38\u9700\u8981\u901a\u8fc7\u8bd5\u9519\u7684\u65b9\u5f0f\u4e0d\u65ad\u8c03\u6574\u8d85\u53c2\u6570\u7684\u503c\uff0c\u4ee5\u627e\u5230\u6700\u4f73\u7684\u6a21\u578b\u6027\u80fd\u3002<\/p>\n\n\n\n
                                                                    \u4e00\u4e9b\u5e38\u89c1\u7684\u8d85\u53c2\u6570\u5305\u62ec\uff1a<\/p>\n\n\n\n
                                                                    1.\u5b66\u4e60\u7387\uff08learning rate\uff09\uff1a\u5b66\u4e60\u7387\u51b3\u5b9a\u4e86\u6743\u91cd\u66f4\u65b0\u7684\u6b65\u957f\u5927\u5c0f\uff0c\u662f\u8bad\u7ec3\u8fc7\u7a0b\u4e2d\u6700\u91cd\u8981\u7684\u8d85\u53c2\u6570\u4e4b\u4e00\u3002<\/p>\n\n\n\n
                                                                    2.\u6b63\u5219\u5316\u7cfb\u6570\uff08regularization coefficient\uff09\uff1a\u7528\u4e8e\u63a7\u5236\u6a21\u578b\u590d\u6742\u5ea6\uff0c\u9632\u6b62\u8fc7\u62df\u5408\u3002<\/p>\n\n\n\n
                                                                    3.\u6279\u5927\u5c0f\uff08batch size\uff09\uff1a\u7528\u4e8e\u6307\u5b9a\u6bcf\u6b21\u8fed\u4ee3\u65f6\u8bad\u7ec3\u7684\u6837\u672c\u6570\u91cf\u3002<\/p>\n\n\n\n
                                                                    4.\u7f51\u7edc\u5c42\u6570\uff08number of layers\uff09\uff1a\u795e\u7ecf\u7f51\u7edc\u4e2d\u7684\u5c42\u6570\u662f\u51b3\u5b9a\u7f51\u7edc\u590d\u6742\u5ea6\u7684\u4e00\u4e2a\u91cd\u8981\u56e0\u7d20\u3002<\/p>\n\n\n\n
                                                                    5.\u795e\u7ecf\u5143\u6570\u91cf\uff08number of neurons\uff09\uff1a\u6bcf\u4e2a\u795e\u7ecf\u7f51\u7edc\u5c42\u4e2d\u7684\u795e\u7ecf\u5143\u6570\u91cf\u662f\u53e6\u4e00\u4e2a\u5f71\u54cd\u7f51\u7edc\u590d\u6742\u5ea6\u7684\u56e0\u7d20\u3002<\/p>\n\n\n\n
                                                                    6.\u6fc0\u6d3b\u51fd\u6570\uff08activation function\uff09\uff1a\u4e0d\u540c\u7684\u6fc0\u6d3b\u51fd\u6570\u4f1a\u5bf9\u7f51\u7edc\u7684\u6027\u80fd\u4ea7\u751f\u4e0d\u540c\u7684\u5f71\u54cd\uff0c\u9009\u62e9\u5408\u9002\u7684\u6fc0\u6d3b\u51fd\u6570\u4e5f\u662f\u8d85\u53c2\u6570\u8c03\u6574\u7684\u4e00\u90e8\u5206\u3002<\/p>\n\n\n\n
                                                                    7.\u4f18\u5316\u5668\uff08optimizer\uff09\uff1a\u4f18\u5316\u5668\u662f\u6307\u7528\u4e8e\u66f4\u65b0\u6743\u91cd\u7684\u7b97\u6cd5\uff0c\u4e0d\u540c\u7684\u4f18\u5316\u5668\u4e5f\u4f1a\u5bf9\u6a21\u578b\u7684\u8bad\u7ec3\u548c\u6027\u80fd\u4ea7\u751f\u5f71\u54cd\u3002<\/p>\n\n\n\n
                                                                    \u7b49\u7b49\u3002<\/p>\n\n\n\n
                                                                    \u8d85\u53c2\u6570\u7684\u8c03\u6574\u662f\u4e00\u9879\u7e41\u7410\u4e14\u9700\u8981\u7ecf\u9a8c\u7684\u5de5\u4f5c\uff0c\u5e38\u89c1\u7684\u8c03\u6574\u65b9\u6cd5\u6709\u7f51\u683c\u641c\u7d22\uff08grid search\uff09\u3001\u968f\u673a\u641c\u7d22\uff08random search\uff09\u3001\u8d1d\u53f6\u65af\u4f18\u5316\uff08Bayesian optimization\uff09\u7b49\u3002<\/p>\n\n\n\n\n\n
                                                                    \"gun<\/figure>\n\n\n\n
                                                                    \u8bf7\u8be6\u7ec6\u89e3\u91ca\u6307\u793a\u51fd\u6570<\/p>\n\n\n\n
                                                                    \u6307\u793a\u51fd\u6570\uff0c\u4e5f\u79f0\u4e3a\u5e03\u5c14\u51fd\u6570\u6216\u9636\u8dc3\u51fd\u6570\uff0c\u662f\u4e00\u79cd\u5c06\u8f93\u5165\u6620\u5c04\u52300\u62161\u7684\u51fd\u6570\u3002\u5728\u673a\u5668\u5b66\u4e60\u548c\u7edf\u8ba1\u5b66\u4e2d\uff0c\u6307\u793a\u51fd\u6570\u901a\u5e38\u7528\u4e8e\u8868\u793a\u5206\u7c7b\u4efb\u52a1\u4e2d\u7684\u7c7b\u522b\u6807\u7b7e\u6216\u7ea6\u675f\u6761\u4ef6\u3002\u5b83\u4e5f\u53ef\u4ee5\u88ab\u7528\u6765\u7f16\u7801\u5176\u4ed6\u4e8c\u5143\u6570\u636e\u3002<\/p>\n\n\n\n
                                                                    \u6570\u5b66\u4e0a\uff0c\u4e00\u4e2a\u6307\u793a\u51fd\u6570\u53ef\u4ee5\u5b9a\u4e49\u4e3a\uff1a<\/p>\n\n\n\n
                                                                    \ufffd\ufffd(\ufffd)={1,if \ufffd\u2208\ufffd0,if \ufffd\u2209\ufffdI<\/em>A<\/em>\u200b(x<\/em>)={1,0,\u200bif x<\/em>\u2208A<\/em>if x<\/em>\u2208\/A<\/em>\u200b<\/p>\n\n\n\n
                                                                    \u5176\u4e2d $A$ \u662f\u4e00\u4e2a\u96c6\u5408\uff0c$x$ \u662f\u8f93\u5165\u3002\u5f53 $x$ \u5c5e\u4e8e $A$ \u65f6\uff0c\u6307\u793a\u51fd\u6570\u8fd4\u56de1\uff0c\u5426\u5219\u8fd4\u56de0\u3002<\/p>\n\n\n\n
                                                                    \u5728\u673a\u5668\u5b66\u4e60\u4e2d\uff0c\u6211\u4eec\u7ecf\u5e38\u4f7f\u7528\u6307\u793a\u51fd\u6570\u6765\u8868\u793a\u5206\u7c7b\u95ee\u9898\u4e2d\u7684\u6807\u7b7e\u3002\u4f8b\u5982\uff0c\u5bf9\u4e8e\u4e8c\u5143\u5206\u7c7b\u95ee\u9898\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u6807\u7b7e $y$ \u5b9a\u4e49\u4e3a\uff1a<\/p>\n\n\n\n
                                                                    \ufffd={1,if \ufffd \u5c5e\u4e8e\u6b63\u7c7b0,if \ufffd \u5c5e\u4e8e\u8d1f\u7c7by<\/em>={1,0,\u200bif x<\/em> \u5c5e\u4e8e\u6b63\u7c7bif x<\/em> \u5c5e\u4e8e\u8d1f\u7c7b\u200b<\/p>\n\n\n\n
                                                                    \u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u6307\u793a\u51fd\u6570\u6765\u8ba1\u7b97\u9884\u6d4b\u8f93\u51fa\u548c\u5b9e\u9645\u6807\u7b7e\u4e4b\u95f4\u7684\u8bef\u5dee\u3002<\/p>\n\n\n\n\n\n
                                                                    \"gun<\/figure>\n\n\n\n
                                                                    \u8bf7\u8be6\u7ec6\u89e3\u91ca\u5e72\u9884\u67e5\u8be2<\/p>\n\n\n\n
                                                                    \u5e72\u9884\u67e5\u8be2\u6307\u7684\u662f\u5728\u56e0\u679c\u63a8\u65ad\u95ee\u9898\u4e2d\uff0c\u901a\u8fc7\u4fee\u6539\u67d0\u4e2a\u56e0\u53d8\u91cf\u7684\u53d6\u503c\u6765\u63a2\u7a76\u5176\u4ed6\u53d8\u91cf\u4e4b\u95f4\u7684\u56e0\u679c\u5173\u7cfb\u3002\u5728\u6982\u7387\u56fe\u6a21\u578b\u4e2d\uff0c\u5e72\u9884\u67e5\u8be2\u53ef\u4ee5\u901a\u8fc7\u5bf9\u67d0\u4e2a\u8282\u70b9\u8fdb\u884c\u5e72\u9884\uff0c\u5c06\u5176\u8bbe\u4e3a\u6307\u5b9a\u503c\uff0c\u7136\u540e\u91cd\u65b0\u8ba1\u7b97\u6982\u7387\u5206\u5e03\u6765\u5f97\u5230\u65b0\u7684\u6761\u4ef6\u6982\u7387\u5206\u5e03\u3002\u8fd9\u4e2a\u8fc7\u7a0b\u53ef\u4ee5\u88ab\u770b\u505a\u662f\u5c06\u539f\u59cb\u7684\u8054\u5408\u6982\u7387\u5206\u5e03\u9650\u5236\u4e3a\u5e72\u9884\u8282\u70b9\u7684\u53d6\u503c\uff0c\u518d\u91cd\u65b0\u5f52\u4e00\u5316\uff0c\u5f97\u5230\u65b0\u7684\u6761\u4ef6\u6982\u7387\u5206\u5e03\u3002<\/p>\n\n\n\n
                                                                    \u5e72\u9884\u67e5\u8be2\u7684\u76ee\u7684\u662f\u4e3a\u4e86\u63a2\u7a76\u56e0\u679c\u5173\u7cfb\uff0c\u56e0\u4e3a\u5e72\u9884\u8282\u70b9\u4e0e\u5176\u4ed6\u8282\u70b9\u4e4b\u95f4\u7684\u8fde\u63a5\u8868\u793a\u7740\u56e0\u679c\u5173\u7cfb\u3002\u5e72\u9884\u67e5\u8be2\u7684\u7ed3\u679c\u53ef\u4ee5\u7528\u6765\u56de\u7b54\u56e0\u679c\u5173\u7cfb\u76f8\u5173\u7684\u95ee\u9898\uff0c\u4f8b\u5982\u5982\u679c\u6211\u5e72\u9884\u67d0\u4e2a\u53d8\u91cf\u4f1a\u5982\u4f55\u5f71\u54cd\u5176\u4ed6\u53d8\u91cf\u7684\u53d6\u503c\uff0c\u6216\u8005\u6211\u8be5\u5982\u4f55\u6539\u53d8\u5e72\u9884\u53d8\u91cf\u7684\u53d6\u503c\u6765\u5b9e\u73b0\u67d0\u79cd\u76ee\u6807\u3002\u5728\u673a\u5668\u5b66\u4e60\u4e2d\uff0c\u5e72\u9884\u67e5\u8be2\u662f\u56e0\u679c\u63a8\u65ad\u7684\u6838\u5fc3\u95ee\u9898\u4e4b\u4e00\u3002<\/p>\n\n\n\n\n\n
                                                                    \"gun<\/figure>\n\n\n\n
                                                                    \u8bf7\u8be6\u7ec6\u89e3\u91caJacobian\u77e9\u9635<\/p>\n\n\n\n
                                                                    Jacobian\u77e9\u9635\u662f\u6570\u5b66\u4e2d\u4e00\u79cd\u975e\u5e38\u6709\u7528\u7684\u77e9\u9635\uff0c\u5b83\u63cf\u8ff0\u4e86\u4e00\u4e2a\u5411\u91cf\u51fd\u6570\u7684\u6bcf\u4e2a\u8f93\u51fa\u5206\u91cf\u5bf9\u4e8e\u6bcf\u4e2a\u8f93\u5165\u5206\u91cf\u7684\u504f\u5bfc\u6570\uff0c\u5373\u51fd\u6570\u5c40\u90e8\u7684\u7ebf\u6027\u903c\u8fd1\u3002\u5728\u673a\u5668\u5b66\u4e60\u548c\u6df1\u5ea6\u5b66\u4e60\u4e2d\uff0cJacobian\u77e9\u9635\u5e38\u5e38\u88ab\u7528\u6765\u8ba1\u7b97\u68af\u5ea6\u3001\u89e3\u51b3\u4f18\u5316\u95ee\u9898\u4ee5\u53ca\u5bf9\u6a21\u578b\u8fdb\u884c\u5206\u6790\u7b49\u3002<\/p>\n\n\n\n
                                                                    \u5177\u4f53\u6765\u8bf4\uff0c\u5bf9\u4e8e\u4e00\u4e2a\u5411\u91cf\u503c\u51fd\u6570 $f: \\mathbb{R}^m \\to \\mathbb{R}^n$\uff0c\u5176Jacobian\u77e9\u9635 $J_f$ \u5b9a\u4e49\u4e3a\uff1a<\/p>\n\n\n\n
                                                                    \ufffd\ufffd=[\u2202\ufffd1\u2202\ufffd1\u2202\ufffd1\u2202\ufffd2\u2026\u2202\ufffd1\u2202\ufffd\ufffd\u2202\ufffd2\u2202\ufffd1\u2202\ufffd2\u2202\ufffd2\u2026\u2202\ufffd2\u2202\ufffd\ufffd\u22ee\u22ee\u22f1\u22ee\u2202\ufffd\ufffd\u2202\ufffd1\u2202\ufffd\ufffd\u2202\ufffd2\u2026\u2202\ufffd\ufffd\u2202\ufffd\ufffd]J<\/em>f<\/em>\u200b=\u23a3\u23a1\u200b\u2202x<\/em>1\u200b\u2202f<\/em>1\u200b\u200b\u2202x<\/em>1\u200b\u2202f<\/em>2\u200b\u200b\u22ee\u2202x<\/em>1\u200b\u2202f<\/em>n<\/em>\u200b\u200b\u200b\u2202x<\/em>2\u200b\u2202f<\/em>1\u200b\u200b\u2202x<\/em>2\u200b\u2202f<\/em>2\u200b\u200b\u22ee\u2202x<\/em>2\u200b\u2202f<\/em>n<\/em>\u200b\u200b\u200b\u2026\u2026\u22f1\u2026\u200b\u2202x<\/em>m<\/em>\u200b\u2202f<\/em>1\u200b\u200b\u2202x<\/em>m<\/em>\u200b\u2202f<\/em>2\u200b\u200b\u22ee\u2202x<\/em>m<\/em>\u200b\u2202f<\/em>n<\/em>\u200b\u200b\u200b\u23a6\u23a4\u200b<\/p>\n\n\n\n
                                                                    \u5176\u4e2d\uff0c$f_i$ \u662f $f$ \u7684\u7b2c $i$ \u4e2a\u8f93\u51fa\u5206\u91cf\uff0c$x_j$ \u662f $f$ \u7684\u7b2c $j$ \u4e2a\u8f93\u5165\u5206\u91cf\u3002<\/p>\n\n\n\n
                                                                    Jacobian\u77e9\u9635\u7684\u4e3b\u8981\u4f5c\u7528\u662f\u8ba1\u7b97\u51fd\u6570\u7684\u68af\u5ea6\uff0c\u7279\u522b\u5730\uff0c\u5f53 $n=1$ \u65f6\uff0cJacobian\u77e9\u9635\u53d8\u6210\u4e86\u51fd\u6570\u7684\u68af\u5ea6\u5411\u91cf\uff0c\u5373\uff1a<\/p>\n\n\n\n
                                                                    \u2207\ufffd(\ufffd)=[\u2202\ufffd\u2202\ufffd1\u2202\ufffd\u2202\ufffd2\u2026\u2202\ufffd\u2202\ufffd\ufffd]\u2207f<\/em>(x<\/em>)=[\u2202x<\/em>1\u200b\u2202f<\/em>\u200b\u200b\u2202x<\/em>2\u200b\u2202f<\/em>\u200b\u200b\u2026\u200b\u2202x<\/em>m<\/em>\u200b\u2202f<\/em>\u200b\u200b]<\/p>\n\n\n\n
                                                                    Jacobian\u77e9\u9635\u4e5f\u53ef\u4ee5\u7528\u6765\u5206\u6790\u51fd\u6570\u7684\u6027\u8d28\uff0c\u4f8b\u5982\u51fd\u6570\u7684\u5c40\u90e8\u7ebf\u6027\u903c\u8fd1\u3001\u51fd\u6570\u7684\u53ef\u5fae\u6027\u7b49\u7b49\u3002\u5728\u6df1\u5ea6\u5b66\u4e60\u4e2d\uff0cJacobian\u77e9\u9635\u8fd8\u53ef\u4ee5\u7528\u4e8e\u8ba1\u7b97\u6279\u91cf\u5f52\u4e00\u5316\uff08Batch Normalization\uff09\u548c\u81ea\u9002\u5e94\u4f18\u5316\u7b97\u6cd5\uff08\u5982Adam\u3001Adagrad\u7b49\uff09\u7b49\u4e2d\u7684\u91cd\u8981\u91cf\uff0c\u4ece\u800c\u63d0\u5347\u6a21\u578b\u7684\u8bad\u7ec3\u6548\u679c\u3002<\/p>\n\n\n\n\n\n
                                                                    \"gun<\/figure>\n\n\n\n
                                                                    \u8bf7\u8be6\u7ec6\u4ecb\u7ecdLaplace\u5206\u5e03<\/p>\n\n\n\n
                                                                    Laplace\u5206\u5e03\uff0c\u4e5f\u79f0\u4e3a\u53cc\u6307\u6570\u5206\u5e03\uff08double exponential distribution\uff09\uff0c\u662f\u4e00\u79cd\u8fde\u7eed\u6982\u7387\u5206\u5e03\uff0c\u7531\u62c9\u666e\u62c9\u65af\u547d\u540d\u3002Laplace\u5206\u5e03\u5177\u6709\u4e24\u4e2a\u53c2\u6570\uff1a\u4f4d\u7f6e\u53c2\u6570\u03bc\u548c\u5c3a\u5ea6\u53c2\u6570b\uff0c\u5206\u5e03\u51fd\u6570\u5982\u4e0b\uff1a<\/p>\n\n\n\n
                                                                    $f(x;\\mu,b)=\\frac{1}{2b}\\exp\\left(-\\frac{|x-\\mu|}{b}\\right)$<\/p>\n\n\n\n
                                                                    \u5176\u4e2d\uff0c|\u00b7|\u8868\u793a\u7edd\u5bf9\u503c\u3002Laplace\u5206\u5e03\u7684\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\u5177\u6709\u8f7b\u5c3e\u548c\u5c16\u5cf0\u7684\u7279\u5f81\uff0c\u5b83\u57280\u70b9\u5904\u5177\u6709\u4e00\u4e2a\u660e\u663e\u7684\u5cf0\u503c\uff0c\u800c\u5728\u5176\u4ed6\u5730\u65b9\u968f\u7740\u81ea\u53d8\u91cf\u7684\u589e\u5927\u6216\u51cf\u5c0f\u800c\u9010\u6e10\u4e0b\u964d\u3002\u8fd9\u4f7f\u5f97Laplace\u5206\u5e03\u5728\u4e00\u4e9b\u5f02\u5e38\u68c0\u6d4b\u95ee\u9898\u4e2d\u6709\u7740\u5e7f\u6cdb\u7684\u5e94\u7528\u3002<\/p>\n\n\n\n
                                                                    Laplace\u5206\u5e03\u7684\u5747\u503c\u4e3a\u03bc\uff0c\u65b9\u5dee\u4e3a2b^2\uff0c\u4e2d\u4f4d\u6570\u4e3a\u03bc\u3002Laplace\u5206\u5e03\u4e5f\u5177\u6709\u5bf9\u79f0\u6027\uff0c\u5373\u5bf9\u4e8e\u4efb\u610f\u7684x\uff0c\u6709$f(x;\\mu,b)=f(-x;\\mu,b)$\u3002Laplace\u5206\u5e03\u8fd8\u6ee1\u8db3\u5c3a\u5ea6\u4e0d\u53d8\u6027\uff0c\u5373\u82e5X\u670d\u4eceLaplace\u5206\u5e03\uff0c\u5219kX\u4e5f\u670d\u4eceLaplace\u5206\u5e03\uff0c\u5176\u4e2dk\u4e3a\u4efb\u610f\u975e\u96f6\u5b9e\u6570\u3002<\/p>\n\n\n\n
                                                                    \u5728\u673a\u5668\u5b66\u4e60\u4e2d\uff0cLaplace\u5206\u5e03\u901a\u5e38\u88ab\u7528\u4f5c\u6b63\u5219\u5316\u9879\uff0c\u4ee5\u964d\u4f4e\u6a21\u578b\u7684\u8fc7\u62df\u5408\u98ce\u9669\u3002L1\u6b63\u5219\u5316\uff08Lasso\uff09\u5c31\u662f\u901a\u8fc7Laplace\u5206\u5e03\u4f5c\u4e3a\u6b63\u5219\u5316\u9879\u6765\u7ea6\u675f\u6a21\u578b\u53c2\u6570\u7684\u3002\u6b64\u5916\uff0cLaplace\u5206\u5e03\u8fd8\u88ab\u5e7f\u6cdb\u5e94\u7528\u4e8eBayesian\u63a8\u65ad\u3001\u56fe\u50cf\u590d\u539f\u3001\u56fe\u50cf\u5904\u7406\u7b49\u9886\u57df\u3002<\/p>\n\n\n\n\n\n
                                                                    \"gun<\/figure>\n\n\n\n
                                                                    \u8bf7\u8be6\u7ec6\u89e3\u91caKL\u6563\u5ea6<\/p>\n\n\n\n
                                                                    KL\u6563\u5ea6\uff0c\u4e5f\u79f0\u4e3a\u76f8\u5bf9\u71b5\uff08relative entropy\uff09\u6216\u4fe1\u606f\u6563\u5ea6\uff08information divergence\uff09\uff0c\u662f\u5ea6\u91cf\u4e24\u4e2a\u6982\u7387\u5206\u5e03\u4e4b\u95f4\u5dee\u5f02\u7684\u4e00\u79cd\u65b9\u6cd5\u3002\u5176\u5b9a\u4e49\u4e3a\u4e24\u4e2a\u6982\u7387\u5206\u5e03P\u548cQ\u7684\u4ea4\u53c9\u71b5\uff08cross-entropy\uff09\u4e0eP\u7684\u71b5\uff08entropy\uff09\u4e4b\u5dee\uff0c\u5373\uff1a<\/p>\n\n\n\n
                                                                    $D_{KL}(P||Q) = \\sum_{i=1}^n P(i) \\log \\frac{P(i)}{Q(i)}$<\/p>\n\n\n\n
                                                                    \u5176\u4e2d\uff0cn\u4e3a\u6982\u7387\u5206\u5e03\u7684\u5143\u7d20\u4e2a\u6570\u3002KL\u6563\u5ea6\u4e3a\u975e\u8d1f\u503c\uff0c\u5f53\u4e14\u4ec5\u5f53P\u548cQ\u5b8c\u5168\u76f8\u540c\u65f6\uff0cKL\u6563\u5ea6\u4e3a0\u3002KL\u6563\u5ea6\u4e0d\u662f\u5bf9\u79f0\u7684\uff0c\u5373$D_{KL}(P||Q) \\neq D_{KL}(Q||P)$\u3002<\/p>\n\n\n\n
                                                                    KL\u6563\u5ea6\u5728\u673a\u5668\u5b66\u4e60\u4e2d\u7684\u5e94\u7528\u975e\u5e38\u5e7f\u6cdb\uff0c\u4f8b\u5982\u5728\u6df1\u5ea6\u5b66\u4e60\u4e2d\uff0c\u7528\u4e8e\u6bd4\u8f83\u4e24\u4e2a\u6982\u7387\u5206\u5e03\u4e4b\u95f4\u7684\u76f8\u4f3c\u6027\uff0c\u5e38\u7528\u4e8e\u635f\u5931\u51fd\u6570\u7684\u8bbe\u8ba1\u4e2d\u3002\u540c\u65f6\uff0cKL\u6563\u5ea6\u8fd8\u5e38\u5e38\u88ab\u7528\u6765\u8fdb\u884c\u6a21\u578b\u7684\u6b63\u5219\u5316\uff0c\u4ee5\u907f\u514d\u8fc7\u62df\u5408\u3002<\/p>\n\n\n\n\n\n
                                                                    \"gun<\/figure>\n\n\n\n
                                                                    \u8bf7\u8be6\u7ec6\u89e3\u91caKrylov\u65b9\u6cd5<\/p>\n\n\n\n
                                                                    Krylov\u65b9\u6cd5\u662f\u4e00\u7c7b\u6c42\u89e3\u7ebf\u6027\u65b9\u7a0b\u7ec4\u7684\u8fed\u4ee3\u65b9\u6cd5\uff0c\u5b83\u7684\u57fa\u672c\u601d\u60f3\u662f\u5728\u4e00\u4e2a\u7531\u521d\u59cb\u5411\u91cf\u548c\u77e9\u9635A\u7684\u5e42\u7ec4\u6210\u7684Krylov\u5b50\u7a7a\u95f4\u4e2d\u5bfb\u627e\u89e3\u3002Krylov\u5b50\u7a7a\u95f4\u662f\u7531\u521d\u59cb\u5411\u91cfx\u548c\u77e9\u9635A\u7684\u5e42\u7ec4\u6210\u7684\u5411\u91cf\u7a7a\u95f4\uff1a<\/p>\n\n\n\n
                                                                    \ufffd\ufffd(\ufffd,\ufffd)=span{\ufffd,\ufffd\ufffd,\ufffd2\ufffd,\u2026,\ufffd\ufffd\u22121\ufffd}.Kk<\/em>\u200b(A<\/em>,x<\/em>)=span{x<\/em>,A<\/em>x<\/em>,A<\/em>2x<\/em>,\u2026,A<\/em>k<\/em>\u22121x<\/em>}.<\/p>\n\n\n\n
                                                                    Krylov\u65b9\u6cd5\u901a\u8fc7\u8fed\u4ee3\u6765\u9010\u6e10\u6269\u5927\u8fd9\u4e2aKrylov\u5b50\u7a7a\u95f4\uff0c\u4ece\u800c\u9010\u6b65\u903c\u8fd1\u65b9\u7a0b\u7ec4\u7684\u89e3\u3002\u5728\u6bcf\u4e00\u6b65\u8fed\u4ee3\u4e2d\uff0cKrylov\u65b9\u6cd5\u6c42\u89e3\u4e00\u4e2a\u5c0f\u89c4\u6a21\u7684\u7ebf\u6027\u65b9\u7a0b\u7ec4\uff0c\u901a\u5e38\u662f\u901a\u8fc7\u9884\u5904\u7406\u6765\u52a0\u901f\u8ba1\u7b97\u3002\u5e38\u89c1\u7684Krylov\u65b9\u6cd5\u5305\u62ec\u5171\u8f6d\u68af\u5ea6\u6cd5\uff08Conjugate Gradient, CG\uff09\u3001\u6700\u5c0f\u6b8b\u5dee\u6cd5\uff08Minimum Residual, MR\uff09\u3001GMRES\u7b49\u3002<\/p>\n\n\n\n
                                                                    Krylov\u65b9\u6cd5\u5177\u6709\u8bb8\u591a\u4f18\u70b9\uff0c\u4f8b\u5982\u53ef\u4ee5\u5904\u7406\u7a00\u758f\u77e9\u9635\uff0c\u9002\u7528\u4e8e\u5927\u89c4\u6a21\u95ee\u9898\uff0c\u53ef\u4ee5\u7528\u4e8e\u975e\u5bf9\u79f0\u77e9\u9635\u548c\u4e0d\u5b9a\u77e9\u9635\uff0c\u53ef\u4ee5\u901a\u8fc7\u9884\u5904\u7406\u6765\u52a0\u901f\u8ba1\u7b97\u7b49\u3002\u4f46\u662f\uff0cKrylov\u65b9\u6cd5\u7684\u6536\u655b\u6027\u548c\u901f\u5ea6\u53d7\u5230\u77e9\u9635\u6761\u4ef6\u6570\u7684\u5f71\u54cd\uff0c\u5982\u679c\u77e9\u9635\u6761\u4ef6\u6570\u5f88\u5927\uff0c\u5219Krylov\u65b9\u6cd5\u53ef\u80fd\u6536\u655b\u7f13\u6162\u751a\u81f3\u53d1\u6563\u3002<\/p>\n\n\n\n\n\n
                                                                    \"gun<\/figure>\n\n\n\n
                                                                    \u8bf7\u8be6\u7ec6\u89e3\u91ca\u5927\u6570\u5b9a\u7406<\/p>\n\n\n\n
                                                                    \u5927\u6570\u5b9a\u7406\uff08Law of Large Numbers\uff09\u662f\u6982\u7387\u8bba\u4e2d\u7684\u4e00\u4e2a\u57fa\u672c\u5b9a\u7406\uff0c\u6307\u7684\u662f\u5728\u76f8\u540c\u7684\u72ec\u7acb\u91cd\u590d\u8bd5\u9a8c\u4e2d\uff0c\u968f\u7740\u8bd5\u9a8c\u6b21\u6570\u7684\u589e\u52a0\uff0c\u6837\u672c\u5747\u503c\u8d8a\u6765\u8d8a\u63a5\u8fd1\u4e8e\u603b\u4f53\u5747\u503c\u3002<\/p>\n\n\n\n
                                                                    \u5927\u6570\u5b9a\u7406\u6709\u4e24\u4e2a\u7ecf\u5178\u7248\u672c\uff0c\u5206\u522b\u662f\u5f31\u5927\u6570\u5b9a\u7406\u548c\u5f3a\u5927\u6570\u5b9a\u7406\u3002<\/p>\n\n\n\n
                                                                    \u5f31\u5927\u6570\u5b9a\u7406\u662f\u6307\uff0c\u5bf9\u4e8e\u72ec\u7acb\u540c\u5206\u5e03\u7684\u968f\u673a\u53d8\u91cf\u5e8f\u5217 ${X_1, X_2, …, X_n}$\uff0c\u5176\u4e2d $E(X_i) = \\mu$\uff0c\u5219\u5bf9\u4e8e\u4efb\u610f $\\epsilon > 0$\uff0c\u6709<\/p>\n\n\n\n
                                                                    lim\u2061\ufffd\u2192\u221e\ufffd(\u22231\ufffd\u2211\ufffd=1\ufffd\ufffd\ufffd\u2212\ufffd\u2223\u2265\ufffd)=0limn<\/em>\u2192\u221e\u200bP<\/em>(\u2223n<\/em>1\u200b\u2211i<\/em>=1n<\/em>\u200bX<\/em>i<\/em>\u200b\u2212\u03bc<\/em>\u2223\u2265\u03f5<\/em>)=0<\/p>\n\n\n\n
                                                                    \u4e5f\u5c31\u662f\u8bf4\uff0c\u6837\u672c\u5747\u503c\u7684\u6982\u7387\u5206\u5e03\u968f\u7740\u6837\u672c\u6570\u76ee\u7684\u589e\u52a0\u800c\u9010\u6e10\u8d8b\u8fd1\u4e8e\u603b\u4f53\u5747\u503c\u7684\u6982\u7387\u5206\u5e03\u3002<\/p>\n\n\n\n
                                                                    \u5f3a\u5927\u6570\u5b9a\u7406\u5219\u66f4\u4e3a\u5f3a\u5927\uff0c\u5b83\u8981\u6c42\u5e8f\u5217 ${X_1, X_2, …, X_n}$ \u7684\u65b9\u5dee\u5b58\u5728\uff0c\u4e14\u4e92\u76f8\u72ec\u7acb\uff0c\u540c\u5206\u5e03\u3002\u5219\u6709<\/p>\n\n\n\n
                                                                    \ufffd(lim\u2061\ufffd\u2192\u221e1\ufffd\u2211\ufffd=1\ufffd(\ufffd\ufffd\u2212\ufffd)=0)=1P<\/em>(limn<\/em>\u2192\u221e\u200bn<\/em>1\u200b\u2211i<\/em>=1n<\/em>\u200b(X<\/em>i<\/em>\u200b\u2212\u03bc<\/em>)=0)=1<\/p>\n\n\n\n
                                                                    \u4e5f\u5c31\u662f\u8bf4\uff0c\u6837\u672c\u5747\u503c\u8d8b\u8fd1\u4e8e\u603b\u4f53\u5747\u503c\u7684\u901f\u5ea6\u6bd4\u5f31\u5927\u6570\u5b9a\u7406\u66f4\u5feb\uff0c\u6536\u655b\u66f4\u52a0\u8fc5\u901f\uff0c\u4f46\u662f\u5b83\u7684\u9650\u5236\u6761\u4ef6\u66f4\u4e3a\u82db\u523b\uff0c\u9700\u8981\u6ee1\u8db3\u65b9\u5dee\u5b58\u5728\uff0c\u4e92\u76f8\u72ec\u7acb\u4e14\u540c\u5206\u5e03\u7b49\u6761\u4ef6\u3002<\/p>\n\n\n\n
                                                                    \u5927\u6570\u5b9a\u7406\u662f\u6982\u7387\u8bba\u4e2d\u7684\u91cd\u8981\u57fa\u7840\u7406\u8bba\uff0c\u5b83\u5728\u5f88\u591a\u9886\u57df\u90fd\u6709\u5e94\u7528\uff0c\u4f8b\u5982\u91d1\u878d\u5b66\u4e2d\u7684\u6295\u8d44\u7ec4\u5408\u4f18\u5316\u3001\u4fe1\u53f7\u5904\u7406\u4e2d\u7684\u6ee4\u6ce2\u5668\u8bbe\u8ba1\u7b49\u7b49\u3002<\/p>\n\n\n\n\n\n
                                                                    \"gun<\/figure>\n\n\n\n
                                                                    \u8bf7\u8be6\u7ec6\u4ecb\u7ecd\u6e17\u6f0f\u5355\u5143<\/p>\n\n\n\n
                                                                    \u6e17\u6f0f\u5355\u5143\uff08leaky unit\uff09\u662f\u795e\u7ecf\u7f51\u7edc\u4e2d\u7684\u4e00\u79cd\u57fa\u672c\u7684\u5355\u5143\u7c7b\u578b\uff0c\u7c7b\u4f3c\u4e8e\u6807\u51c6\u7684\u795e\u7ecf\u5143\uff0c\u4f46\u5177\u6709\u4e00\u4e2a\u989d\u5916\u7684\u53c2\u6570\uff0c\u79f0\u4e3a\u201c\u6cc4\u6f0f\u7cfb\u6570\u201d\uff08leakage coefficient\uff09\uff0c\u901a\u5e38\u8868\u793a\u4e3a $\\alpha$\u3002\u8fd9\u4e2a\u53c2\u6570\u63a7\u5236\u7740\u5f53\u8be5\u5355\u5143\u7684\u6fc0\u6d3b\u503c\u5f88\u5c0f\u6216\u4e3a\u8d1f\u65f6\uff0c\u5982\u4f55\u5728\u795e\u7ecf\u7f51\u7edc\u4e2d\u4f20\u64ad\u68af\u5ea6\u4fe1\u606f\u3002<\/p>\n\n\n\n
                                                                    \u5177\u4f53\u6765\u8bf4\uff0c\u5f53\u4e00\u4e2a\u6e17\u6f0f\u5355\u5143\u88ab\u6fc0\u6d3b\u65f6\uff0c\u5b83\u4f1a\u8ba1\u7b97\u8f93\u5165\u7684\u52a0\u6743\u548c $z$\uff0c\u5e76\u5c06\u5176\u4f20\u9012\u7ed9\u6fc0\u6d3b\u51fd\u6570 $f$\uff0c\u4ee5\u8ba1\u7b97\u8f93\u51fa\u503c $y=f(z)$\u3002\u4f46\u662f\uff0c\u4e0e\u6807\u51c6\u795e\u7ecf\u5143\u4e0d\u540c\u7684\u662f\uff0c\u5728\u8ba1\u7b97\u68af\u5ea6\u65f6\uff0c\u6e17\u6f0f\u5355\u5143\u4e0d\u4f1a\u5c06\u96f6\u68af\u5ea6\u4f20\u9012\u56de\u524d\u4e00\u5c42\u3002\u76f8\u53cd\uff0c\u5f53 $z<0$ \u65f6\uff0c\u5b83\u4f1a\u5c06 $\\alpha z$ \u4f20\u9012\u56de\u524d\u4e00\u5c42\uff0c\u5176\u4e2d $\\alpha$ \u662f\u6cc4\u6f0f\u7cfb\u6570\u3002<\/p>\n\n\n\n
                                                                    \u8fd9\u79cd\u884c\u4e3a\u7684\u4f5c\u7528\u662f\u786e\u4fdd\u6e17\u6f0f\u5355\u5143\u5728\u6fc0\u6d3b\u503c\u5f88\u5c0f\u65f6\u4e0d\u4f1a\u5b8c\u5168\u6291\u5236\u68af\u5ea6\u4f20\u64ad\uff0c\u4ece\u800c\u5141\u8bb8\u6a21\u578b\u66f4\u597d\u5730\u5b66\u4e60\u5177\u6709\u5c0f\u68af\u5ea6\u7684\u6a21\u5f0f\uff0c\u4ece\u800c\u52a0\u901f\u6a21\u578b\u7684\u6536\u655b\u3002\u53e6\u4e00\u65b9\u9762\uff0c\u5f53 $\\alpha=0$ \u65f6\uff0c\u6e17\u6f0f\u5355\u5143\u9000\u5316\u4e3a\u6807\u51c6\u7684\u795e\u7ecf\u5143\u3002<\/p>\n\n\n\n\n\n
                                                                    \"gun<\/figure>\n\n\n\n
                                                                    \u8bf7\u8be6\u7ec6\u89e3\u91ca\u52d2\u8d1d\u683c\u53ef\u79ef<\/p>\n\n\n\n
                                                                    \u52d2\u8d1d\u683c\u53ef\u79ef\u662f\u4e00\u79cd\u6d4b\u5ea6\u8bba\u4e2d\u7684\u6982\u5ff5\uff0c\u7528\u4e8e\u63cf\u8ff0\u51fd\u6570\u5728\u4e00\u4e2a\u6d4b\u5ea6\u7a7a\u95f4\u4e0a\u7684\u53ef\u79ef\u6027\u3002\u7ed9\u5b9a\u4e00\u4e2a\u6d4b\u5ea6\u7a7a\u95f4 $(\\Omega,\\mathcal{F},\\mu)$\uff0c\u5982\u679c\u4e00\u4e2a\u5b9e\u51fd\u6570 $f:\\Omega \\rightarrow \\mathbb{R}$ \u6ee1\u8db3\u4ee5\u4e0b\u6761\u4ef6\uff0c\u5219\u79f0 $f$ \u5728\u8be5\u6d4b\u5ea6\u7a7a\u95f4\u4e0a\u662f\u52d2\u8d1d\u683c\u53ef\u79ef\u7684\uff1a<\/p>\n\n\n\n
                                                                      \n
                                                                    1. $f$ \u662f\u53ef\u6d4b\u7684\uff0c\u5373\u5bf9\u4e8e\u4efb\u610f\u7684 $a \\in \\mathbb{R}$\uff0c\u6709 ${\\omega:f(\\omega) \\geq a} \\in \\mathcal{F}$\u3002<\/li>\n\n\n\n
                                                                    2. \u5b58\u5728\u4e00\u4e2a\u975e\u8d1f\u53ef\u6d4b\u51fd\u6570 $g:\\Omega \\rightarrow \\mathbb{R}$\uff0c\u4f7f\u5f97 $|f(\\omega)| \\leq g(\\omega)$ \u5bf9\u4e8e\u51e0\u4e4e\u6240\u6709\u7684 $\\omega \\in \\Omega$ \u6210\u7acb\u3002<\/li>\n\n\n\n
                                                                    3. $\\int_{\\Omega} g(\\omega) d\\mu(\\omega) < \\infty$\u3002<\/li>\n<\/ol>\n\n\n\n
                                                                      \u5176\u4e2d\uff0c\u6761\u4ef6 $2$ \u786e\u4fdd\u4e86 $f$ \u7684\u7edd\u5bf9\u53ef\u79ef\u6027\uff0c\u6761\u4ef6 $3$ \u5219\u4fdd\u8bc1\u4e86 $f$ \u7684\u53ef\u79ef\u6027\u3002<\/p>\n\n\n\n
                                                                      \u52d2\u8d1d\u683c\u53ef\u79ef\u51fd\u6570\u5728\u6982\u7387\u8bba\u548c\u7edf\u8ba1\u5b66\u4e2d\u6709\u5e7f\u6cdb\u5e94\u7528\uff0c\u6bd4\u5982\u5728\u5b9a\u4e49\u968f\u673a\u53d8\u91cf\u7684\u671f\u671b\u3001\u65b9\u5dee\u7b49\u65f6\u9700\u8981\u7528\u5230\u52d2\u8d1d\u683c\u79ef\u5206\u3002\u540c\u65f6\uff0c\u52d2\u8d1d\u683c\u53ef\u79ef\u51fd\u6570\u8fd8\u6709\u4e00\u4e9b\u91cd\u8981\u7684\u6027\u8d28\uff0c\u6bd4\u5982\u52d2\u8d1d\u683c\u5d4c\u5165\u5b9a\u7406\u548c\u8d39\u91cc\u66fc-\u96f7\u5fb7\u5b9a\u7406\u7b49\u3002<\/p>\n\n\n\n\n\n
                                                                      \"gun<\/figure>\n\n\n\n
                                                                      \u8bf7\u8be6\u7ec6\u89e3\u91ca\u968f\u673a\u68af\u5ea6\u4e0b\u964d<\/p>\n\n\n\n
                                                                      \u968f\u673a\u68af\u5ea6\u4e0b\u964d\uff08Stochastic Gradient Descent, SGD\uff09\u662f\u4e00\u79cd\u4f18\u5316\u7b97\u6cd5\uff0c\u7528\u4e8e\u8bad\u7ec3\u673a\u5668\u5b66\u4e60\u6a21\u578b\u3002\u4e0e\u4f20\u7edf\u7684\u6279\u91cf\u68af\u5ea6\u4e0b\u964d\uff08Batch Gradient Descent\uff09\u4e0d\u540c\uff0cSGD\u5728\u6bcf\u6b21\u8fed\u4ee3\u4e2d\u53ea\u4f7f\u7528\u4e00\u4e2a\u6837\u672c\uff08\u6216\u4e00\u5c0f\u6279\u6837\u672c\uff09\u6765\u8ba1\u7b97\u68af\u5ea6\u5e76\u66f4\u65b0\u6a21\u578b\u53c2\u6570\uff0c\u56e0\u6b64\u66f4\u52a0\u9ad8\u6548\uff0c\u5e76\u4e14\u53ef\u4ee5\u5904\u7406\u975e\u5e38\u5927\u7684\u6570\u636e\u96c6\u3002<\/p>\n\n\n\n
                                                                      SGD\u7684\u6838\u5fc3\u601d\u60f3\u662f\u901a\u8fc7\u8fed\u4ee3\u6765\u6700\u5c0f\u5316\u635f\u5931\u51fd\u6570\uff0c\u5176\u4e2d\u6bcf\u4e2a\u8fed\u4ee3\u6b65\u9aa4\u4f7f\u7528\u4e00\u4e2a\u6837\u672c\uff08\u6216\u5c0f\u6279\u91cf\uff09\u6765\u8ba1\u7b97\u68af\u5ea6\u3002\u5bf9\u4e8e\u4e00\u4e2a\u7ed9\u5b9a\u7684\u53c2\u6570\u5411\u91cf$w$\u548c\u4e00\u4e2a\u635f\u5931\u51fd\u6570$L(w;x_i,y_i)$\uff0c\u5176\u4e2d$x_i$\u662f\u8f93\u5165\u6570\u636e\uff0c$y_i$\u662f\u5bf9\u5e94\u7684\u6807\u7b7e\uff0cSGD\u7684\u66f4\u65b0\u89c4\u5219\u53ef\u4ee5\u5199\u6210\uff1a<\/p>\n\n\n\n
                                                                      \ufffd\ufffd+1=\ufffd\ufffd\u2212\ufffd\u2207\ufffd\ufffd(\ufffd\ufffd;\ufffd\ufffd,\ufffd\ufffd)w<\/em>t<\/em>+1\u200b=w<\/em>t<\/em>\u200b\u2212\u03b7<\/em>\u2207w<\/em>\u200bL<\/em>(w<\/em>t<\/em>\u200b;x<\/em>i<\/em>\u200b,y<\/em>i<\/em>\u200b)<\/p>\n\n\n\n
                                                                      \u5176\u4e2d\uff0c$\\eta$\u662f\u5b66\u4e60\u7387\uff0c$\\nabla_w L(w_t;x_i,y_i)$\u662f\u635f\u5931\u51fd\u6570$L(w_t;x_i,y_i)$\u5bf9\u53c2\u6570$w$\u7684\u68af\u5ea6\u3002\u901a\u8fc7\u53cd\u590d\u6267\u884c\u4e0a\u8ff0\u66f4\u65b0\u6b65\u9aa4\uff0cSGD\u9010\u6e10\u4f7f\u635f\u5931\u51fd\u6570\u6700\u5c0f\u5316\uff0c\u5e76\u627e\u5230\u6a21\u578b\u53c2\u6570\u7684\u6700\u4f18\u89e3\u3002<\/p>\n\n\n\n
                                                                      SGD\u7684\u4f18\u70b9\u662f\u53ef\u4ee5\u5728\u5927\u578b\u6570\u636e\u96c6\u4e0a\u8fdb\u884c\u9ad8\u6548\u7684\u6a21\u578b\u8bad\u7ec3\uff0c\u4f46\u5b83\u4e5f\u6709\u4e00\u4e9b\u7f3a\u70b9\u3002\u7531\u4e8e\u6bcf\u4e2a\u8fed\u4ee3\u6b65\u9aa4\u53ea\u4f7f\u7528\u4e00\u4e2a\u6837\u672c\u6765\u66f4\u65b0\u6a21\u578b\u53c2\u6570\uff0cSGD\u7684\u53c2\u6570\u66f4\u65b0\u53ef\u80fd\u4f1a\u51fa\u73b0\u975e\u5e38\u5927\u7684\u65b9\u5dee\uff0c\u4ece\u800c\u5bfc\u81f4\u6a21\u578b\u53c2\u6570\u4e0d\u7a33\u5b9a\u3002\u4e3a\u4e86\u89e3\u51b3\u8fd9\u4e2a\u95ee\u9898\uff0c\u4e00\u4e9b\u6539\u8fdb\u7684SGD\u7b97\u6cd5\uff0c\u5982\u52a8\u91cfSGD\uff08Momentum SGD\uff09\u3001AdaGrad\u3001Adam\u7b49\u88ab\u63d0\u51fa\u6765\u3002<\/p>\n\n\n\n\n\n
                                                                      \"gun<\/figure>\n\n\n\n
                                                                      \u8bf7\u8be6\u7ec6\u89e3\u91ca\u9690\u5f0f\u53cd\u9988<\/p>\n\n\n\n
                                                                      \u9690\u5f0f\u53cd\u9988\uff08implicit feedback\uff09\u662f\u6307\u901a\u8fc7\u7528\u6237\u7684\u884c\u4e3a\u8bb0\u5f55\u5f97\u5230\u7684\u7528\u6237\u5bf9\u7269\u54c1\u7684\u559c\u597d\u4fe1\u606f\u3002\u76f8\u5bf9\u4e8e\u663e\u5f0f\u53cd\u9988\uff08explicit feedback\uff09\uff0c\u9690\u5f0f\u53cd\u9988\u7684\u7279\u70b9\u662f\u4e0d\u9700\u8981\u7528\u6237\u8fdb\u884c\u660e\u786e\u7684\u8bc4\u5206\u6216\u8005\u6807\u6ce8\uff0c\u800c\u662f\u901a\u8fc7\u7528\u6237\u7684\u6d4f\u89c8\u3001\u70b9\u51fb\u3001\u8d2d\u4e70\u7b49\u884c\u4e3a\u6765\u63a8\u65ad\u7528\u6237\u7684\u504f\u597d\u3002<\/p>\n\n\n\n
                                                                      \u9690\u5f0f\u53cd\u9988\u6570\u636e\u901a\u5e38\u5177\u6709\u4ee5\u4e0b\u7279\u70b9\uff1a<\/p>\n\n\n\n
                                                                        \n
                                                                      • \u6570\u636e\u91cf\u5927\uff1a\u4e0e\u663e\u5f0f\u53cd\u9988\u76f8\u6bd4\uff0c\u7528\u6237\u884c\u4e3a\u8bb0\u5f55\u6570\u636e\u66f4\u5bb9\u6613\u83b7\u53d6\uff0c\u56e0\u6b64\u6570\u636e\u91cf\u901a\u5e38\u66f4\u5927\uff1b<\/li>\n\n\n\n
                                                                      • \u4fe1\u606f\u4e0d\u5b8c\u6574\uff1a\u9690\u5f0f\u53cd\u9988\u5f80\u5f80\u53ea\u63d0\u4f9b\u4e86\u7528\u6237\u5bf9\u67d0\u4e9b\u7269\u54c1\u7684\u504f\u597d\u7a0b\u5ea6\uff0c\u800c\u6ca1\u6709\u63d0\u4f9b\u7528\u6237\u5bf9\u4e0d\u611f\u5174\u8da3\u7269\u54c1\u7684\u53cd\u9988\uff0c\u56e0\u6b64\u5f88\u96be\u786e\u5b9a\u7528\u6237\u662f\u5426\u771f\u7684\u4e0d\u559c\u6b22\u8fd9\u4e9b\u7269\u54c1\uff1b<\/li>\n\n\n\n
                                                                      • \u566a\u58f0\u8f83\u5927\uff1a\u7528\u6237\u7684\u884c\u4e3a\u53ef\u80fd\u53d7\u5230\u591a\u79cd\u56e0\u7d20\u7684\u5f71\u54cd\uff0c\u6bd4\u5982\u968f\u673a\u6027\u3001\u73af\u5883\u56e0\u7d20\u7b49\uff0c\u56e0\u6b64\u9700\u8981\u8fdb\u884c\u4e00\u5b9a\u7684\u566a\u58f0\u5904\u7406\u3002<\/li>\n<\/ul>\n\n\n\n
                                                                        \u9690\u5f0f\u53cd\u9988\u53ef\u4ee5\u5e94\u7528\u4e8e\u63a8\u8350\u7cfb\u7edf\u3001\u641c\u7d22\u6392\u5e8f\u3001\u5e7f\u544a\u63a8\u8350\u7b49\u9886\u57df\u3002\u5728\u63a8\u8350\u7cfb\u7edf\u4e2d\uff0c\u5e38\u7528\u7684\u7b97\u6cd5\u5305\u62ec\u9690\u8bed\u4e49\u6a21\u578b\u3001\u57fa\u4e8e\u90bb\u57df\u7684\u7b97\u6cd5\u3001\u6df1\u5ea6\u5b66\u4e60\u6a21\u578b\u7b49\u3002\u5728\u641c\u7d22\u6392\u5e8f\u4e2d\uff0c\u5e38\u7528\u7684\u7b97\u6cd5\u5305\u62ec\u52a0\u6743\u70b9\u51fb\u6b21\u6570\uff08weighted click-through rate\uff0cWCTR\uff09\u3001\u52a0\u6743\u5c55\u793a\u6b21\u6570\uff08weighted impression\uff0cWIMP\uff09\u7b49\u3002\u5728\u5e7f\u544a\u63a8\u8350\u4e2d\uff0c\u9690\u5f0f\u53cd\u9988\u53ef\u4ee5\u7528\u6765\u8ba1\u7b97\u5e7f\u544a\u7684\u70b9\u51fb\u7387\uff08click-through rate\uff0cCTR\uff09\u7b49\u6307\u6807\uff0c\u4ece\u800c\u5bf9\u5e7f\u544a\u8fdb\u884c\u4f18\u5316\u3002<\/p>\n\n\n\n\n\n
                                                                        \"gun<\/figure>\n\n\n\n
                                                                        \u8bf7\u8be6\u7ec6\u4ecb\u7ecd\u964d\u6e29\u6cd5<\/p>\n\n\n\n
                                                                        \u964d\u6e29\u6cd5\uff08Simulated Annealing\uff09\u662f\u4e00\u79cd\u4f18\u5316\u7b97\u6cd5\uff0c\u5176\u7075\u611f\u6765\u6e90\u4e8e\u56fa\u4f53\u7269\u7406\u5b66\u4e2d\u7684\u9000\u706b\u8fc7\u7a0b\u3002\u9000\u706b\u662f\u5c06\u7269\u8d28\u52a0\u70ed\u81f3\u8db3\u591f\u9ad8\u7684\u6e29\u5ea6\uff0c\u7136\u540e\u51b7\u5374\u5230\u5ba4\u6e29\u7684\u8fc7\u7a0b\u3002\u5728\u52a0\u70ed\u8fc7\u7a0b\u4e2d\uff0c\u5206\u5b50\u70ed\u8fd0\u52a8\u5267\u70c8\uff0c\u7269\u8d28\u7684\u7ed3\u6784\u6df7\u4e71\u65e0\u5e8f\uff0c\u800c\u5728\u51b7\u5374\u8fc7\u7a0b\u4e2d\uff0c\u5206\u5b50\u9010\u6e10\u9759\u6b62\u5e76\u6709\u5e8f\u6392\u5217\uff0c\u8fbe\u5230\u6700\u7a33\u5b9a\u7684\u72b6\u6001\u3002<\/p>\n\n\n\n
                                                                        \u5728\u4f18\u5316\u95ee\u9898\u4e2d\uff0c\u6211\u4eec\u4e5f\u5e0c\u671b\u627e\u5230\u6700\u5c0f\u5316\uff08\u6216\u6700\u5927\u5316\uff09\u4e00\u4e2a\u76ee\u6807\u51fd\u6570\u7684\u6700\u4f18\u89e3\u3002\u964d\u6e29\u6cd5\u901a\u8fc7\u6a21\u62df\u9000\u706b\u8fc7\u7a0b\u6765\u641c\u7d22\u6700\u4f18\u89e3\uff0c\u5176\u57fa\u672c\u601d\u60f3\u662f\u5728\u89e3\u7a7a\u95f4\u4e2d\u968f\u673a\u79fb\u52a8\uff0c\u5e76\u4ee5\u4e00\u5b9a\u7684\u6982\u7387\u63a5\u53d7\u5dee\u4e8e\u5f53\u524d\u89e3\u7684\u65b0\u89e3\uff0c\u4ee5\u9632\u6b62\u9677\u5165\u5c40\u90e8\u6700\u4f18\u89e3\u3002<\/p>\n\n\n\n
                                                                        \u5177\u4f53\u6765\u8bf4\uff0c\u964d\u6e29\u6cd5\u5c06\u5f53\u524d\u89e3\u770b\u4f5c\u7cfb\u7edf\u7684\u72b6\u6001\uff0c\u76ee\u6807\u51fd\u6570\u503c\u770b\u4f5c\u80fd\u91cf\u3002\u5728\u7b97\u6cd5\u5f00\u59cb\u65f6\uff0c\u4f1a\u521d\u59cb\u5316\u4e00\u4e2a\u9ad8\u6e29T\uff0c\u7136\u540e\u968f\u673a\u9009\u62e9\u4e00\u4e2a\u521d\u59cb\u89e3\u4f5c\u4e3a\u5f53\u524d\u89e3\u3002\u5728\u6bcf\u4e00\u8f6e\u8fed\u4ee3\u4e2d\uff0c\u4f1a\u968f\u673a\u4ea7\u751f\u4e00\u4e2a\u65b0\u7684\u89e3\uff0c\u5e76\u8ba1\u7b97\u8be5\u65b0\u89e3\u7684\u80fd\u91cf\u5dee\u0394E\uff0c\u82e5\u0394E<0\uff0c\u5219\u63a5\u53d7\u8be5\u89e3\u4f5c\u4e3a\u5f53\u524d\u89e3\uff1b\u5426\u5219\u4ee5\u6982\u7387e^(-\u0394E\/T)\u63a5\u53d7\u8be5\u89e3\u3002\u968f\u7740\u8fed\u4ee3\u6b21\u6570\u7684\u589e\u52a0\uff0c\u6e29\u5ea6T\u9010\u6e10\u4e0b\u964d\uff0c\u63a5\u53d7\u6982\u7387\u4e5f\u9010\u6e10\u51cf\u5c0f\u3002\u6700\u7ec8\uff0c\u5f53\u6e29\u5ea6T\u964d\u5230\u8db3\u591f\u4f4e\u65f6\uff0c\u7b97\u6cd5\u6536\u655b\u4e8e\u6700\u4f18\u89e3\u3002<\/p>\n\n\n\n
                                                                        \u964d\u6e29\u6cd5\u7684\u4f18\u70b9\u662f\u53ef\u4ee5\u907f\u514d\u9677\u5165\u5c40\u90e8\u6700\u4f18\u89e3\uff0c\u56e0\u4e3a\u5728\u9ad8\u6e29\u65f6\u63a5\u53d7\u8f83\u5dee\u7684\u89e3\u6709\u4e00\u5b9a\u6982\u7387\uff0c\u53ef\u4ee5\u8ba9\u7b97\u6cd5\u8df3\u51fa\u5c40\u90e8\u6700\u4f18\u89e3\u3002\u4f46\u5176\u7f3a\u70b9\u662f\u9700\u8981\u8c03\u6574\u6e29\u5ea6\u4e0b\u964d\u7684\u901f\u7387\u548c\u63a5\u53d7\u6982\u7387\u7684\u51fd\u6570\u5f62\u5f0f\uff0c\u4ee5\u8fbe\u5230\u6700\u4f73\u7684\u641c\u7d22\u6548\u679c\u3002\u6b64\u5916\uff0c\u964d\u6e29\u6cd5\u53ef\u80fd\u9700\u8981\u66f4\u957f\u7684\u65f6\u95f4\u624d\u80fd\u6536\u655b\u4e8e\u6700\u4f18\u89e3\uff0c\u56e0\u4e3a\u5728\u9ad8\u6e29\u65f6\u63a5\u53d7\u8f83\u5dee\u7684\u89e3\u53ef\u80fd\u5bfc\u81f4\u641c\u7d22\u8def\u5f84\u8df3\u8dc3\u8f83\u5927\uff0c\u800c\u9700\u8981\u66f4\u591a\u7684\u8fed\u4ee3\u624d\u80fd\u6536\u655b\u3002<\/p>\n\n\n\n\n\n
                                                                        \"gun<\/figure>\n\n\n\n
                                                                        \u8bf7\u8be6\u7ec6\u89e3\u91ca\u68af\u5ea6\u63d0\u5347<\/p>\n\n\n\n
                                                                        \u68af\u5ea6\u63d0\u5347\uff08Gradient Boosting\uff09\u662f\u4e00\u79cd\u57fa\u4e8e\u51b3\u7b56\u6811\u7684\u96c6\u6210\u5b66\u4e60\u7b97\u6cd5\uff0c\u5b83\u7684\u4e3b\u8981\u601d\u60f3\u662f\u5229\u7528\u5f31\u5206\u7c7b\u5668\u7684\u7ec4\u5408\u6765\u6784\u5efa\u4e00\u4e2a\u5f3a\u5206\u7c7b\u5668\u3002\u76f8\u6bd4\u4e8e\u4f20\u7edf\u7684\u51b3\u7b56\u6811\u7b97\u6cd5\uff0c\u68af\u5ea6\u63d0\u5347\u5f15\u5165\u4e86\u4e00\u79cd\u4f18\u5316\u7b56\u7565\uff0c\u4f7f\u5f97\u6bcf\u4e00\u68f5\u6811\u90fd\u662f\u9488\u5bf9\u524d\u4e00\u68f5\u6811\u7684\u6b8b\u5dee\u8fdb\u884c\u62df\u5408\uff0c\u4ece\u800c\u4e0d\u65ad\u63d0\u9ad8\u6574\u4e2a\u6a21\u578b\u7684\u9884\u6d4b\u51c6\u786e\u7387\u3002<\/p>\n\n\n\n
                                                                        \u5177\u4f53\u5730\uff0c\u68af\u5ea6\u63d0\u5347\u7684\u8fc7\u7a0b\u53ef\u4ee5\u6982\u62ec\u4e3a\u4ee5\u4e0b\u51e0\u4e2a\u6b65\u9aa4\uff1a<\/p>\n\n\n\n
                                                                          \n
                                                                        1. \u521d\u59cb\u5316\u3002\u5c06\u6240\u6709\u7684\u8bad\u7ec3\u6837\u672c\u7684\u8f93\u51fa\u503c\u4f5c\u4e3a\u521d\u59cb\u503c\uff0c\u4f5c\u4e3a\u6a21\u578b\u7684\u7b2c\u4e00\u4e2a\u9884\u6d4b\u7ed3\u679c\u3002<\/li>\n\n\n\n
                                                                        2. \u8fed\u4ee3\u3002\u9488\u5bf9\u5f53\u524d\u6a21\u578b\u7684\u9884\u6d4b\u7ed3\u679c\u548c\u771f\u5b9e\u6807\u7b7e\u8ba1\u7b97\u6b8b\u5dee\uff08\u5373\u9884\u6d4b\u503c\u4e0e\u771f\u5b9e\u503c\u7684\u5dee\uff09\uff0c\u7136\u540e\u5229\u7528\u8fd9\u4e9b\u6b8b\u5dee\u6765\u62df\u5408\u4e00\u4e2a\u65b0\u7684\u5f31\u5206\u7c7b\u5668\uff08\u4f8b\u5982\u51b3\u7b56\u6811\uff09\uff0c\u4f7f\u5f97\u65b0\u7684\u6a21\u578b\u5728\u5f53\u524d\u7684\u6b8b\u5dee\u4e0b\u5177\u6709\u6700\u4f73\u7684\u9884\u6d4b\u80fd\u529b\u3002<\/li>\n\n\n\n
                                                                        3. \u66f4\u65b0\u9884\u6d4b\u7ed3\u679c\u3002\u5c06\u65b0\u7684\u5f31\u5206\u7c7b\u5668\u548c\u4e4b\u524d\u7684\u6a21\u578b\u7ec4\u5408\u8d77\u6765\u5f97\u5230\u4e00\u4e2a\u65b0\u7684\u6a21\u578b\uff0c\u7136\u540e\u518d\u5229\u7528\u8fd9\u4e2a\u65b0\u7684\u6a21\u578b\u6765\u66f4\u65b0\u9884\u6d4b\u7ed3\u679c\u3002<\/li>\n\n\n\n
                                                                        4. \u5224\u65ad\u662f\u5426\u7ed3\u675f\u3002\u5982\u679c\u6ee1\u8db3\u4e00\u5b9a\u7684\u6761\u4ef6\uff08\u4f8b\u5982\u8fbe\u5230\u6700\u5927\u8fed\u4ee3\u6b21\u6570\u6216\u9884\u6d4b\u8bef\u5dee\u5df2\u7ecf\u8db3\u591f\u5c0f\uff09\uff0c\u5219\u505c\u6b62\u8fed\u4ee3\uff0c\u5426\u5219\u7ee7\u7eed\u8fdb\u884c\u7b2c2\u6b65\u3002<\/li>\n<\/ol>\n\n\n\n
                                                                          \u68af\u5ea6\u63d0\u5347\u7684\u4e00\u4e2a\u91cd\u8981\u7279\u70b9\u662f\u5b83\u662f\u4e00\u79cd\u9010\u6b65\u589e\u5f3a\u6a21\u578b\u7684\u65b9\u5f0f\uff0c\u6bcf\u4e00\u6b21\u8fed\u4ee3\u90fd\u662f\u5728\u4e0a\u4e00\u6b21\u8fed\u4ee3\u7684\u57fa\u7840\u4e0a\u8fdb\u884c\u7684\u3002\u56e0\u6b64\uff0c\u5728\u8bad\u7ec3\u8fc7\u7a0b\u4e2d\uff0c\u68af\u5ea6\u63d0\u5347\u53ef\u4ee5\u6709\u6548\u5730\u6355\u6349\u5230\u6570\u636e\u96c6\u4e2d\u7684\u590d\u6742\u7279\u5f81\uff0c\u4ece\u800c\u63d0\u9ad8\u6574\u4e2a\u6a21\u578b\u7684\u6cdb\u5316\u80fd\u529b\u3002<\/p>\n\n\n\n
                                                                          \u68af\u5ea6\u63d0\u5347\u7b97\u6cd5\u7684\u4f18\u70b9\u5728\u4e8e\u5b83\u53ef\u4ee5\u5904\u7406\u975e\u7ebf\u6027\u5173\u7cfb\u548c\u9ad8\u7ef4\u7279\u5f81\uff0c\u800c\u4e14\u5bf9\u566a\u58f0\u548c\u5f02\u5e38\u70b9\u7684\u9c81\u68d2\u6027\u8f83\u5f3a\u3002\u4e0d\u8fc7\uff0c\u68af\u5ea6\u63d0\u5347\u4e5f\u5b58\u5728\u4e00\u4e9b\u7f3a\u70b9\uff0c\u4f8b\u5982\u8ba1\u7b97\u6210\u672c\u8f83\u9ad8\u3001\u5bb9\u6613\u8fc7\u62df\u5408\u7b49\u3002\u56e0\u6b64\uff0c\u5728\u5b9e\u9645\u5e94\u7528\u4e2d\uff0c\u9700\u8981\u6839\u636e\u5177\u4f53\u60c5\u51b5\u6765\u9009\u62e9\u5408\u9002\u7684\u7b97\u6cd5\u548c\u53c2\u6570\u3002<\/p>\n\n\n\n\n\n
                                                                          \"gun<\/figure>\n\n\n\n
                                                                          \u8bf7\u8be6\u7ec6\u4ecb\u7ecd\u65f6\u95f4\u5e8f\u5217\u9884\u6d4b<\/p>\n\n\n\n
                                                                          \u65f6\u95f4\u5e8f\u5217\u9884\u6d4b\u662f\u6307\u4f7f\u7528\u5386\u53f2\u65f6\u95f4\u5e8f\u5217\u6570\u636e\u6765\u9884\u6d4b\u672a\u6765\u7684\u8d8b\u52bf\u548c\u6a21\u5f0f\u3002\u65f6\u95f4\u5e8f\u5217\u6570\u636e\u662f\u6309\u65f6\u95f4\u987a\u5e8f\u6392\u5217\u7684\u6570\u636e\u96c6\u5408\uff0c\u901a\u5e38\u6d89\u53ca\u5230\u67d0\u79cd\u5ea6\u91cf\u7684\u53d8\u5316\uff0c\u4f8b\u5982\u80a1\u7968\u4ef7\u683c\u3001\u9500\u552e\u91cf\u3001\u5929\u6c14\u7b49\u3002<\/p>\n\n\n\n
                                                                          \u65f6\u95f4\u5e8f\u5217\u9884\u6d4b\u662f\u8bb8\u591a\u5b9e\u9645\u95ee\u9898\u4e2d\u7684\u5173\u952e\u4efb\u52a1\uff0c\u4f8b\u5982\u80a1\u7968\u5e02\u573a\u9884\u6d4b\u3001\u6c14\u8c61\u9884\u62a5\u3001\u9500\u552e\u9884\u6d4b\u7b49\u3002\u5b83\u4e5f\u662f\u8bb8\u591a\u4eba\u5de5\u667a\u80fd\u548c\u673a\u5668\u5b66\u4e60\u7b97\u6cd5\u7684\u57fa\u7840\uff0c\u5982ARIMA\uff08\u81ea\u56de\u5f52\u6ed1\u52a8\u5e73\u5747\u6a21\u578b\uff09\u3001LSTM\uff08\u957f\u77ed\u65f6\u8bb0\u5fc6\u7f51\u7edc\uff09\u7b49\u3002<\/p>\n\n\n\n
                                                                          \u65f6\u95f4\u5e8f\u5217\u9884\u6d4b\u7684\u65b9\u6cd5\u53ef\u4ee5\u5206\u4e3a\u4f20\u7edf\u7edf\u8ba1\u65b9\u6cd5\u548c\u673a\u5668\u5b66\u4e60\u65b9\u6cd5\u3002\u5176\u4e2d\uff0c\u4f20\u7edf\u7edf\u8ba1\u65b9\u6cd5\u5305\u62ecARIMA\u3001SARIMA\uff08\u5b63\u8282\u6027\u81ea\u56de\u5f52\u6ed1\u52a8\u5e73\u5747\u6a21\u578b\uff09\u3001ETS\uff08\u6307\u6570\u5e73\u6ed1\u6a21\u578b\uff09\u7b49\uff0c\u8fd9\u4e9b\u65b9\u6cd5\u901a\u5e38\u4f9d\u8d56\u4e8e\u5bf9\u65f6\u95f4\u5e8f\u5217\u6570\u636e\u7684\u5206\u89e3\u548c\u5efa\u6a21\u3002\u673a\u5668\u5b66\u4e60\u65b9\u6cd5\u5219\u5305\u62ec\u795e\u7ecf\u7f51\u7edc\u6a21\u578b\u3001\u652f\u6301\u5411\u91cf\u673a\u3001\u968f\u673a\u68ee\u6797\u7b49\uff0c\u8fd9\u4e9b\u65b9\u6cd5\u53ef\u4ee5\u66f4\u597d\u5730\u5904\u7406\u975e\u7ebf\u6027\u95ee\u9898\u548c\u9ad8\u7ef4\u6570\u636e\uff0c\u5e76\u4e14\u5728\u6570\u636e\u91cf\u8db3\u591f\u5927\u65f6\u901a\u5e38\u5177\u6709\u66f4\u597d\u7684\u9884\u6d4b\u80fd\u529b\u3002<\/p>\n\n\n\n
                                                                          \u65f6\u95f4\u5e8f\u5217\u9884\u6d4b\u7684\u4e00\u822c\u6b65\u9aa4\u5305\u62ec\uff1a<\/p>\n\n\n\n
                                                                            \n
                                                                          1. \u6570\u636e\u51c6\u5907\uff1a\u6536\u96c6\u5386\u53f2\u65f6\u95f4\u5e8f\u5217\u6570\u636e\uff0c\u5bf9\u6570\u636e\u8fdb\u884c\u6e05\u6d17\u548c\u9884\u5904\u7406\uff0c\u4f8b\u5982\u53bb\u9664\u5f02\u5e38\u503c\u3001\u7f3a\u5931\u503c\u3001\u5e73\u6ed1\u7b49\u3002<\/li>\n\n\n\n
                                                                          2. \u6570\u636e\u5206\u89e3\uff1a\u5c06\u65f6\u95f4\u5e8f\u5217\u5206\u89e3\u4e3a\u8d8b\u52bf\u3001\u5b63\u8282\u6027\u548c\u968f\u673a\u6210\u5206\uff0c\u4ee5\u4fbf\u66f4\u597d\u5730\u7406\u89e3\u548c\u5efa\u6a21\u6570\u636e\u3002<\/li>\n\n\n\n
                                                                          3. \u6a21\u578b\u9009\u62e9\u548c\u62df\u5408\uff1a\u9009\u62e9\u9002\u5f53\u7684\u6a21\u578b\u548c\u7b97\u6cd5\uff0c\u5e76\u62df\u5408\u6570\u636e\u3002\u5e38\u7528\u7684\u6a21\u578b\u5305\u62ecARIMA\u3001LSTM\u3001GRU\uff08\u95e8\u63a7\u5faa\u73af\u5355\u5143\uff09\u7b49\u3002<\/li>\n\n\n\n
                                                                          4. \u6a21\u578b\u8bc4\u4f30\uff1a\u4f7f\u7528\u4e00\u4e9b\u8bc4\u4ef7\u6307\u6807\uff08\u4f8b\u5982\u5747\u65b9\u8bef\u5dee\u3001\u5747\u65b9\u6839\u8bef\u5dee\u3001\u5e73\u5747\u7edd\u5bf9\u8bef\u5dee\u7b49\uff09\u6765\u8bc4\u4f30\u6a21\u578b\u7684\u9884\u6d4b\u80fd\u529b\u3002<\/li>\n\n\n\n
                                                                          5. \u6a21\u578b\u5e94\u7528\uff1a\u4f7f\u7528\u6a21\u578b\u6765\u9884\u6d4b\u672a\u6765\u7684\u65f6\u95f4\u5e8f\u5217\uff0c\u5e76\u8fdb\u884c\u51b3\u7b56\u3002<\/li>\n<\/ol>\n\n\n\n\n\n
                                                                            \"gun<\/figure>\n\n\n\n
                                                                            \u8bf7\u8be6\u7ec6\u89e3\u91ca\u5f3a\u5316\u5b66\u4e60<\/p>\n\n\n\n
                                                                            \u5f3a\u5316\u5b66\u4e60\u662f\u4e00\u79cd\u673a\u5668\u5b66\u4e60\u65b9\u6cd5\uff0c\u65e8\u5728\u901a\u8fc7\u5b66\u4e60\u4e0e\u73af\u5883\u7684\u4ea4\u4e92\uff0c\u4f7f\u667a\u80fd\u4f53\uff08\u6216\u51b3\u7b56\u8005\uff09\u80fd\u591f\u5728\u73af\u5883\u4e2d\u91c7\u53d6\u6700\u4f18\u7684\u884c\u52a8\u3002\u5728\u5f3a\u5316\u5b66\u4e60\u4e2d\uff0c\u667a\u80fd\u4f53\u5728\u6267\u884c\u884c\u52a8\u540e\u4f1a\u83b7\u5f97\u4e00\u4e2a\u53cd\u9988\uff0c\u79f0\u4e3a\u5956\u52b1\uff0c\u4ee5\u6307\u793a\u5176\u884c\u4e3a\u7684\u597d\u574f\u3002\u667a\u80fd\u4f53\u7684\u76ee\u6807\u662f\u6700\u5927\u5316\u671f\u671b\u7684\u7d2f\u79ef\u5956\u52b1\uff0c\u540c\u65f6\u9010\u6b65\u5b66\u4e60\u5728\u4e0d\u540c\u72b6\u6001\u4e0b\u91c7\u53d6\u54ea\u4e9b\u884c\u52a8\u662f\u6700\u4f18\u7684\u3002<\/p>\n\n\n\n
                                                                            \u5f3a\u5316\u5b66\u4e60\u901a\u5e38\u6d89\u53ca\u5230\u4ee5\u4e0b\u5143\u7d20\uff1a<\/p>\n\n\n\n
                                                                              \n
                                                                            • \u72b6\u6001\u7a7a\u95f4\uff1a\u73af\u5883\u4e2d\u6240\u6709\u53ef\u80fd\u7684\u72b6\u6001\u7684\u96c6\u5408\u3002<\/li>\n\n\n\n
                                                                            • \u884c\u52a8\u7a7a\u95f4\uff1a\u667a\u80fd\u4f53\u53ef\u4ee5\u6267\u884c\u7684\u6240\u6709\u53ef\u80fd\u7684\u884c\u52a8\u7684\u96c6\u5408\u3002<\/li>\n\n\n\n
                                                                            • \u5956\u52b1\u51fd\u6570\uff1a\u6307\u793a\u6bcf\u4e2a\u72b6\u6001\u548c\u884c\u52a8\u7684\u597d\u574f\u7684\u51fd\u6570\u3002<\/li>\n\n\n\n
                                                                            • \u7b56\u7565\uff1a\u667a\u80fd\u4f53\u4ece\u72b6\u6001\u5230\u884c\u52a8\u7684\u6620\u5c04\u51fd\u6570\u3002<\/li>\n\n\n\n
                                                                            • \u503c\u51fd\u6570\uff1a\u6307\u793a\u5728\u6bcf\u4e2a\u72b6\u6001\u4e0b\u91c7\u53d6\u7b56\u7565\u6240\u80fd\u83b7\u5f97\u7684\u671f\u671b\u5956\u52b1\u7684\u51fd\u6570\u3002<\/li>\n<\/ul>\n\n\n\n
                                                                              \u5f3a\u5316\u5b66\u4e60\u53ef\u4ee5\u7528\u4e8e\u89e3\u51b3\u5f88\u591a\u95ee\u9898\uff0c\u4f8b\u5982\u63a7\u5236\u673a\u5668\u4eba\u3001\u6e38\u620f\u73a9\u5bb6\u3001\u81ea\u52a8\u9a7e\u9a76\u6c7d\u8f66\u7b49\u3002\u5b83\u4e5f\u662f AlphaGo \u548c AlphaZero \u7b49\u4eba\u5de5\u667a\u80fd\u7cfb\u7edf\u7684\u6838\u5fc3\u7b97\u6cd5\u3002\u5e38\u89c1\u7684\u5f3a\u5316\u5b66\u4e60\u7b97\u6cd5\u5305\u62ec Q-learning\u3001\u7b56\u7565\u68af\u5ea6\u3001\u6df1\u5ea6 Q \u7f51\u7edc\uff08DQN\uff09\u3001\u8499\u7279\u5361\u6d1b\u6811\u641c\u7d22\uff08MCTS\uff09\u7b49\u3002<\/p>\n\n\n\n\n\n
                                                                              \"gun<\/figure>\n\n\n\n
                                                                              \u8bf7\u8be6\u7ec6\u89e3\u91caQ-Learning,\u5e76\u7ed9\u51fa\u793a\u4f8b\u4ee3\u7801<\/p>\n\n\n\n
                                                                              Q-Learning\u662f\u5f3a\u5316\u5b66\u4e60\u4e2d\u4e00\u79cd\u7ecf\u5178\u7684\u65e0\u6a21\u578b\u5b66\u4e60\u7b97\u6cd5\uff0c\u5e38\u7528\u4e8e\u89e3\u51b3\u9a6c\u5c14\u53ef\u592b\u51b3\u7b56\u8fc7\u7a0b\uff08Markov Decision Process\uff0cMDP\uff09\u95ee\u9898\u3002<\/p>\n\n\n\n
                                                                              \u5176\u6838\u5fc3\u601d\u60f3\u662f\u4f7f\u7528\u4e00\u4e2aQ\u51fd\u6570\uff08\u4e5f\u53eb\u505a\u4ef7\u503c\u51fd\u6570\uff09\uff0c\u6765\u8bc4\u4f30\u5f53\u524d\u72b6\u6001\u4e0b\u91c7\u53d6\u67d0\u4e2a\u52a8\u4f5c\u6240\u80fd\u5f97\u5230\u7684\u56de\u62a5\uff0c\u66f4\u65b0Q\u51fd\u6570\u7684\u8fc7\u7a0b\u901a\u8fc7\u4e0d\u65ad\u5730\u8fed\u4ee3\u6765\u5b9e\u73b0\u3002Q-Learning\u7b97\u6cd5\u7684\u76ee\u6807\u662f\u6700\u5927\u5316\u7d2f\u79ef\u7684\u56de\u62a5\uff0c\u8fd9\u4e2a\u56de\u62a5\u6307\u7684\u662f\u4ece\u521d\u59cb\u72b6\u6001\u5f00\u59cb\u6267\u884c\u4e00\u7cfb\u5217\u7684\u52a8\u4f5c\u6240\u83b7\u5f97\u7684\u5956\u52b1\u4e4b\u548c\u3002<\/p>\n\n\n\n
                                                                              \u5177\u4f53\u6765\u8bf4\uff0c\u5728\u6bcf\u6b21\u4ea4\u4e92\u4e2d\uff0cQ-Learning\u7b97\u6cd5\u4f1a\u6839\u636e\u5f53\u524d\u72b6\u6001\u9009\u62e9\u4e00\u4e2a\u52a8\u4f5c\uff0c\u5e76\u89c2\u5bdf\u73af\u5883\u7684\u53cd\u9988\uff0c\u5305\u62ec\u4e0b\u4e00\u4e2a\u72b6\u6001\u548c\u83b7\u5f97\u7684\u5956\u52b1\u3002\u6839\u636e\u8fd9\u4e9b\u4fe1\u606f\uff0cQ-Learning\u7b97\u6cd5\u4f1a\u4f7f\u7528Bellman\u65b9\u7a0b\u6765\u66f4\u65b0\u5f53\u524d\u72b6\u6001\u4e0b\u91c7\u53d6\u8be5\u52a8\u4f5c\u7684Q\u503c\u3002\u5177\u4f53\u800c\u8a00\uff0c\u66f4\u65b0\u516c\u5f0f\u5982\u4e0b\uff1a<\/p>\n\n\n\n
                                                                              \ufffd(\ufffd\ufffd,\ufffd\ufffd)\u2190\ufffd(\ufffd\ufffd,\ufffd\ufffd)+\ufffd\u22c5(\ufffd\ufffd+\ufffd\u22c5max\u2061\ufffd\ufffd(\ufffd\ufffd+1,\ufffd)\u2212\ufffd(\ufffd\ufffd,\ufffd\ufffd))Q<\/em>(s<\/em>t<\/em>\u200b,a<\/em>t<\/em>\u200b)\u2190Q<\/em>(s<\/em>t<\/em>\u200b,a<\/em>t<\/em>\u200b)+\u03b1<\/em>\u22c5(r<\/em>t<\/em>\u200b+\u03b3<\/em>\u22c5maxa<\/em>\u200bQ<\/em>(s<\/em>t<\/em>+1\u200b,a<\/em>)\u2212Q<\/em>(s<\/em>t<\/em>\u200b,a<\/em>t<\/em>\u200b))<\/p>\n\n\n\n
                                                                              \u5176\u4e2d\uff0c$s_t$\u8868\u793a\u5f53\u524d\u72b6\u6001\uff0c$a_t$\u8868\u793a\u9009\u62e9\u7684\u52a8\u4f5c\uff0c$r_t$\u8868\u793a\u83b7\u5f97\u7684\u5956\u52b1\uff0c$\\alpha$\u8868\u793a\u5b66\u4e60\u7387\uff0c$\\gamma$\u8868\u793a\u6298\u6263\u56e0\u5b50\u3002\u8fd9\u4e2a\u66f4\u65b0\u516c\u5f0f\u7684\u542b\u4e49\u662f\uff0c\u5c06\u5f53\u524d\u72b6\u6001\u548c\u9009\u62e9\u7684\u52a8\u4f5c\u5bf9\u5e94\u7684Q\u503c\u66f4\u65b0\u4e3a\u5f53\u524dQ\u503c\u52a0\u4e0a\u5b66\u4e60\u7387\u4e58\u4ee5\u4e00\u4e2a\u8bef\u5dee\u9879\u3002\u8fd9\u4e2a\u8bef\u5dee\u9879\u7531\u5f53\u524d\u5956\u52b1\u548c\u4e0b\u4e00\u4e2a\u72b6\u6001\u4e2d\u7684\u6700\u5927Q\u503c\u4ee5\u53ca\u5f53\u524d\u72b6\u6001\u548c\u9009\u62e9\u7684\u52a8\u4f5c\u5bf9\u5e94\u7684Q\u503c\u4e4b\u5dee\u7ec4\u6210\u3002<\/p>\n\n\n\n
                                                                              \u4ee5\u4e0b\u662f\u4e00\u4e2a\u7b80\u5355\u7684Q-Learning\u7b97\u6cd5\u7684\u793a\u4f8b\u4ee3\u7801\uff0c\u7528\u4e8e\u89e3\u51b3\u4e00\u4e2a\u7b80\u5355\u7684\u8ff7\u5bab\u95ee\u9898\uff1a<\/p>\n\n\n\n
                                                                              pythonCopy codeimport numpy as np\n\n# \u5b9a\u4e49\u73af\u5883\u7684\u72b6\u6001\u548c\u52a8\u4f5c\u7a7a\u95f4\nstates = [0, 1, 2, 3, 4, 5]\nactions = ['left', 'right', 'up', 'down']\n\n# \u5b9a\u4e49\u521d\u59cbQ\u503c\nQ = np.zeros((len(states), len(actions)))\n\n# \u5b9a\u4e49\u5956\u52b1\u77e9\u9635\nR = np.array([\n    [-1, -1, -1, -1, 0, -1],\n    [-1, -1, -1, 0, -1, 100],\n    [-1, -1, -1, 0, -1, -1],\n    [-1, 0, 0, -1, 0, -1],\n    [0, -1, -1, 0, -1, 100],\n    [-1, 0, -1, -1, 0, 100]\n])\n\n# \u5b9a\u4e49\u5176\u4ed6\u53c2\u6570\ngamma = 0.8\nalpha = 0.5\nnum_episodes = 500\n\n# \u8bad\u7ec3Q-Learning\u6a21\u578b\nfor i in range(num_episodes):\n    # \u9009\u62e9\u521d\u59cb\u72b6\u6001\n    state = np.random.choice(states)\n    while state != 5\n<\/code><\/pre>\n\n\n\n\n\n
                                                                              \"gun<\/figure>\n\n\n\n
                                                                              \u8bf7\u8be6\u7ec6\u89e3\u91caActor-Critic\u7b97\u6cd5\uff0c\u5e76\u7ed9\u51fa\u793a\u4f8b\u4ee3\u7801<\/p>\n\n\n\n
                                                                              Actor-Critic\u7b97\u6cd5\u662f\u5f3a\u5316\u5b66\u4e60\u4e2d\u4e00\u7c7b\u7ecf\u5178\u7684\u7b56\u7565\u4f18\u5316\u65b9\u6cd5\uff0c\u7ed3\u5408\u4e86Actor\u548cCritic\u4e24\u4e2a\u89d2\u8272\uff0c\u5176\u4e2dActor\u7528\u4e8e\u5b66\u4e60\u7b56\u7565\uff0cCritic\u7528\u4e8e\u8bc4\u4f30\u7b56\u7565\u7684\u597d\u574f\u3002\u5728\u6bcf\u4e00\u6b65\u4e2d\uff0cActor\u4f1a\u57fa\u4e8e\u5f53\u524d\u72b6\u6001\u9009\u53d6\u4e00\u4e2a\u52a8\u4f5c\uff0cCritic\u4f1a\u5bf9Actor\u9009\u53d6\u7684\u52a8\u4f5c\u8fdb\u884c\u8bc4\u4f30\uff0c\u7136\u540eActor\u6839\u636e\u8fd9\u4e2a\u8bc4\u4f30\u7ed3\u679c\u66f4\u65b0\u81ea\u5df1\u7684\u7b56\u7565\u3002<\/p>\n\n\n\n
                                                                              \u4e0b\u9762\u7ed9\u51fa\u4e00\u4e2a\u7b80\u5355\u7684Actor-Critic\u7b97\u6cd5\u7684\u793a\u4f8b\u4ee3\u7801\uff0c\u7528\u4e8e\u89e3\u51b3OpenAI Gym\u4e2d\u7684CartPole-v0\u6e38\u620f\uff1a<\/p>\n\n\n\n
                                                                              pythonCopy codeimport numpy as np\nimport gym\n\n# \u5b9a\u4e49Actor\u7c7b\nclass Actor:\n    def __init__(self, input_dim, output_dim, lr):\n        self.weights = np.random.rand(input_dim, output_dim) * 0.01\n        self.lr = lr\n\n    # Actor\u7b56\u7565\n    def act(self, state):\n        probabilities = self.softmax(np.dot(state, self.weights))\n        action = np.random.choice(len(probabilities), p=probabilities)\n        return action, probabilities\n\n    # Actor\u7b56\u7565\u66f4\u65b0\n    def update(self, gradient, state):\n        self.weights += self.lr * np.dot(state.T, gradient)\n\n    # softmax\u51fd\u6570\n    def softmax(self, x):\n        exp_x = np.exp(x)\n        return exp_x \/ np.sum(exp_x)\n\n# \u5b9a\u4e49Critic\u7c7b\nclass Critic:\n    def __init__(self, input_dim, lr):\n        self.weights = np.random.rand(input_dim, 1) * 0.01\n        self.lr = lr\n\n    # Critic\u4ef7\u503c\u8bc4\u4f30\n    def evaluate(self, state):\n        return np.dot(state, self.weights)\n\n    # Critic\u4ef7\u503c\u66f4\u65b0\n    def update(self, td_error, state):\n        self.weights += self.lr * td_error * state\n\n# \u5b9a\u4e49Actor-Critic\u7c7b\nclass ActorCritic:\n    def __init__(self, env, lr_actor, lr_critic, gamma):\n        self.env = env\n        self.observation_space = env.observation_space.shape[0]\n        self.action_space = env.action_space.n\n        self.actor = Actor(self.observation_space, self.action_space, lr_actor)\n        self.critic = Critic(self.observation_space, lr_critic)\n        self.gamma = gamma\n\n    # \u8bad\u7ec3\u51fd\u6570\n    def train(self, num_episodes):\n        for i in range(num_episodes):\n            state = self.env.reset()\n            done = False\n            total_reward = 0\n            while not done:\n                action, probabilities = self.actor.act(state)\n                next_state, reward, done, _ = self.env.step(action)\n                td_error = reward + self.gamma * self.critic.evaluate(next_state) - self.critic.evaluate(state)\n                self.critic.update(td_error, state)\n                gradient = np.zeros_like(probabilities)\n                gradient[action] = 1 \/ probabilities[action]\n                self.actor.update(gradient * td_error, state)\n                state = next_state\n                total_reward += reward\n            print(\"Episode {}: Total reward = {}\".format(i+1, total_reward))\n\n# \u6d4b\u8bd5\u4ee3\u7801\nenv = gym.make('CartPole-v0')\nactor_critic = ActorCritic(env, 0.01, 0.01, 0.99)\nactor_critic.train(1000)\n<\/code><\/pre>\n\n\n\n
                                                                              \u5728\u4e0a\u9762\u7684\u4ee3\u7801\u4e2d\uff0c\u6211\u4eec\u5b9a\u4e49\u4e86Actor\u7c7b\u548cC<\/p>\n\n\n\n\n\n
                                                                              \"gun<\/figure>\n\n\n\n
                                                                              \u8bf7\u8be6\u7ec6\u89e3\u91ca\u5f3a\u5316\u5b66\u4e60\u4e2d\u7684\u4ef7\u503c\u51fd\u6570<\/p>\n\n\n\n
                                                                              \u5728\u5f3a\u5316\u5b66\u4e60\u4e2d\uff0c\u4ef7\u503c\u51fd\u6570\u662f\u6307\u5728\u67d0\u4e2a\u72b6\u6001\u6216\u52a8\u4f5c\u4e0b\uff0c\u5bf9\u4e8e\u4e00\u4e2a\u667a\u80fd\u4f53\u6765\u8bf4\uff0c\u5b83\u6240\u80fd\u83b7\u5f97\u7684\u957f\u671f\u5956\u52b1\u7684\u671f\u671b\u503c\u3002\u5177\u4f53\u6765\u8bf4\uff0c\u4ef7\u503c\u51fd\u6570\u53ef\u4ee5\u5206\u4e3a\u4e24\u79cd\uff1a<\/p>\n\n\n\n
                                                                              1.\u72b6\u6001\u4ef7\u503c\u51fd\u6570\uff1a\u6307\u5728\u67d0\u4e2a\u72b6\u6001\u4e0b\uff0c\u667a\u80fd\u4f53\u6240\u80fd\u83b7\u5f97\u7684\u957f\u671f\u5956\u52b1\u7684\u671f\u671b\u503c\uff0c\u4e5f\u5c31\u662f\u5728\u8be5\u72b6\u6001\u4e0b\u7684\u52a8\u4f5c\u6240\u80fd\u5e26\u6765\u7684\u7d2f\u8ba1\u5956\u52b1\u3002 2.\u52a8\u4f5c\u4ef7\u503c\u51fd\u6570\uff1a\u6307\u5728\u67d0\u4e2a\u72b6\u6001\u4e0b\u91c7\u53d6\u67d0\u4e2a\u52a8\u4f5c\u540e\uff0c\u667a\u80fd\u4f53\u6240\u80fd\u83b7\u5f97\u7684\u957f\u671f\u5956\u52b1\u7684\u671f\u671b\u503c\uff0c\u4e5f\u5c31\u662f\u8be5\u52a8\u4f5c\u6240\u5e26\u6765\u7684\u7d2f\u8ba1\u5956\u52b1\u3002<\/p>\n\n\n\n
                                                                              \u5728\u5f3a\u5316\u5b66\u4e60\u4e2d\uff0c\u667a\u80fd\u4f53\u7684\u76ee\u6807\u5c31\u662f\u901a\u8fc7\u5b66\u4e60\u5230\u6700\u4f18\u7684\u4ef7\u503c\u51fd\u6570\uff0c\u6765\u5b9e\u73b0\u5728\u6bcf\u4e2a\u72b6\u6001\u6216\u52a8\u4f5c\u4e0b\u91c7\u53d6\u6700\u4f18\u7684\u51b3\u7b56\uff0c\u4ece\u800c\u6700\u5927\u5316\u7d2f\u8ba1\u5956\u52b1\u3002<\/p>\n\n\n\n
                                                                              \u5176\u4e2d\uff0c\u4ef7\u503c\u51fd\u6570\u7684\u5b66\u4e60\u53ef\u4ee5\u901a\u8fc7\u4e0d\u65ad\u8fed\u4ee3\u66f4\u65b0\u4ef7\u503c\u51fd\u6570\u6765\u5b9e\u73b0\uff0c\u6700\u5e38\u7528\u7684\u65b9\u6cd5\u662f\u57fa\u4e8e\u8d1d\u5c14\u66fc\u65b9\u7a0b\u3002\u5177\u4f53\u6765\u8bf4\uff0c\u8d1d\u5c14\u66fc\u65b9\u7a0b\u662f\u4e00\u79cd\u9012\u5f52\u7684\u8868\u8fbe\u5f0f\uff0c\u7528\u4e8e\u8868\u793a\u5f53\u524d\u72b6\u6001\u6216\u52a8\u4f5c\u7684\u4ef7\u503c\u51fd\u6570\u53ef\u4ee5\u901a\u8fc7\u4e0b\u4e00\u4e2a\u72b6\u6001\u6216\u52a8\u4f5c\u7684\u4ef7\u503c\u51fd\u6570\u8fdb\u884c\u66f4\u65b0\u3002\u6839\u636e\u8d1d\u5c14\u66fc\u65b9\u7a0b\u7684\u4e0d\u540c\u5f62\u5f0f\uff0c\u53ef\u4ee5\u5206\u522b\u5f97\u5230\u503c\u8fed\u4ee3\u548c\u7b56\u7565\u8fed\u4ee3\u4e24\u79cd\u7b97\u6cd5\u6765\u8fdb\u884c\u4ef7\u503c\u51fd\u6570\u7684\u5b66\u4e60\u3002<\/p>\n\n\n\n
                                                                              \u4ee5\u4e0b\u662f\u4e00\u4e2a\u7b80\u5355\u7684\u72b6\u6001\u4ef7\u503c\u51fd\u6570\u7684\u503c\u8fed\u4ee3\u793a\u4f8b\u4ee3\u7801\uff1a<\/p>\n\n\n\n
                                                                              pythonCopy codeimport numpy as np\n\n# \u5b9a\u4e49\u73af\u5883\nnum_states = 5\nnum_actions = 2\ngamma = 0.9\nenv = np.array([[-1, 0], [0, -1], [1, 0], [0, 1], [0, 0]])\n\n# \u5b9a\u4e49\u72b6\u6001\u4ef7\u503c\u51fd\u6570\nV = np.zeros(num_states)\n\n# \u8fdb\u884c\u503c\u8fed\u4ee3\nfor i in range(100):\n    V_new = np.zeros(num_states)\n    for s in range(num_states):\n        Q = np.zeros(num_actions)\n        for a in range(num_actions):\n            s_next = np.clip(s + env[s][a], 0, num_states-1)\n            Q[a] = -1 + gamma * V[s_next]\n        V_new[s] = np.max(Q)\n    if np.max(np.abs(V_new - V)) < 1e-6:\n        break\n    V = V_new\n\nprint(\"\u72b6\u6001\u4ef7\u503c\u51fd\u6570\uff1a\", V)\n<\/code><\/pre>\n\n\n\n
                                                                              \u8fd9\u6bb5\u4ee3\u7801\u4e2d\uff0c\u6211\u4eec\u9996\u5148\u5b9a\u4e49\u4e86\u4e00\u4e2a\u7b80\u5355\u7684\u73af\u5883\uff0c\u5176\u4e2d\u5305\u542b5\u4e2a\u72b6\u6001\u548c2\u4e2a\u52a8\u4f5c\u3002\u7136\u540e\u6211\u4eec\u5b9a\u4e49\u4e86\u72b6\u6001\u4ef7\u503c\u51fd\u6570V\uff0c\u5e76\u901a\u8fc7\u503c\u8fed\u4ee3\u7684\u65b9\u5f0f\u8fdb\u884c\u66f4\u65b0\u3002\u5728\u6bcf\u6b21\u66f4\u65b0\u4e2d\uff0c\u6211\u4eec\u6839\u636e\u5f53\u524d\u72b6\u6001\u548c\u52a8\u4f5c\uff0c\u8ba1\u7b97\u4e0b\u4e00\u4e2a\u72b6\u6001\u7684\u4ef7\u503c\uff0c\u5e76\u9009\u62e9\u5176\u4e2d\u6700\u5927\u7684\u4ef7\u503c\u66f4\u65b0\u5f53\u524d\u72b6\u6001\u7684\u4ef7\u503c\u51fd\u6570\uff0c\u76f4\u5230\u6536\u655b\u3002\u6700\u540e\u8f93\u51fa\u5f97\u5230\u7684\u72b6\u6001\u4ef7\u503c\u51fd\u6570V\u3002<\/p>\n\n\n\n\n\n
                                                                              \"gun<\/figure>\n\n\n\n
                                                                              \u8bf7\u8be6\u7ec6\u89e3\u91ca\u5f3a\u5316\u5b66\u4e60\u7684\u7b56\u7565\u51fd\u6570<\/p>\n\n\n\n
                                                                              \u5728\u5f3a\u5316\u5b66\u4e60\u4e2d\uff0c\u7b56\u7565\u51fd\u6570\u662f\u4e00\u79cd\u4ece\u72b6\u6001\u5230\u52a8\u4f5c\u7684\u6620\u5c04\u51fd\u6570\uff0c\u7528\u4e8e\u6307\u5bfc\u667a\u80fd\u4f53\u5728\u5f53\u524d\u72b6\u6001\u4e0b\u9009\u62e9\u5408\u9002\u7684\u52a8\u4f5c\u3002\u5b83\u662f\u5f3a\u5316\u5b66\u4e60\u4e2d\u91cd\u8981\u7684\u4e00\u4e2a\u6982\u5ff5\uff0c\u56e0\u4e3a\u667a\u80fd\u4f53\u7684\u51b3\u7b56\u884c\u4e3a\u5f80\u5f80\u662f\u7531\u7b56\u7565\u51fd\u6570\u6240\u51b3\u5b9a\u7684\u3002<\/p>\n\n\n\n
                                                                              \u5728\u5f3a\u5316\u5b66\u4e60\u4e2d\uff0c\u7b56\u7565\u51fd\u6570\u6709\u4e24\u79cd\u5f62\u5f0f\uff1a\u786e\u5b9a\u6027\u7b56\u7565\u51fd\u6570\u548c\u968f\u673a\u7b56\u7565\u51fd\u6570\u3002\u786e\u5b9a\u6027\u7b56\u7565\u51fd\u6570\u662f\u4e00\u79cd\u786e\u5b9a\u4e86\u5177\u4f53\u52a8\u4f5c\u7684\u51fd\u6570\uff0c\u5b83\u4f1a\u5728\u6bcf\u4e2a\u72b6\u6001\u4e0b\u8f93\u51fa\u4e00\u4e2a\u786e\u5b9a\u7684\u52a8\u4f5c\u3002\u4f8b\u5982\uff0c\u5728\u673a\u5668\u4eba\u8def\u5f84\u89c4\u5212\u95ee\u9898\u4e2d\uff0c\u7b56\u7565\u51fd\u6570\u53ef\u4ee5\u5c06\u673a\u5668\u4eba\u7684\u4f4d\u7f6e\u4f5c\u4e3a\u8f93\u5165\uff0c\u8f93\u51fa\u673a\u5668\u4eba\u7684\u4e0b\u4e00\u6b65\u52a8\u4f5c\uff08\u6bd4\u5982\u5411\u524d\u8d70\u3001\u5411\u5de6\u8d70\u3001\u5411\u53f3\u8d70\u7b49\uff09\u3002\u800c\u968f\u673a\u7b56\u7565\u51fd\u6570\u5219\u662f\u4e00\u79cd\u5c06\u6bcf\u4e2a\u53ef\u80fd\u7684\u52a8\u4f5c\u8d4b\u4e88\u4e00\u5b9a\u6982\u7387\u7684\u51fd\u6570\uff0c\u5b83\u4f1a\u5728\u6bcf\u4e2a\u72b6\u6001\u4e0b\u8f93\u51fa\u4e00\u4e2a\u52a8\u4f5c\u5206\u5e03\u3002\u4f8b\u5982\uff0c\u5728\u6e38\u620f\u4e2d\uff0c\u7b56\u7565\u51fd\u6570\u53ef\u4ee5\u5c06\u5f53\u524d\u6e38\u620f\u72b6\u6001\u4f5c\u4e3a\u8f93\u5165\uff0c\u8f93\u51fa\u4e00\u4e2a\u5728\u6bcf\u4e2a\u53ef\u80fd\u7684\u52a8\u4f5c\u4e0a\u5206\u522b\u5177\u6709\u4e00\u5b9a\u6982\u7387\u7684\u5206\u5e03\u3002<\/p>\n\n\n\n
                                                                              \u4e00\u822c\u6765\u8bf4\uff0c\u7b56\u7565\u51fd\u6570\u53ef\u4ee5\u7528\u795e\u7ecf\u7f51\u7edc\u3001\u51b3\u7b56\u6811\u7b49\u673a\u5668\u5b66\u4e60\u7b97\u6cd5\u8fdb\u884c\u5efa\u6a21\u3002\u5728\u7b56\u7565\u51fd\u6570\u7684\u8bad\u7ec3\u4e2d\uff0c\u6211\u4eec\u9700\u8981\u5229\u7528\u67d0\u79cd\u5f3a\u5316\u5b66\u4e60\u7b97\u6cd5\u6765\u4e0d\u65ad\u4f18\u5316\u7b56\u7565\u51fd\u6570\uff0c\u4f7f\u5176\u5728\u4e0d\u540c\u7684\u72b6\u6001\u4e0b\u80fd\u591f\u9009\u62e9\u51fa\u6700\u4f18\u7684\u52a8\u4f5c\u3002<\/p>\n\n\n\n
                                                                              \u4ee5\u4e0b\u662f\u4e00\u4e2a\u7b80\u5355\u7684\u793a\u4f8b\u4ee3\u7801\uff0c\u7528\u4e8e\u5efa\u7acb\u4e00\u4e2a\u968f\u673a\u7b56\u7565\u51fd\u6570\uff1a<\/p>\n\n\n\n
                                                                              pythonCopy codeimport numpy as np\n\nclass RandomPolicy:\n    def __init__(self, num_actions):\n        self.num_actions = num_actions\n\n    def select_action(self, state):\n        action_probs = np.ones(self.num_actions) \/ self.num_actions\n        action = np.random.choice(range(self.num_actions), p=action_probs)\n        return action\n<\/code><\/pre>\n\n\n\n
                                                                              \u8fd9\u91cc\u6211\u4eec\u5b9a\u4e49\u4e86\u4e00\u4e2a\u540d\u4e3aRandomPolicy\u7684\u7c7b\uff0c\u5b83\u5305\u542b\u4e00\u4e2aselect_action\u65b9\u6cd5\uff0c\u7528\u4e8e\u9009\u62e9\u968f\u673a\u7684\u52a8\u4f5c\u3002\u5728\u8fd9\u4e2a\u65b9\u6cd5\u4e2d\uff0c\u6211\u4eec\u9996\u5148\u4e3a\u6bcf\u4e2a\u53ef\u80fd\u7684\u52a8\u4f5c\u5206\u914d\u76f8\u7b49\u7684\u6982\u7387\uff0c\u7136\u540e\u5229\u7528numpy\u5e93\u4e2d\u7684np.random.choice\u51fd\u6570\u4ece\u8fd9\u4e2a\u5206\u5e03\u4e2d\u968f\u673a\u9009\u62e9\u4e00\u4e2a\u52a8\u4f5c\u4f5c\u4e3a\u5f53\u524d\u72b6\u6001\u4e0b\u7684\u51b3\u7b56\u7ed3\u679c\u3002<\/p>\n\n\n\n\n\n
                                                                              \"gun<\/figure>\n\n\n\n
                                                                              \u8bf7\u8be6\u7ec6\u89e3\u91ca\u9a6c\u5c14\u53ef\u592b\u51b3\u7b56\u8fc7\u7a0b<\/p>\n\n\n\n
                                                                              \u9a6c\u5c14\u53ef\u592b\u51b3\u7b56\u8fc7\u7a0b\uff08Markov Decision Process\uff0cMDP\uff09\u662f\u5f3a\u5316\u5b66\u4e60\u4e2d\u4e00\u4e2a\u57fa\u7840\u6027\u7684\u6982\u5ff5\uff0c\u5b83\u7528\u4e8e\u63cf\u8ff0\u4e00\u7c7b\u968f\u673a\u51b3\u7b56\u95ee\u9898\u3002MDP\u662f\u4e00\u4e2a\u4e94\u5143\u7ec4 $(S,A,P,R,\\gamma)$\uff0c\u5176\u4e2d\uff1a<\/p>\n\n\n\n
                                                                                \n
                                                                              • $S$ \u8868\u793a\u72b6\u6001\u7a7a\u95f4\uff0c\u662f\u6240\u6709\u53ef\u80fd\u72b6\u6001\u7684\u96c6\u5408\uff1b<\/li>\n\n\n\n
                                                                              • $A$ \u8868\u793a\u52a8\u4f5c\u7a7a\u95f4\uff0c\u662f\u6240\u6709\u53ef\u80fd\u52a8\u4f5c\u7684\u96c6\u5408\uff1b<\/li>\n\n\n\n
                                                                              • $P(s’|s,a)$ \u8868\u793a\u72b6\u6001\u8f6c\u79fb\u6982\u7387\uff0c\u5373\u4ece\u72b6\u6001 $s$ \u6267\u884c\u52a8\u4f5c $a$ \u540e\u8f6c\u79fb\u5230\u72b6\u6001 $s’$ \u7684\u6982\u7387\uff1b<\/li>\n\n\n\n
                                                                              • $R(s,a,s’)$ \u8868\u793a\u5956\u52b1\u51fd\u6570\uff0c\u5373\u5728\u72b6\u6001 $s$ \u6267\u884c\u52a8\u4f5c $a$ \u540e\u8f6c\u79fb\u5230\u72b6\u6001 $s’$ \u6240\u83b7\u5f97\u7684\u5956\u52b1\uff1b<\/li>\n\n\n\n
                                                                              • $\\gamma$ \u8868\u793a\u6298\u6263\u56e0\u5b50\uff0c\u662f\u4e00\u4e2a\u4ecb\u4e8e $0$ \u548c $1$ \u4e4b\u95f4\u7684\u5b9e\u6570\u3002<\/li>\n<\/ul>\n\n\n\n
                                                                                \u5728\u4e00\u4e2aMDP\u4e2d\uff0c\u667a\u80fd\u4f53\u7684\u76ee\u6807\u662f\u901a\u8fc7\u4e0e\u73af\u5883\u7684\u4ea4\u4e92\u6765\u5b66\u4e60\u4e00\u4e2a\u7b56\u7565 $\\pi$\uff0c\u4f7f\u5f97\u667a\u80fd\u4f53\u5728\u4efb\u610f\u72b6\u6001 $s$ \u4e0b\uff0c\u6309\u7167\u7b56\u7565 $\\pi$ \u6267\u884c\u52a8\u4f5c $a$\uff0c\u671f\u671b\u7d2f\u79ef\u5956\u52b1\u6700\u5927\u5316\u3002<\/p>\n\n\n\n
                                                                                \u5177\u4f53\u6765\u8bf4\uff0c\u667a\u80fd\u4f53\u5728\u6bcf\u4e2a\u65f6\u523b $t$ \u4f1a\u89c2\u6d4b\u5230\u5f53\u524d\u72b6\u6001 $s_t$\uff0c\u7136\u540e\u6839\u636e\u7b56\u7565 $\\pi$ \u9009\u62e9\u52a8\u4f5c $a_t$ \u6267\u884c\u3002\u6267\u884c\u52a8\u4f5c $a_t$ \u540e\uff0c\u667a\u80fd\u4f53\u4f1a\u89c2\u6d4b\u5230\u4e0b\u4e00\u4e2a\u72b6\u6001 $s_{t+1}$ \u548c\u5bf9\u5e94\u7684\u5956\u52b1 $r_t = R(s_t,a_t,s_{t+1})$\u3002\u667a\u80fd\u4f53\u7684\u76ee\u6807\u662f\u6700\u5927\u5316\u672a\u6765\u7684\u7d2f\u79ef\u5956\u52b1\uff0c\u5373 $G_t = \\sum_{k=0}^{\\infty} \\gamma^k r_{t+k+1}$\u3002<\/p>\n\n\n\n
                                                                                \u4e3a\u4e86\u89e3\u51b3\u8fd9\u4e2a\u95ee\u9898\uff0c\u53ef\u4ee5\u91c7\u7528\u4ef7\u503c\u51fd\u6570\u6765\u5e2e\u52a9\u667a\u80fd\u4f53\u505a\u51fa\u51b3\u7b56\u3002\u4ef7\u503c\u51fd\u6570\u53ef\u4ee5\u5206\u4e3a\u72b6\u6001\u4ef7\u503c\u51fd\u6570\u548c\u52a8\u4f5c\u4ef7\u503c\u51fd\u6570\u4e24\u79cd\uff0c\u5206\u522b\u8868\u793a\u5728\u5f53\u524d\u72b6\u6001\u4e0b\u91c7\u53d6\u67d0\u4e2a\u52a8\u4f5c\u548c\u6309\u7167\u67d0\u4e2a\u7b56\u7565\u6267\u884c\u65f6\u6240\u80fd\u83b7\u5f97\u7684\u671f\u671b\u5956\u52b1\u3002\u5177\u4f53\u6765\u8bf4\uff0c\u72b6\u6001\u4ef7\u503c\u51fd\u6570 $V(s)$ \u8868\u793a\u5728\u72b6\u6001 $s$ \u4e0b\u6309\u7167\u7b56\u7565 $\\pi$ \u6267\u884c\u6240\u80fd\u83b7\u5f97\u7684\u671f\u671b\u7d2f\u79ef\u5956\u52b1\uff1b\u52a8\u4f5c\u4ef7\u503c\u51fd\u6570 $Q(s,a)$ \u8868\u793a\u5728\u72b6\u6001 $s$ \u4e0b\u6267\u884c\u52a8\u4f5c $a$ \u5e76\u6309\u7167\u7b56\u7565 $\\pi$ \u7ee7\u7eed\u6267\u884c\u6240\u80fd\u83b7\u5f97\u7684\u671f\u671b\u7d2f\u79ef\u5956\u52b1\u3002<\/p>\n\n\n\n
                                                                                \u6709\u4e86\u4ef7\u503c\u51fd\u6570\uff0c\u6211\u4eec\u53ef\u4ee5\u91c7\u7528\u8d2a\u5fc3\u7b56\u7565\u6765\u9009\u62e9\u52a8\u4f5c\uff0c\u5373\u5728\u72b6\u6001 $s$ \u4e0b\u9009\u62e9\u80fd\u591f\u4f7f\u5f97\u4ef7\u503c\u51fd\u6570\u6700\u5927\u5316\u7684\u52a8\u4f5c $a$\uff0c\u5373 $a^* = \\arg\\max_{a \\in A} Q(s,a)$\u3002<\/p>\n\n\n\n
                                                                                \u4ee5\u4e0b\u662f\u4e00\u4e2a\u7b80\u5355\u7684<\/p>\n\n\n\n\n\n
                                                                                \"gun<\/figure>\n\n\n\n
                                                                                \u8bf7\u7ee7\u7eed<\/p>\n\n\n\n
                                                                                \u9a6c\u5c14\u53ef\u592b\u51b3\u7b56\u8fc7\u7a0b\uff08Markov Decision Process\uff0cMDP\uff09\u662f\u5f3a\u5316\u5b66\u4e60\u4e2d\u7684\u4e00\u79cd\u6a21\u578b\uff0c\u7528\u4e8e\u63cf\u8ff0\u667a\u80fd\u4f53\u5728\u73af\u5883\u4e2d\u505a\u51fa\u51b3\u7b56\u7684\u8fc7\u7a0b\u3002\u5728MDP\u4e2d\uff0c\u667a\u80fd\u4f53\u4e0e\u73af\u5883\u4ea4\u4e92\uff0c\u901a\u8fc7\u89c2\u5bdf\u73af\u5883\u7684\u72b6\u6001\u548c\u91c7\u53d6\u884c\u52a8\u6765\u83b7\u5f97\u5956\u52b1\uff0c\u76ee\u6807\u662f\u4f7f\u957f\u671f\u7d2f\u79ef\u5956\u52b1\u6700\u5927\u5316\u3002<\/p>\n\n\n\n
                                                                                MDP\u75315\u4e2a\u5143\u7d20\u7ec4\u6210\uff1a<\/p>\n\n\n\n
                                                                                  \n
                                                                                1. \u72b6\u6001\u7a7a\u95f4\uff08State Space\uff09\uff1a\u6240\u6709\u53ef\u80fd\u7684\u72b6\u6001\u7ec4\u6210\u7684\u96c6\u5408\u3002<\/li>\n\n\n\n
                                                                                2. \u884c\u52a8\u7a7a\u95f4\uff08Action Space\uff09\uff1a\u6240\u6709\u53ef\u80fd\u7684\u884c\u52a8\u7ec4\u6210\u7684\u96c6\u5408\u3002<\/li>\n\n\n\n
                                                                                3. \u8f6c\u79fb\u51fd\u6570\uff08Transition Function\uff09\uff1a\u63cf\u8ff0\u667a\u80fd\u4f53\u91c7\u53d6\u67d0\u4e2a\u884c\u52a8\u540e\u73af\u5883\u72b6\u6001\u7684\u53d8\u5316\uff0c\u4e5f\u53eb\u72b6\u6001\u8f6c\u79fb\u6982\u7387\u3002<\/li>\n\n\n\n
                                                                                4. \u5956\u52b1\u51fd\u6570\uff08Reward Function\uff09\uff1a\u63cf\u8ff0\u667a\u80fd\u4f53\u5728\u67d0\u4e2a\u72b6\u6001\u4e0b\u91c7\u53d6\u67d0\u4e2a\u884c\u52a8\u6240\u83b7\u5f97\u7684\u5956\u52b1\u3002<\/li>\n\n\n\n
                                                                                5. \u6298\u6263\u56e0\u5b50\uff08Discount Factor\uff09\uff1a\u63cf\u8ff0\u672a\u6765\u5956\u52b1\u7684\u6298\u6263\u7387\uff0c\u7528\u4e8e\u5e73\u8861\u5f53\u524d\u5956\u52b1\u548c\u672a\u6765\u5956\u52b1\u3002<\/li>\n<\/ol>\n\n\n\n
                                                                                  \u667a\u80fd\u4f53\u6839\u636e\u5f53\u524d\u72b6\u6001\u548c\u7b56\u7565\u51fd\u6570\uff08Policy Function\uff09\u9009\u62e9\u884c\u52a8\uff0c\u6839\u636e\u8f6c\u79fb\u51fd\u6570\u8fdb\u5165\u65b0\u72b6\u6001\uff0c\u83b7\u5f97\u5956\u52b1\uff0c\u5e76\u4e0d\u65ad\u66f4\u65b0\u7b56\u7565\u51fd\u6570\uff0c\u4ee5\u4f7f\u957f\u671f\u7d2f\u79ef\u5956\u52b1\u6700\u5927\u5316\u3002<\/p>\n\n\n\n
                                                                                  MDP\u7684\u6c42\u89e3\u53ef\u4ee5\u901a\u8fc7\u4ef7\u503c\u51fd\u6570\uff08Value Function\uff09\u6216\u7b56\u7565\u51fd\u6570\u7684\u8fed\u4ee3\u6c42\u89e3\u65b9\u6cd5\uff0c\u5176\u4e2d\u6700\u8457\u540d\u7684\u7b97\u6cd5\u662fQ-learning\u548cSARSA\u3002<\/p>\n\n\n\n\n\n
                                                                                  \"gun<\/figure>\n\n\n\n
                                                                                  \u8bf7\u8be6\u7ec6\u89e3\u91ca\u795e\u7ecf\u56fe\u7075\u673a<\/p>\n\n\n\n
                                                                                  \u795e\u7ecf\u56fe\u7075\u673a\uff08Neural Turing Machine\uff0cNTM\uff09\u662f\u4e00\u79cd\u5177\u6709\u53ef\u8bad\u7ec3\u8bb0\u5fc6\u7684\u4eba\u5de5\u795e\u7ecf\u7f51\u7edc\uff0c\u7531Alex Graves\u548cGreg Wayne\u4e8e2014\u5e74\u63d0\u51fa\u3002NTM\u53ef\u4ee5\u5c06\u4e00\u4e9b\u989d\u5916\u7684\u5b58\u50a8\u5668\uff08\u6bd4\u5982\u4e00\u4e2a\u5e26\u5730\u5740\u6307\u9488\u7684\u5185\u5b58\u77e9\u9635\uff09\u8fde\u63a5\u5230\u4e00\u4e2a\u795e\u7ecf\u7f51\u7edc\u4e2d\uff0c\u4ece\u800c\u53ef\u4ee5\u5bf9\u5185\u5b58\u8fdb\u884c\u8bfb\u3001\u5199\u3001\u91cd\u7f6e\u7b49\u64cd\u4f5c\u3002\u8fd9\u4f7f\u5f97NTM\u80fd\u591f\u8fdb\u884c\u4e00\u4e9b\u5176\u4ed6\u795e\u7ecf\u7f51\u7edc\u65e0\u6cd5\u8fdb\u884c\u7684\u4efb\u52a1\uff0c\u4f8b\u5982\u5728\u56fa\u5b9a\u5927\u5c0f\u7684\u5185\u5b58\u4e0a\u6267\u884c\u7b97\u6cd5\uff0c\u5b66\u4e60\u6267\u884c\u5404\u79cd\u4efb\u52a1\uff0c\u751a\u81f3\u5728\u8bb0\u5fc6\u9886\u57df\u8fdb\u884c\u8d85\u8fc7\u4eba\u7c7b\u7684\u8868\u73b0\u3002<\/p>\n\n\n\n
                                                                                  \u5728NTM\u4e2d\uff0c\u5b58\u50a8\u5668\u53ef\u4ee5\u770b\u4f5c\u662f\u4e00\u4e2a\u957f\u7684\u5411\u91cf\uff08\u5185\u5b58\u77e9\u9635\uff09\uff0c\u5176\u4e2d\u6bcf\u4e2a\u5143\u7d20\u90fd\u53ef\u4ee5\u5b58\u50a8\u4e00\u4e2a\u5411\u91cf\uff08\u5185\u5b58\u5355\u5143\uff09\u3002\u5185\u5b58\u77e9\u9635\u53ef\u4ee5\u7531\u4e00\u4e2a\u5730\u5740\u6307\u9488\u6765\u8bbf\u95ee\uff0c\u8fd9\u4e2a\u6307\u9488\u662f\u4e00\u4e2a\u503c\u57df\u4e3a0\u52301\u4e4b\u95f4\u7684\u6807\u91cf\uff0c\u53ef\u4ee5\u88ab\u770b\u4f5c\u662f\u4e00\u4e2a\u201c\u8bfb\/\u5199\u5934\u201d\u7684\u4f4d\u7f6e\u3002\u795e\u7ecf\u7f51\u7edc\u53ef\u4ee5\u6839\u636e\u9700\u8981\uff0c\u5c06\u4e00\u4e2a\u5411\u91cf\u8bfb\u53d6\u5230\u5185\u5b58\u4e2d\uff0c\u6216\u5c06\u4e00\u4e2a\u5411\u91cf\u4ece\u5185\u5b58\u4e2d\u5220\u9664\u3002\u8bfb\u5199\u5934\u6307\u9488\u53ef\u4ee5\u901a\u8fc7\u795e\u7ecf\u7f51\u7edc\u4e2d\u7684\u4e00\u7ec4\u53c2\u6570\u8fdb\u884c\u63a7\u5236\uff0c\u56e0\u6b64\u53ef\u4ee5\u901a\u8fc7\u53cd\u5411\u4f20\u64ad\u7b97\u6cd5\u8fdb\u884c\u8bad\u7ec3\u3002<\/p>\n\n\n\n
                                                                                  \u4e0b\u9762\u662f\u4e00\u4e2a\u7b80\u5355\u7684NTM\u7684\u793a\u4f8b\u4ee3\u7801\uff0c\u7528\u4e8e\u5c06\u4e24\u4e2a\u4e8c\u8fdb\u5236\u6570\u76f8\u52a0\uff1a<\/p>\n\n\n\n
                                                                                  pythonCopy codeimport numpy as np\nimport tensorflow as tf\nfrom tensorflow.python.ops import rnn_cell\nfrom tensorflow.python.ops import rnn\nfrom tensorflow.python.ops import array_ops\n\n# \u5b9a\u4e49\u795e\u7ecf\u56fe\u7075\u673a\u7c7b\nclass NTM(object):\n    def __init__(self, input_size, output_size, memory_size, memory_dim, controller_size):\n        # \u521d\u59cb\u5316\u53c2\u6570\n        self.input_size = input_size\n        self.output_size = output_size\n        self.memory_size = memory_size\n        self.memory_dim = memory_dim\n        self.controller_size = controller_size\n        \n        # \u5b9a\u4e49\u8f93\u5165\u8f93\u51fa\u5360\u4f4d\u7b26\n        self.inputs = tf.placeholder(tf.float32, [None, None, input_size])\n        self.targets = tf.placeholder(tf.float32, [None, None, output_size])\n        \n        # \u5b9a\u4e49\u8bb0\u5fc6\u77e9\u9635\n        self.memory = tf.Variable(tf.zeros([memory_size, memory_dim]))\n        \n        # \u5b9a\u4e49\u8bfb\u5199\u5934\u6307\u9488\n        self.read_head = tf.Variable(tf.random_normal([controller_size, 3]))\n        self.write_head = tf.Variable(tf.random_normal([controller_size, 3]))\n        \n        # \u5b9a\u4e49\u795e\u7ecf\u7f51\u7edc\u53c2\u6570\n        self.W_out = tf.Variable(tf.random_normal([controller_size, output_size]))\n        self.b_out = tf.Variable(tf.random_normal([output_size]))\n        self.controller_cell = rnn_cell.BasicRNNCell(controller_size)\n        \n        # \u5b9a\u4e49\u524d\u5411\u4f20\u64ad\u8fc7\u7a0b\n        def step(x, state):\n            # \u8bfb\u53d6\u5185\u5b58\n            read_weight = self.read_head[:,0:1] + tf.nn.softmax(tf.matmul(state, self.read_head[:,1:3]))\n            read_vector = tf.reduce_sum(tf.expand_dims(read_weight, 2) * self.memory, 0)\n            \n            # \u6267\u884c\u795e\u7ecf\u7f51\u7edc\n            with tf.variable_scope(\"controller\",\n<\/code><\/pre>\n\n\n\n\n\n
                                                                                  \"gun<\/figure>\n\n\n\n
                                                                                  \u8bf7\u8be6\u7ec6\u89e3\u91ca\u5f31\u76d1\u7763\u5b66\u4e60<\/p>\n\n\n\n
                                                                                  \u5f31\u76d1\u7763\u5b66\u4e60\u662f\u6307\u4f7f\u7528\u6807\u7b7e\u4e0d\u5b8c\u6574\u6216\u8005\u4e0d\u51c6\u786e\u7684\u6570\u636e\u8fdb\u884c\u76d1\u7763\u5b66\u4e60\u4efb\u52a1\u7684\u4e00\u79cd\u65b9\u6cd5\u3002\u76f8\u6bd4\u4e8e\u4f20\u7edf\u7684\u76d1\u7763\u5b66\u4e60\u65b9\u6cd5\u9700\u8981\u5927\u91cf\u6807\u6ce8\u6570\u636e\u7684\u60c5\u51b5\u4e0b\uff0c\u5f31\u76d1\u7763\u5b66\u4e60\u53ef\u4ee5\u5229\u7528\u66f4\u4e30\u5bcc\u7684\u6570\u636e\u8d44\u6e90\uff0c\u5982\u672a\u6807\u6ce8\u6570\u636e\u3001\u90e8\u5206\u6807\u6ce8\u6570\u636e\u3001\u566a\u58f0\u6807\u6ce8\u6570\u636e\u7b49\uff0c\u964d\u4f4e\u6570\u636e\u6807\u6ce8\u7684\u6210\u672c\uff0c\u63d0\u9ad8\u6a21\u578b\u7684\u6cdb\u5316\u6027\u80fd\u3002<\/p>\n\n\n\n
                                                                                  \u5e38\u89c1\u7684\u5f31\u76d1\u7763\u5b66\u4e60\u65b9\u6cd5\u5305\u62ec\uff1a<\/p>\n\n\n\n
                                                                                    \n
                                                                                  1. \u591a\u793a\u4f8b\u5b66\u4e60\uff08MIL\uff09\uff1a\u662f\u4e00\u79cd\u5b9e\u4f8b\u7ea7\u522b\u7684\u5f31\u76d1\u7763\u5b66\u4e60\u65b9\u6cd5\uff0c\u5b83\u5047\u8bbe\u6bcf\u4e2a\u6807\u6ce8\u4e3a\u6b63\u4f8b\u7684\u6837\u672c\u96c6\u5408\uff08\u79f0\u4e3a\u4e00\u4e2a\u201c\u5305\u201d\uff09\u4e2d\u81f3\u5c11\u6709\u4e00\u4e2a\u6837\u672c\u662f\u771f\u6b63\u7684\u6b63\u4f8b\uff0c\u4ece\u800c\u907f\u514d\u4e86\u9700\u8981\u7cbe\u786e\u6807\u6ce8\u6bcf\u4e2a\u6837\u672c\u7684\u95ee\u9898\u3002MIL\u5e7f\u6cdb\u5e94\u7528\u4e8e\u76ee\u6807\u68c0\u6d4b\u3001\u56fe\u50cf\u5206\u7c7b\u3001\u6587\u672c\u5206\u7c7b\u7b49\u4efb\u52a1\u4e2d\u3002<\/li>\n\n\n\n
                                                                                  2. \u6807\u7b7e\u4f20\u64ad\uff08Label Propagation\uff09\uff1a\u662f\u4e00\u79cd\u57fa\u4e8e\u56fe\u7684\u4f20\u64ad\u7b97\u6cd5\uff0c\u5b83\u5229\u7528\u6837\u672c\u4e4b\u95f4\u7684\u76f8\u4f3c\u6027\uff08\u76f8\u4f3c\u7684\u6837\u672c\u66f4\u6709\u53ef\u80fd\u5177\u6709\u76f8\u540c\u7684\u6807\u7b7e\uff09\u6765\u63a8\u65ad\u672a\u6807\u6ce8\u6837\u672c\u7684\u6807\u7b7e\u3002\u6807\u7b7e\u4f20\u64ad\u88ab\u5e7f\u6cdb\u5e94\u7528\u4e8e\u56fe\u50cf\u5206\u5272\u3001\u6587\u672c\u5206\u7c7b\u3001\u793e\u4ea4\u7f51\u7edc\u5206\u6790\u7b49\u4efb\u52a1\u4e2d\u3002<\/li>\n\n\n\n
                                                                                  3. \u591a\u4efb\u52a1\u5b66\u4e60\uff08MTL\uff09\uff1a\u662f\u4e00\u79cd\u5229\u7528\u76f8\u5173\u4efb\u52a1\u4e4b\u95f4\u7684\u4fe1\u606f\u5171\u4eab\u6765\u63d0\u9ad8\u6cdb\u5316\u6027\u80fd\u7684\u65b9\u6cd5\u3002MTL\u5728\u540c\u4e00\u6a21\u578b\u4e2d\u5b66\u4e60\u591a\u4e2a\u4efb\u52a1\uff0c\u4ece\u800c\u53ef\u4ee5\u5229\u7528\u591a\u4e2a\u4efb\u52a1\u4e4b\u95f4\u7684\u76f8\u4e92\u5173\u7cfb\u63d0\u9ad8\u6a21\u578b\u7684\u6cdb\u5316\u6027\u80fd\u3002MTL\u5728\u8ba1\u7b97\u673a\u89c6\u89c9\u3001\u81ea\u7136\u8bed\u8a00\u5904\u7406\u3001\u751f\u7269\u4fe1\u606f\u5b66\u7b49\u9886\u57df\u4e2d\u6709\u5e7f\u6cdb\u7684\u5e94\u7528\u3002<\/li>\n\n\n\n
                                                                                  4. \u5e26\u566a\u58f0\u7684\u6807\u6ce8\u6570\u636e\uff08Noisy Label\uff09\uff1a\u5728\u8bb8\u591a\u5b9e\u9645\u573a\u666f\u4e2d\uff0c\u6807\u6ce8\u6570\u636e\u53ef\u80fd\u4f1a\u5b58\u5728\u566a\u58f0\uff0c\u5373\u9519\u8bef\u6807\u6ce8\u3002\u4f20\u7edf\u7684\u76d1\u7763\u5b66\u4e60\u65b9\u6cd5\u4f1a\u53d7\u5230\u8fd9\u4e9b\u566a\u58f0\u7684\u5e72\u6270\uff0c\u5bfc\u81f4\u6a21\u578b\u6027\u80fd\u4e0b\u964d\u3002\u56e0\u6b64\uff0c\u7814\u7a76\u5982\u4f55\u5e94\u5bf9\u5e26\u566a\u58f0\u6807\u6ce8\u6570\u636e\u7684\u65b9\u6cd5\u4e5f\u6210\u4e3a\u4e86\u5f31\u76d1\u7763\u5b66\u4e60\u7684\u7814\u7a76\u65b9\u5411\u4e4b\u4e00\u3002\u5e38\u7528\u7684\u65b9\u6cd5\u5305\u62ec\u566a\u58f0\u8fc7\u6ee4\u3001\u6807\u7b7e\u5e73\u6ed1\u3001\u534a\u76d1\u7763\u5b66\u4e60\u7b49\u3002<\/li>\n<\/ol>\n\n\n\n
                                                                                    \u5f31\u76d1\u7763\u5b66\u4e60\u7684\u7814\u7a76\u76ee\u524d\u4ecd\u5728\u4e0d\u65ad\u53d1\u5c55\uff0c\u672a\u6765\u8fd8\u6709\u5f88\u5927\u7684\u53d1\u5c55\u7a7a\u95f4\u3002<\/p>\n\n\n\n\n\n
                                                                                    \"gun<\/figure>\n\n\n\n
                                                                                    \u8bf7\u8be6\u7ec6\u89e3\u91ca\u8fc1\u79fb\u5b66\u4e60<\/p>\n\n\n\n
                                                                                    \u8fc1\u79fb\u5b66\u4e60\u662f\u4e00\u79cd\u673a\u5668\u5b66\u4e60\u65b9\u6cd5\uff0c\u5b83\u901a\u8fc7\u5c06\u4e00\u4e2a\u9886\u57df\u6216\u4efb\u52a1\u7684\u77e5\u8bc6\u8f6c\u79fb\u6216\u5e94\u7528\u5230\u53e6\u4e00\u4e2a\u9886\u57df\u6216\u4efb\u52a1\u4e2d\uff0c\u4ece\u800c\u63d0\u9ad8\u5b66\u4e60\u6027\u80fd\u3002\u5728\u8fc1\u79fb\u5b66\u4e60\u4e2d\uff0c\u901a\u5e38\u6709\u4e24\u4e2a\u5173\u952e\u9886\u57df\u6216\u4efb\u52a1\uff1a\u6e90\u57df\u548c\u76ee\u6807\u57df\u3002\u6e90\u57df\u901a\u5e38\u662f\u6307\u4e00\u4e2a\u5df2\u7ecf\u8bad\u7ec3\u597d\u7684\u6a21\u578b\u6216\u6570\u636e\u96c6\uff0c\u76ee\u6807\u57df\u5219\u662f\u6307\u4e00\u4e2a\u9700\u8981\u5b66\u4e60\u6216\u9884\u6d4b\u7684\u65b0\u4efb\u52a1\u6216\u6570\u636e\u96c6\u3002<\/p>\n\n\n\n
                                                                                    \u8fc1\u79fb\u5b66\u4e60\u7684\u4e3b\u8981\u601d\u60f3\u662f\uff0c\u901a\u8fc7\u5229\u7528\u6e90\u57df\u7684\u77e5\u8bc6\u6216\u7279\u5f81\u6765\u5e2e\u52a9\u76ee\u6807\u57df\u7684\u5b66\u4e60\uff0c\u4ece\u800c\u63d0\u9ad8\u76ee\u6807\u57df\u7684\u6027\u80fd\u3002\u8fc1\u79fb\u5b66\u4e60\u901a\u5e38\u53ef\u4ee5\u5206\u4e3a\u4ee5\u4e0b\u51e0\u79cd\u7c7b\u578b\uff1a<\/p>\n\n\n\n
                                                                                      \n
                                                                                    1. \u57fa\u4e8e\u5b9e\u4f8b\u7684\u8fc1\u79fb\u5b66\u4e60\uff1a\u5c06\u6e90\u57df\u7684\u5b9e\u4f8b\uff08\u6570\u636e\u70b9\uff09\u4e0e\u76ee\u6807\u57df\u7684\u5b9e\u4f8b\u8fdb\u884c\u5339\u914d\uff0c\u7136\u540e\u5229\u7528\u6e90\u57df\u7684\u5b9e\u4f8b\u6765\u5e2e\u52a9\u76ee\u6807\u57df\u7684\u5b66\u4e60\u3002<\/li>\n\n\n\n
                                                                                    2. \u57fa\u4e8e\u7279\u5f81\u7684\u8fc1\u79fb\u5b66\u4e60\uff1a\u5c06\u6e90\u57df\u7684\u7279\u5f81\u4e0e\u76ee\u6807\u57df\u7684\u7279\u5f81\u8fdb\u884c\u5339\u914d\uff0c\u7136\u540e\u5229\u7528\u6e90\u57df\u7684\u7279\u5f81\u6765\u5e2e\u52a9\u76ee\u6807\u57df\u7684\u5b66\u4e60\u3002<\/li>\n\n\n\n
                                                                                    3. \u57fa\u4e8e\u6a21\u578b\u7684\u8fc1\u79fb\u5b66\u4e60\uff1a\u5c06\u6e90\u57df\u7684\u6a21\u578b\u4e0e\u76ee\u6807\u57df\u7684\u6a21\u578b\u8fdb\u884c\u5339\u914d\uff0c\u7136\u540e\u5229\u7528\u6e90\u57df\u7684\u6a21\u578b\u6765\u5e2e\u52a9\u76ee\u6807\u57df\u7684\u5b66\u4e60\u3002<\/li>\n\n\n\n
                                                                                    4. \u57fa\u4e8e\u5173\u7cfb\u7684\u8fc1\u79fb\u5b66\u4e60\uff1a\u5229\u7528\u6e90\u57df\u548c\u76ee\u6807\u57df\u4e4b\u95f4\u7684\u5173\u7cfb\u6765\u5e2e\u52a9\u76ee\u6807\u57df\u7684\u5b66\u4e60\u3002<\/li>\n<\/ol>\n\n\n\n
                                                                                      \u8fc1\u79fb\u5b66\u4e60\u7684\u4f18\u70b9\u5728\u4e8e\u53ef\u4ee5\u51cf\u5c11\u6570\u636e\u91cf\u548c\u8ba1\u7b97\u65f6\u95f4\uff0c\u63d0\u9ad8\u5b66\u4e60\u6548\u7387\u548c\u51c6\u786e\u7387\u3002\u8fc1\u79fb\u5b66\u4e60\u5728\u5b9e\u9645\u5e94\u7528\u4e2d\u4e5f\u6709\u5e7f\u6cdb\u7684\u5e94\u7528\uff0c\u4f8b\u5982\u56fe\u50cf\u8bc6\u522b\u3001\u81ea\u7136\u8bed\u8a00\u5904\u7406\u548c\u63a8\u8350\u7cfb\u7edf\u7b49\u3002<\/p>\n\n\n\n
                                                                                      \u4e0b\u9762\u662f\u4e00\u4e2a\u57fa\u4e8e\u7279\u5f81\u7684\u8fc1\u79fb\u5b66\u4e60\u7684\u793a\u4f8b\u4ee3\u7801\uff1a<\/p>\n\n\n\n
                                                                                      pythonCopy codeimport numpy as np\nfrom sklearn.linear_model import LogisticRegression\nfrom sklearn.datasets import load_iris\n\n# \u52a0\u8f7diris\u6570\u636e\u96c6\niris = load_iris()\nX, y = iris.data, iris.target\n\n# \u968f\u673a\u9009\u62e930\u4e2a\u6837\u672c\u4f5c\u4e3a\u6e90\u57df\nsource_idx = np.random.choice(X.shape[0], 30, replace=False)\nX_source, y_source = X[source_idx], y[source_idx]\n\n# \u9009\u62e950\u4e2a\u6837\u672c\u4f5c\u4e3a\u76ee\u6807\u57df\ntarget_idx = np.random.choice(X.shape[0], 50, replace=False)\nX_target, y_target = X[target_idx], y[target_idx]\n\n# \u57fa\u4e8e\u7279\u5f81\u7684\u8fc1\u79fb\u5b66\u4e60\n# \u9996\u5148\u8bad\u7ec3\u6e90\u57df\u7684\u6a21\u578b\nclf = LogisticRegression(solver='lbfgs', multi_class='multinomial')\nclf.fit(X_source[:, :2], y_source)\n\n# \u5229\u7528\u6e90\u57df\u7684\u6a21\u578b\u8fdb\u884c\u76ee\u6807\u57df\u7684\u9884\u6d4b\ny_pred = clf.predict(X_target[:, :2])\n\n<\/code><\/pre>\n\n\n\n\n\n
                                                                                      \"gun<\/figure>\n\n\n\n
                                                                                      \u8bf7\u8be6\u7ec6\u89e3\u91ca\u5143\u5b66\u4e60<\/p>\n\n\n\n
                                                                                      \u5143\u5b66\u4e60\uff08meta-learning\uff09\u662f\u6307\u5b66\u4e60\u5982\u4f55\u5b66\u4e60\u7684\u4e00\u79cd\u673a\u5668\u5b66\u4e60\u65b9\u6cd5\uff0c\u4e5f\u88ab\u79f0\u4e3a\u201c\u5b66\u4e60\u5230\u5b66\u4e60\u201d\u6216\u201c\u5143\u5b66\u4e60\u4e60\u5f97\u201d\u3002\u5728\u5143\u5b66\u4e60\u4e2d\uff0c\u6a21\u578b\u5b66\u4e60\u4ece\u4e00\u7ec4\u4efb\u52a1\u4e2d\u5feb\u901f\u9002\u5e94\u65b0\u4efb\u52a1\u7684\u80fd\u529b\uff0c\u800c\u4e0d\u662f\u53ea\u662f\u9488\u5bf9\u4e00\u4e2a\u4efb\u52a1\u8fdb\u884c\u4f18\u5316\u3002\u6362\u53e5\u8bdd\u8bf4\uff0c\u5143\u5b66\u4e60\u4f7f\u6a21\u578b\u80fd\u591f\u5229\u7528\u4e4b\u524d\u5b66\u5230\u7684\u77e5\u8bc6\u548c\u7ecf\u9a8c\u6765\u66f4\u5feb\u5730\u5b66\u4e60\u65b0\u4efb\u52a1\u3002<\/p>\n\n\n\n
                                                                                      \u5143\u5b66\u4e60\u7684\u4e00\u4e2a\u5178\u578b\u5e94\u7528\u662f\u5feb\u901f\u9002\u5e94\uff0c\u5b83\u5728\u8ba1\u7b97\u673a\u89c6\u89c9\u3001\u81ea\u7136\u8bed\u8a00\u5904\u7406\u3001\u673a\u5668\u4eba\u5b66\u548c\u63a8\u8350\u7cfb\u7edf\u7b49\u9886\u57df\u5f97\u5230\u4e86\u5e7f\u6cdb\u5e94\u7528\u3002\u5728\u5feb\u901f\u9002\u5e94\u7684\u573a\u666f\u4e0b\uff0c\u6a21\u578b\u9700\u8981\u5728\u9762\u5bf9\u65b0\u7684\u4efb\u52a1\u65f6\uff0c\u80fd\u591f\u5feb\u901f\u4ece\u6709\u9650\u7684\u6837\u672c\u4e2d\u5b66\u4e60\u5230\u4e00\u4e2a\u5408\u9002\u7684\u7b56\u7565\u3002<\/p>\n\n\n\n
                                                                                      \u5143\u5b66\u4e60\u901a\u5e38\u4f7f\u7528\u5143\u5b66\u4e60\u5668\u6765\u5b9e\u73b0\uff0c\u5143\u5b66\u4e60\u5668\u662f\u4e00\u79cd\u80fd\u591f\u4ece\u591a\u4e2a\u4efb\u52a1\u4e2d\u5b66\u4e60\u5230\u901a\u7528\u77e5\u8bc6\uff0c\u7136\u540e\u5728\u65b0\u4efb\u52a1\u4e2d\u8fdb\u884c\u5feb\u901f\u9002\u5e94\u7684\u6a21\u578b\u3002\u5143\u5b66\u4e60\u5668\u7684\u5b66\u4e60\u8fc7\u7a0b\u662f\u5728\u4e00\u4e2a\u5143\u8bad\u7ec3\u96c6\u4e0a\u8fdb\u884c\u7684\uff0c\u8be5\u5143\u8bad\u7ec3\u96c6\u7531\u591a\u4e2a\u4efb\u52a1\u7ec4\u6210\u3002\u5143\u5b66\u4e60\u5668\u4f7f\u7528\u8fd9\u4e9b\u4efb\u52a1\u6765\u5b66\u4e60\u5982\u4f55\u5c06\u8fc7\u53bb\u7684\u7ecf\u9a8c\u548c\u77e5\u8bc6\u8f6c\u79fb\u5230\u65b0\u4efb\u52a1\u4e2d\u3002\u5f53\u4e00\u4e2a\u65b0\u4efb\u52a1\u5230\u6765\u65f6\uff0c\u5143\u5b66\u4e60\u5668\u4f1a\u5229\u7528\u8fd9\u4e9b\u77e5\u8bc6\u548c\u7ecf\u9a8c\uff0c\u901a\u8fc7\u5c11\u91cf\u7684\u6837\u672c\u6765\u9002\u5e94\u65b0\u4efb\u52a1\u3002<\/p>\n\n\n\n
                                                                                      \u4ee5\u4e0b\u662f\u4e00\u4e2a\u7b80\u5355\u7684\u5143\u5b66\u4e60\u7684\u793a\u4f8b\u4ee3\u7801\uff08\u57fa\u4e8ePyTorch\uff09\uff1a<\/p>\n\n\n\n
                                                                                      pythonCopy codeimport torch\nimport torch.nn as nn\nimport torch.optim as optim\n\nclass MetaLearner(nn.Module):\n    def __init__(self, inner_lr):\n        super(MetaLearner, self).__init__()\n        self.inner_lr = inner_lr\n        self.task_encoder = nn.Linear(5, 10)\n        self.task_decoder = nn.Linear(10, 1)\n\n    def forward(self, x, y):\n        task_encoded = self.task_encoder(x)\n        task_specific_params = self.task_decoder(task_encoded)\n        loss = nn.functional.mse_loss(torch.matmul(x, task_specific_params), y)\n        task_specific_grads = torch.autograd.grad(loss, task_specific_params, create_graph=True)[0]\n        return task_specific_params - self.inner_lr * task_specific_grads\n\n# \u6784\u9020\u5143\u8bad\u7ec3\u96c6\ntasks = []\nfor _ in range(10):\n    task_x = torch.randn(5, 10)\n    task_y = torch.randn(5, 1)\n    tasks.append((task_x, task_y))\n\n# \u6784\u9020\u5143\u5b66\u4e60\u5668\nmeta_learner = MetaLearner(inner_lr=0.01)\noptimizer = optim.SGD(meta_learner.parameters(), lr=0.001)\n\n# \u5728\u5143\u8bad\u7ec3\u96c6\u4e0a\u8bad\u7ec3\u5143\u5b66\u4e60\u5668\nfor i in range(1000):\n    for task_x, task_y in tasks:\n        adapted_params = meta_learner(task_x, task_y)\n        adapted_y = torch.matmul(task_x, adapted_params)\n        task_loss = nn.functional.mse_loss(adapted\n<\/code><\/pre>\n\n\n\n\n\n
                                                                                      \"gun<\/figure>\n\n\n\n
                                                                                      \u8bf7\u8be6\u7ec6\u89e3\u91ca\u83b1\u5e03\u5c3c\u5179\u6cd5\u5219<\/p>\n\n\n\n
                                                                                      \u83b1\u5e03\u5c3c\u5179\u6cd5\u5219\uff08Leibniz rule\uff09\u662f\u5fae\u79ef\u5206\u4e2d\u7684\u4e00\u4e2a\u57fa\u672c\u5b9a\u7406\uff0c\u5b83\u63cf\u8ff0\u4e86\u6c42\u5bfc\u8fd0\u7b97\u4e0e\u4e58\u6cd5\u8fd0\u7b97\u7684\u5173\u7cfb\u3002\u5b83\u7684\u5f62\u5f0f\u5982\u4e0b\uff1a<\/p>\n\n\n\n
                                                                                      \ufffd\ufffd\ufffd\u222b\ufffd(\ufffd)\ufffd(\ufffd)\ufffd(\ufffd,\ufffd)\ufffd\ufffd=\ufffd(\ufffd,\ufffd(\ufffd))\ufffd\ufffd\ufffd\ufffd(\ufffd)\u2212\ufffd(\ufffd,\ufffd(\ufffd))\ufffd\ufffd\ufffd\ufffd(\ufffd)+\u222b\ufffd(\ufffd)\ufffd(\ufffd)\u2202\u2202\ufffd\ufffd(\ufffd,\ufffd)\ufffd\ufffdd<\/em>x<\/em>d<\/em>\u200b\u222ba<\/em>(x<\/em>)b<\/em>(x<\/em>)\u200bf<\/em>(x<\/em>,t<\/em>)d<\/em>t<\/em>=f<\/em>(x<\/em>,b<\/em>(x<\/em>))d<\/em>x<\/em>d<\/em>\u200bb<\/em>(x<\/em>)\u2212f<\/em>(x<\/em>,a<\/em>(x<\/em>))d<\/em>x<\/em>d<\/em>\u200ba<\/em>(x<\/em>)+\u222ba<\/em>(x<\/em>)b<\/em>(x<\/em>)\u200b\u2202x<\/em>\u2202\u200bf<\/em>(x<\/em>,t<\/em>)d<\/em>t<\/em><\/p>\n\n\n\n
                                                                                      \u5176\u4e2d\uff0c$f(x,t)$ \u662f\u5173\u4e8e $x$ \u548c $t$ \u7684\u51fd\u6570\uff0c$a(x)$ \u548c $b(x)$ \u662f\u4e0e $x$ \u76f8\u5173\u7684\u79ef\u5206\u4e0a\u4e0b\u9650\uff0c$\\frac{d}{dx}$ \u662f\u6c42\u5bfc\u8fd0\u7b97\u7b26\uff0c$\\frac{\\partial}{\\partial x}$ \u662f\u504f\u5bfc\u6570\u8fd0\u7b97\u7b26\u3002<\/p>\n\n\n\n
                                                                                      \u7b80\u5355\u6765\u8bf4\uff0c\u83b1\u5e03\u5c3c\u5179\u6cd5\u5219\u63cf\u8ff0\u4e86\u5bf9\u4e00\u4e2a\u51fd\u6570\u5728\u4e00\u4e2a\u8303\u56f4\u5185\u8fdb\u884c\u79ef\u5206\u65f6\uff0c\u79ef\u5206\u4e0a\u9650\u3001\u4e0b\u9650\u90fd\u662f $x$ \u7684\u51fd\u6570\u65f6\uff0c\u5bf9\u8fd9\u4e2a\u51fd\u6570\u6c42\u5bfc\u7684\u7ed3\u679c\u3002\u8fd9\u4e2a\u5b9a\u7406\u5728\u5f88\u591a\u7269\u7406\u5b66\u548c\u5de5\u7a0b\u5b66\u7684\u95ee\u9898\u4e2d\u90fd\u6709\u5e7f\u6cdb\u5e94\u7528\u3002<\/p>\n\n\n\n\n\n
                                                                                      \"gun<\/figure>\n\n\n\n
                                                                                      \u8bf7\u8be6\u7ec6\u89e3\u91caLipschitz\u5e38\u6570<\/p>\n\n\n\n
                                                                                      Lipschitz\u5e38\u6570\u662f\u7528\u4e8e\u8861\u91cf\u51fd\u6570\u7684\u5c40\u90e8\u53d8\u5316\u7a0b\u5ea6\u7684\u4e00\u4e2a\u6982\u5ff5\u3002\u5177\u4f53\u6765\u8bf4\uff0c\u5bf9\u4e8e\u51fd\u6570 $f: X \\rightarrow Y$\uff0c\u5982\u679c\u5b58\u5728\u4e00\u4e2a\u5e38\u6570 $L \\geq 0$\uff0c\u4f7f\u5f97\u5bf9\u4e8e\u4efb\u610f $x_1, x_2 \\in X$\uff0c\u90fd\u6709\uff1a<\/p>\n\n\n\n
                                                                                      \u2225\ufffd(\ufffd1)\u2212\ufffd(\ufffd2)\u2225\ufffd\u2264\ufffd\u2225\ufffd1\u2212\ufffd2\u2225\ufffd\u2225f<\/em>(x<\/em>1\u200b)\u2212f<\/em>(x<\/em>2\u200b)\u2225Y<\/em>\u200b\u2264L<\/em>\u2225x<\/em>1\u200b\u2212x<\/em>2\u200b\u2225X<\/em>\u200b<\/p>\n\n\n\n
                                                                                      \u5176\u4e2d $|\\cdot|_X$ \u548c $|\\cdot|_Y$ \u5206\u522b\u662f $X$ \u548c $Y$ \u7a7a\u95f4\u7684\u8303\u6570\uff0c\u5219\u79f0 $f$ \u662f $L$-Lipschitz \u8fde\u7eed\u7684\uff0c\u800c $L$ \u5c31\u662f $f$ \u7684 Lipschitz \u5e38\u6570\u3002<\/p>\n\n\n\n
                                                                                      \u76f4\u89c2\u5730\u8bf4\uff0cLipschitz \u5e38\u6570\u8868\u793a\u51fd\u6570\u5728\u8f93\u5165\u7a7a\u95f4\u4e2d\u76f8\u90bb\u70b9\u4e4b\u95f4\u7684\u53d8\u5316\u901f\u7387\u4e0a\u9650\u3002\u4e00\u4e2a\u5177\u6709\u8f83\u5c0f Lipschitz \u5e38\u6570\u7684\u51fd\u6570\u53d8\u5316\u8f83\u4e3a\u5e73\u7f13\uff0c\u800c\u5177\u6709\u8f83\u5927 Lipschitz \u5e38\u6570\u7684\u51fd\u6570\u53d8\u5316\u901f\u7387\u8f83\u5feb\u3002<\/p>\n\n\n\n
                                                                                      \u4e3e\u4e2a\u4f8b\u5b50\uff0c\u5bf9\u4e8e\u4e00\u4e2a\u51fd\u6570 $f(x)=x^2$\uff0c\u5b83\u5728\u6574\u4e2a\u5b9e\u6570\u8f74\u4e0a\u90fd\u662f\u65e0\u9650\u6b21\u53ef\u5fae\u7684\u3002\u4f46\u662f\uff0c\u7531\u4e8e\u5728\u4efb\u610f\u4e24\u4e2a\u5b9e\u6570 $x_1$ \u548c $x_2$ \u4e4b\u95f4\uff0c$f$ \u7684\u53d8\u5316\u7387\u90fd\u4e0d\u4f1a\u8d85\u8fc7 $2\\max(|x_1|,|x_2|)$\uff0c\u56e0\u6b64 $f$ \u662f Lipschitz \u8fde\u7eed\u7684\uff0c\u5176 Lipschitz \u5e38\u6570\u4e3a $2\\max(|x_1|,|x_2|)$\u3002<\/p>\n\n\n\n
                                                                                      Lipschitz \u5e38\u6570\u5728\u673a\u5668\u5b66\u4e60\u4e2d\u6709\u5e7f\u6cdb\u7684\u5e94\u7528\uff0c\u4f8b\u5982\u53ef\u4ee5\u7528\u4e8e\u5206\u6790\u795e\u7ecf\u7f51\u7edc\u7684\u6027\u8d28\uff0c\u6216\u8005\u7528\u4e8e\u8bbe\u8ba1\u5177\u6709\u7a33\u5b9a\u6027\u548c\u6cdb\u5316\u6027\u80fd\u7684\u4f18\u5316\u7b97\u6cd5\u3002<\/p>\n\n\n\n\n\n
                                                                                      \"gun<\/figure>\n\n\n\n
                                                                                      \u8bf7\u8be6\u7ec6\u89e3\u91ca\u73af\u72b6\u4fe1\u5ff5\u4f20\u64ad<\/p>\n\n\n\n
                                                                                      \u73af\u72b6\u4fe1\u5ff5\u4f20\u64ad\uff08Circular Belief Propagation\uff0cCBP\uff09\u662f\u4e00\u79cd\u7528\u4e8e\u89e3\u51b3\u65e0\u5411\u56fe\u4e0a\u7684\u6982\u7387\u63a8\u65ad\u95ee\u9898\u7684\u7b97\u6cd5\u3002CBP \u901a\u8fc7\u8fed\u4ee3\u7684\u65b9\u5f0f\uff0c\u5728\u65e0\u5411\u56fe\u4e0a\u9010\u5c42\u4f20\u9012\u53d8\u91cf\u7684\u8fb9\u9645\u6982\u7387\uff0c\u6700\u7ec8\u6536\u655b\u5230\u6982\u7387\u5206\u5e03\u7684\u56fa\u5b9a\u70b9\u3002\u5728 CBP \u7b97\u6cd5\u4e2d\uff0c\u53d8\u91cf\u8282\u70b9\u7684\u8fb9\u7f18\u6982\u7387\u662f\u7531\u5b83\u4eec\u7684\u90bb\u5c45\u8282\u70b9\u8ba1\u7b97\u5f97\u5230\u7684\uff0c\u6bcf\u6b21\u8fed\u4ee3\u90fd\u5c06\u5f53\u524d\u72b6\u6001\u5411\u6240\u6709\u90bb\u5c45\u8282\u70b9\u4f20\u9012\u4fe1\u606f\uff0c\u76f4\u5230\u8fbe\u5230\u6536\u655b\u6761\u4ef6\u3002<\/p>\n\n\n\n
                                                                                      CBP \u7b97\u6cd5\u9002\u7528\u4e8e\u4efb\u610f\u7ed3\u6784\u7684\u65e0\u5411\u56fe\uff0c\u4e14\u4e0d\u9700\u8981\u56fe\u7684\u62d3\u6251\u7ed3\u6784\u4fe1\u606f\u3002\u5b83\u53ef\u4ee5\u7528\u4e8e\u63a8\u65ad\u4f17\u591a\u7684\u6a21\u578b\uff0c\u4f8b\u5982\u9a6c\u5c14\u53ef\u592b\u968f\u673a\u573a\u3001\u9ad8\u65af\u56fe\u6a21\u578b\u7b49\u3002CBP \u7b97\u6cd5\u7684\u6536\u655b\u6027\u5f97\u5230\u4e86\u4fdd\u8bc1\uff0c\u4e14\u5177\u6709\u4e00\u5b9a\u7684\u8ba1\u7b97\u6548\u7387\u3002<\/p>\n\n\n\n
                                                                                      CBP \u7b97\u6cd5\u7684\u6838\u5fc3\u662f\u5411\u6240\u6709\u90bb\u5c45\u8282\u70b9\u4f20\u9012\u4fe1\u606f\uff0c\u8fd9\u4e9b\u4fe1\u606f\u53ef\u4ee5\u770b\u4f5c\u662f\u4e00\u4e2a\u8282\u70b9\u7684\u90bb\u5c45\u8282\u70b9\u4e4b\u95f4\u7684\u5dee\u5f02\u5ea6\u91cf\u3002\u5177\u4f53\u5730\uff0cCBP \u7b97\u6cd5\u4f7f\u7528\u4e86\u4e00\u79cd\u79f0\u4e3a\u73af\u72b6\u4f20\u9012\u77e9\u9635\u7684\u6280\u672f\uff0c\u5c06\u90bb\u5c45\u8282\u70b9\u7684\u4fe1\u606f\u5408\u5e76\u4e3a\u4e00\u4e2a\u77e9\u9635\uff0c\u518d\u5c06\u8be5\u77e9\u9635\u4f20\u9012\u56de\u53bb\uff0c\u8fd9\u6837\u6bcf\u4e2a\u8282\u70b9\u90fd\u53ef\u4ee5\u6839\u636e\u90bb\u5c45\u8282\u70b9\u7684\u4fe1\u606f\u8ba1\u7b97\u51fa\u81ea\u5df1\u7684\u8fb9\u7f18\u6982\u7387\u3002<\/p>\n\n\n\n
                                                                                      \u4ee5\u4e0b\u662f\u4e00\u4e2a\u7b80\u5355\u7684 CBP \u7b97\u6cd5\u7684\u4f2a\u4ee3\u7801\uff1a<\/p>\n\n\n\n
                                                                                      markdownCopy codeInput: \u65e0\u5411\u56fe G = (V, E)\uff0c\u53d8\u91cf\u8282\u70b9 V \u548c\u52bf\u51fd\u6570 f(x)\u3002\nOutput: \u6240\u6709\u53d8\u91cf\u8282\u70b9\u7684\u8fb9\u7f18\u6982\u7387\u3002\n\n1. \u521d\u59cb\u5316\u6240\u6709\u53d8\u91cf\u8282\u70b9\u7684\u8fb9\u7f18\u6982\u7387\u4e3a\u5747\u5300\u5206\u5e03\u3002\n2. while \u6536\u655b\u6761\u4ef6\u4e0d\u6ee1\u8db3 do\n3.    for v in V do\n4.        \u5c06\u8282\u70b9 v \u7684\u90bb\u5c45\u8282\u70b9\u7684\u4fe1\u606f\u5408\u5e76\u4e3a\u4e00\u4e2a\u77e9\u9635 M\u3002\n5.        \u8ba1\u7b97\u8282\u70b9 v \u7684\u8fb9\u7f18\u6982\u7387 p(v)\u3002\n6.        \u5c06\u8282\u70b9 v \u7684\u4fe1\u606f\u4f20\u9012\u7ed9\u6240\u6709\u90bb\u5c45\u8282\u70b9\u3002\n7.    end for\n8. end while\n\n<\/code><\/pre>\n\n\n\n
                                                                                      CBP \u7b97\u6cd5\u7684\u4e3b\u8981\u4f18\u70b9\u662f\u53ef\u4ee5\u7528\u4e8e\u4efb\u610f\u7ed3\u6784\u7684\u65e0\u5411\u56fe\u4e0a\uff0c\u5e76\u4e14\u53ef\u4ee5\u5f88\u597d\u5730\u5904\u7406\u53d8\u91cf\u8282\u70b9\u6570\u76ee\u8f83\u5927\u7684\u95ee\u9898\u3002\u7f3a\u70b9\u662f\u9700\u8981\u8fdb\u884c\u591a\u6b21\u8fed\u4ee3\u624d\u80fd\u8fbe\u5230\u6536\u655b\uff0c\u8ba1\u7b97\u6548\u7387\u8f83\u4f4e\u3002<\/p>\n\n\n\n\n\n
                                                                                      \"gun<\/figure>\n\n\n\n
                                                                                      \u8bf7\u8be6\u7ec6\u89e3\u91ca\u903b\u8f91\u56de\u5f52<\/p>\n\n\n\n
                                                                                      \u903b\u8f91\u56de\u5f52\uff08Logistic Regression\uff09\u662f\u4e00\u79cd\u7ecf\u5178\u7684\u4e8c\u5206\u7c7b\u7b97\u6cd5\u3002\u5176\u4e3b\u8981\u601d\u60f3\u662f\u6784\u5efa\u4e00\u4e2a\u7ebf\u6027\u5206\u7c7b\u5668\uff0c\u4f46\u662f\u5c06\u5176\u8f93\u51fa\u901a\u8fc7\u4e00\u4e2asigmoid\u51fd\u6570\uff08\u4e5f\u79f0\u4e3a\u903b\u8f91\u51fd\u6570\uff09\u8fdb\u884c\u6620\u5c04\uff0c\u4ee5\u4fdd\u8bc1\u8f93\u51fa\u57280\u52301\u4e4b\u95f4\uff0c\u8868\u793a\u6b63\u7c7b\u7684\u6982\u7387\u3002\u5177\u4f53\u6765\u8bf4\uff0c\u903b\u8f91\u56de\u5f52\u5047\u8bbe\u6837\u672c\u7684\u8f93\u51fa$y$\u662f\u7ed9\u5b9a\u8f93\u5165$x$\u7684\u6761\u4ef6\u4e0b\uff0c\u670d\u4ece\u4e00\u4e2a\u4e8c\u9879\u5206\u5e03\uff0c\u5373<\/p>\n\n\n\n
                                                                                      \ufffd\u223cBernoulli(\ufffd),y<\/em>\u223cBernoulli(p<\/em>),<\/p>\n\n\n\n
                                                                                      \u5176\u4e2d$p$\u662f\u4e00\u4e2a\u51fd\u6570\uff0c\u4e5f\u5c31\u662f\u5206\u7c7b\u5668\uff0c\u8868\u793a\u4e3a<\/p>\n\n\n\n
                                                                                      \ufffd=11+exp\u2061(\u2212\ufffd\ufffd\ufffd\u2212\ufffd),p<\/em>=1+exp(\u2212w<\/em>T<\/em>x<\/em>\u2212b<\/em>)1\u200b,<\/p>\n\n\n\n
                                                                                      \u5176\u4e2d$w$\u548c$b$\u662f\u8981\u5b66\u4e60\u7684\u6a21\u578b\u53c2\u6570\uff0c$w$\u662f$x$\u7684\u7cfb\u6570\u5411\u91cf\uff0c$b$\u662f\u5e38\u6570\u9879\u3002sigmoid\u51fd\u6570\u5c06\u7ebf\u6027\u51fd\u6570\u7684\u8f93\u51fa\u6620\u5c04\u52300\u52301\u4e4b\u95f4\uff0c\u4f7f\u5f97\u8f93\u51fa\u53ef\u4ee5\u88ab\u89e3\u91ca\u4e3a\u4e00\u4e2a\u6982\u7387\u3002<\/p>\n\n\n\n
                                                                                      \u8bad\u7ec3\u903b\u8f91\u56de\u5f52\u6a21\u578b\u901a\u5e38\u91c7\u7528\u6781\u5927\u4f3c\u7136\u4f30\u8ba1\u7684\u65b9\u6cd5\u3002\u5047\u8bbe\u6709$n$\u4e2a\u6837\u672c\uff0c$y_i \\in {0,1}$\u662f\u6837\u672c$i$\u7684\u8f93\u51fa\uff0c$x_i \\in \\mathbb{R}^d$\u662f\u6837\u672c$i$\u7684\u8f93\u5165\uff0c$p_i$\u662f\u6837\u672c$i$\u5c5e\u4e8e\u6b63\u7c7b\u7684\u6982\u7387\uff0c\u90a3\u4e48\u4f3c\u7136\u51fd\u6570\u4e3a\uff1a<\/p>\n\n\n\n
                                                                                      \ufffd(\ufffd,\ufffd)=\u220f\ufffd=1\ufffd\ufffd\ufffd\ufffd\ufffd(1\u2212\ufffd\ufffd)1\u2212\ufffd\ufffd.L<\/em>(w<\/em>,b<\/em>)=i<\/em>=1\u220fn<\/em>\u200bp<\/em>i<\/em>y<\/em>i<\/em>\u200b\u200b(1\u2212p<\/em>i<\/em>\u200b)1\u2212y<\/em>i<\/em>\u200b.<\/p>\n\n\n\n
                                                                                      \u5bf9\u6570\u4f3c\u7136\u51fd\u6570\u4e3a\uff1a<\/p>\n\n\n\n
                                                                                      \ufffd(\ufffd,\ufffd)=log\u2061\ufffd(\ufffd,\ufffd)=\u2211\ufffd=1\ufffd[\ufffd\ufffdlog\u2061\ufffd\ufffd+(1\u2212\ufffd\ufffd)log\u2061(1\u2212\ufffd\ufffd)].l<\/em>(w<\/em>,b<\/em>)=logL<\/em>(w<\/em>,b<\/em>)=i<\/em>=1\u2211n<\/em>\u200b[y<\/em>i<\/em>\u200blogp<\/em>i<\/em>\u200b+(1\u2212y<\/em>i<\/em>\u200b)log(1\u2212p<\/em>i<\/em>\u200b)].<\/p>\n\n\n\n
                                                                                      \u76ee\u6807\u662f\u6700\u5927\u5316\u5bf9\u6570\u4f3c\u7136\u51fd\u6570\uff0c\u5373<\/p>\n\n\n\n
                                                                                      max\u2061\ufffd,\ufffd\ufffd(\ufffd,\ufffd).w<\/em>,b<\/em>max\u200bl<\/em>(w<\/em>,b<\/em>).<\/p>\n\n\n\n
                                                                                      \u901a\u5e38\u91c7\u7528\u68af\u5ea6\u4e0b\u964d\u7b49\u4f18\u5316\u7b97\u6cd5\u5bf9\u76ee\u6807\u51fd\u6570\u8fdb\u884c\u6c42\u89e3\u3002\u5728\u6c42\u89e3\u65f6\u53ef\u4ee5\u91c7\u7528\u6b63\u5219\u5316\u7684\u65b9\u6cd5\uff0c\u52a0\u5165L1\u6216L2\u6b63\u5219\u9879\u4ee5\u907f\u514d\u8fc7\u62df\u5408\u95ee\u9898\u3002<\/p>\n\n\n\n
                                                                                      \u4ee5\u4e0b\u662f\u4e00\u4e2a\u7b80\u5355\u7684Python\u793a\u4f8b\u4ee3\u7801\uff0c\u4f7f\u7528scikit-learn\u5b9e\u73b0\u903b\u8f91\u56de\u5f52\u6a21\u578b\u7684\u8bad\u7ec3\u548c\u9884\u6d4b\uff1a<\/p>\n\n\n\n
                                                                                      pythonCopy codefrom sklearn.linear_model import LogisticRegression\n\n# \u6784\u9020\u6837\u672c\u6570\u636e\uff0cX\u8868\u793a\u8f93\u5165\u7279\u5f81\uff0cy\u8868\u793a\u8f93\u51fa\nX = [[0.5, 1], [2, 3], [4, 2.5], [3, 5], [5.5, 4], [6, 6.5]]\ny = [0, 0, 0, 1, 1, 1]\n\n# \u8bad\u7ec3\u903b\u8f91\u56de\u5f52\u6a21\u578b\nclf = LogisticRegression(random_state=0).fit(X, y)\n\n# \u9884\u6d4b\u65b0\u6837\u672c\u7684\u8f93\u51fa\nprint(clf.predict([[1, 2], [4.5, 3.5]]))\n<\/code><\/pre>\n\n\n\n
                                                                                      \u8f93\u51fa\u7ed3\u679c\u4e3a\uff1a<\/p>\n\n\n\n
                                                                                      csharpCopy code[0 0]\n<\/code><\/pre>\n\n\n\n\n\n
                                                                                      \"gun<\/figure>\n\n\n\n
                                                                                      \u8bf7\u8be6\u7ec6\u89e3\u91ca\u6d41\u5f62\u5047\u8bbe<\/p>\n\n\n\n
                                                                                      \u6d41\u5f62\u5047\u8bbe\u662f\u6307\u9ad8\u7ef4\u6570\u636e\u5728\u4f4e\u7ef4\u7a7a\u95f4\u4e2d\u5448\u73b0\u51fa\u6d41\u5f62\u7ed3\u6784\uff0c\u5373\u6570\u636e\u70b9\u4e0d\u662f\u968f\u673a\u5206\u5e03\u7684\uff0c\u800c\u662f\u5b58\u5728\u4e00\u4e9b\u89c4\u5f8b\u6027\u7684\u51e0\u4f55\u5f62\u72b6\u3002\u8be5\u5047\u8bbe\u662f\u5f88\u591a\u673a\u5668\u5b66\u4e60\u7b97\u6cd5\u7684\u57fa\u7840\uff0c\u4f8b\u5982\u805a\u7c7b\u3001\u964d\u7ef4\u3001\u5206\u7c7b\u7b49\u3002<\/p>\n\n\n\n
                                                                                      \u5177\u4f53\u5730\uff0c\u6d41\u5f62\u53ef\u4ee5\u7406\u89e3\u4e3a\u5c40\u90e8\u4e0a\u5177\u6709\u6b27\u51e0\u91cc\u5f97\u7a7a\u95f4\u7ed3\u6784\u7684\u7a7a\u95f4\uff0c\u8fd9\u610f\u5473\u7740\u5728\u6bcf\u4e2a\u5c40\u90e8\u533a\u57df\u5185\uff0c\u6570\u636e\u70b9\u7684\u5206\u5e03\u53ef\u4ee5\u88ab\u6b27\u51e0\u91cc\u5f97\u7a7a\u95f4\u7684\u4e00\u4e9b\u51fd\u6570\u8fdb\u884c\u826f\u597d\u5730\u63cf\u8ff0\u3002\u5728\u5b9e\u9645\u95ee\u9898\u4e2d\uff0c\u6d41\u5f62\u53ef\u4ee5\u662f\u5404\u79cd\u5f62\u72b6\uff0c\u4f8b\u5982\u7403\u5f62\u3001\u73af\u5f62\u3001\u87ba\u65cb\u5f62\u7b49\u3002<\/p>\n\n\n\n
                                                                                      \u6d41\u5f62\u5047\u8bbe\u7684\u6838\u5fc3\u5728\u4e8e\u5c06\u9ad8\u7ef4\u6570\u636e\u6620\u5c04\u5230\u4f4e\u7ef4\u7a7a\u95f4\uff0c\u4f7f\u5f97\u539f\u672c\u590d\u6742\u7684\u6570\u636e\u7ed3\u6784\u5728\u4f4e\u7ef4\u7a7a\u95f4\u4e2d\u5448\u73b0\u51fa\u7b80\u5355\u7684\u5f62\u6001\uff0c\u4ece\u800c\u65b9\u4fbf\u8fdb\u884c\u5904\u7406\u548c\u5206\u6790\u3002\u6d41\u5f62\u5b66\u4e60\u7b97\u6cd5\u7684\u76ee\u6807\u5c31\u662f\u901a\u8fc7\u6570\u636e\u70b9\u7684\u5c40\u90e8\u90bb\u57df\u6765\u91cd\u6784\u4f4e\u7ef4\u7a7a\u95f4\u4e2d\u7684\u6d41\u5f62\u7ed3\u6784\uff0c\u4ee5\u6b64\u6765\u8fdb\u884c\u964d\u7ef4\u6216\u5206\u7c7b\u7b49\u4efb\u52a1\u3002<\/p>\n\n\n\n
                                                                                      \u5e38\u89c1\u7684\u6d41\u5f62\u5b66\u4e60\u7b97\u6cd5\u6709\u5c40\u90e8\u7ebf\u6027\u5d4c\u5165\uff08Locally Linear Embedding, LLE\uff09\u3001\u7b49\u8ddd\u6620\u5c04\uff08Isomap\uff09\u3001\u62c9\u666e\u62c9\u65af\u7279\u5f81\u6620\u5c04\uff08Laplacian Eigenmaps\uff09\u3001t-SNE\u7b49\u3002\u8fd9\u4e9b\u7b97\u6cd5\u53ef\u4ee5\u901a\u8fc7\u5bfb\u627e\u6570\u636e\u70b9\u4e4b\u95f4\u7684\u5173\u7cfb\uff0c\u5982\u90bb\u8fd1\u5173\u7cfb\u6216\u76f8\u4f3c\u6027\u5173\u7cfb\uff0c\u6765\u6784\u5efa\u6d41\u5f62\u7ed3\u6784\uff0c\u4ece\u800c\u5b9e\u73b0\u6570\u636e\u7684\u964d\u7ef4\u6216\u5206\u7c7b\u7b49\u4efb\u52a1\u3002<\/p>\n\n\n\n
                                                                                      \u603b\u7684\u6765\u8bf4\uff0c\u6d41\u5f62\u5047\u8bbe\u63d0\u4f9b\u4e86\u4e00\u79cd\u65b0\u7684\u601d\u8def\u6765\u7406\u89e3\u548c\u5904\u7406\u9ad8\u7ef4\u6570\u636e\uff0c\u5176\u5e94\u7528\u8303\u56f4\u5e7f\u6cdb\uff0c\u5e76\u5728\u8bb8\u591a\u9886\u57df\u4e2d\u53d6\u5f97\u4e86\u6210\u529f\u7684\u5e94\u7528\uff0c\u5982\u56fe\u50cf\u8bc6\u522b\u3001\u8bed\u97f3\u5904\u7406\u3001\u81ea\u7136\u8bed\u8a00\u5904\u7406\u7b49\u3002<\/p>\n\n\n\n\n\n
                                                                                      \"gun<\/figure>\n\n\n\n
                                                                                      \u8bf7\u8be6\u7ec6\u89e3\u91ca\u6d4b\u5ea6\u8bba<\/p>\n\n\n\n
                                                                                      \u6d4b\u5ea6\u8bba\u662f\u6570\u5b66\u4e2d\u7684\u4e00\u4e2a\u5206\u652f\uff0c\u4e3b\u8981\u7814\u7a76\u51fd\u6570\u3001\u96c6\u5408\u7b49\u7684\u5ea6\u91cf\u95ee\u9898\u3002\u5176\u7814\u7a76\u5bf9\u8c61\u662f\u6d4b\u5ea6\uff0c\u5373\u5bf9\u96c6\u5408\u8d4b\u4e88\u5927\u5c0f\u7684\u4e00\u79cd\u65b9\u5f0f\u3002\u6d4b\u5ea6\u8bba\u5728\u6982\u7387\u8bba\u3001\u5b9e\u53d8\u51fd\u6570\u8bba\u3001\u6570\u8bba\u7b49\u9886\u57df\u4e2d\u6709\u5e7f\u6cdb\u5e94\u7528\u3002<\/p>\n\n\n\n
                                                                                      \u6d4b\u5ea6\u662f\u5bf9\u96c6\u5408\u5927\u5c0f\u7684\u5ea6\u91cf\uff0c\u901a\u5e38\u7528\u6765\u8861\u91cf\u67d0\u4e2a\u96c6\u5408\u7684\u5927\u5c0f\u6216\u8005\u91cd\u91cf\u3002\u5728\u6d4b\u5ea6\u8bba\u4e2d\uff0c\u9700\u8981\u6ee1\u8db3\u4ee5\u4e0b\u4e09\u4e2a\u6761\u4ef6\uff1a\u975e\u8d1f\u6027\u3001\u53ef\u6570\u53ef\u52a0\u6027\u548c\u89c4\u8303\u6027\u3002<\/p>\n\n\n\n
                                                                                      \u5177\u4f53\u6765\u8bf4\uff0c\u6d4b\u5ea6\u8bba\u4e2d\u7684\u6d4b\u5ea6\u53ef\u4ee5\u5b9a\u4e49\u5728\u5404\u79cd\u96c6\u5408\u4e0a\uff0c\u5305\u62ec\u5b9e\u6570\u3001\u5411\u91cf\u7a7a\u95f4\u3001\u51fd\u6570\u7a7a\u95f4\u7b49\u7b49\u3002\u5e38\u89c1\u7684\u6d4b\u5ea6\u5305\u62ec\u957f\u5ea6\u3001\u9762\u79ef\u3001\u4f53\u79ef\u3001Lebesgue\u6d4b\u5ea6\u7b49\u3002<\/p>\n\n\n\n
                                                                                      \u6d4b\u5ea6\u8bba\u6709\u8bb8\u591a\u57fa\u672c\u5b9a\u7406\uff0c\u5305\u62ec\u6d4b\u5ea6\u7684\u53ef\u5217\u53ef\u52a0\u6027\u3001\u5355\u8c03\u6709\u754c\u6536\u655b\u5b9a\u7406\u3001Fatou\u5f15\u7406\u3001Lebesgue\u63a7\u5236\u6536\u655b\u5b9a\u7406\u7b49\u3002\u8fd9\u4e9b\u5b9a\u7406\u5bf9\u4e8e\u7814\u7a76\u6982\u7387\u8bba\u3001\u5b9e\u5206\u6790\u3001\u8c03\u548c\u5206\u6790\u7b49\u9886\u57df\u90fd\u6709\u5f88\u5927\u7684\u4f5c\u7528\u3002<\/p>\n\n\n\n
                                                                                      \u6d4b\u5ea6\u8bba\u5728\u673a\u5668\u5b66\u4e60\u4e2d\u7684\u5e94\u7528\u4e5f\u5f88\u5e7f\u6cdb\uff0c\u4f8b\u5982\u57fa\u4e8e\u6d4b\u5ea6\u7684\u5f02\u5e38\u68c0\u6d4b\u3001\u57fa\u4e8e\u6d4b\u5ea6\u7684\u5206\u7c7b\u3001\u57fa\u4e8e\u6d4b\u5ea6\u7684\u805a\u7c7b\u7b49\u7b49\u3002\u5728\u6df1\u5ea6\u5b66\u4e60\u4e2d\uff0c\u6d4b\u5ea6\u8bba\u7684\u4e00\u4e9b\u601d\u60f3\u4e5f\u88ab\u7528\u4e8e\u89e3\u51b3\u8bad\u7ec3\u8fc7\u7a0b\u4e2d\u7684\u4e00\u4e9b\u95ee\u9898\uff0c\u4f8b\u5982\u5bf9\u6297\u6837\u672c\u7684\u68c0\u6d4b\u548c\u5bf9\u6297\u8bad\u7ec3\u4e2d\u7684\u6b63\u5219\u5316\u7b49\u3002<\/p>\n\n\n\n\n\n
                                                                                      \"gun<\/figure>\n\n\n\n
                                                                                      \u8bf7\u8be6\u7ec6\u89e3\u91ca\u9053\u5fb7\u5316<\/p>\n\n\n\n
                                                                                      \u9053\u5fb7\u5316\uff08moralization\uff09\u662f\u8d1d\u53f6\u65af\u7f51\u7edc\u4e2d\u7684\u4e00\u4e2a\u64cd\u4f5c\uff0c\u5b83\u5c06\u7f51\u7edc\u4e2d\u7684\u4e00\u4e9b\u8282\u70b9\u8f6c\u6362\u4e3a\u6f5c\u5728\u7684\u539f\u56e0\u8282\u70b9\uff0c\u4ee5\u4fbf\u8fdb\u884c\u56e0\u679c\u63a8\u65ad\u3002<\/p>\n\n\n\n
                                                                                      \u5728\u8d1d\u53f6\u65af\u7f51\u7edc\u4e2d\uff0c\u6709\u4e9b\u8282\u70b9\u662f\u89c2\u6d4b\u53d8\u91cf\uff0c\u5b83\u4eec\u7684\u503c\u662f\u5df2\u77e5\u7684\uff0c\u6709\u4e9b\u8282\u70b9\u662f\u9690\u53d8\u91cf\u6216\u672a\u77e5\u53d8\u91cf\uff0c\u9700\u8981\u901a\u8fc7\u6982\u7387\u63a8\u65ad\u6765\u786e\u5b9a\u5176\u503c\u3002\u5728\u8fdb\u884c\u56e0\u679c\u63a8\u65ad\u65f6\uff0c\u901a\u5e38\u9700\u8981\u5c06\u4e00\u4e9b\u89c2\u6d4b\u53d8\u91cf\u8f6c\u5316\u4e3a\u6f5c\u5728\u7684\u539f\u56e0\u8282\u70b9\uff0c\u8fd9\u4e2a\u64cd\u4f5c\u5c31\u53eb\u505a\u9053\u5fb7\u5316\u3002<\/p>\n\n\n\n
                                                                                      \u5177\u4f53\u6765\u8bf4\uff0c\u5f53\u4e00\u4e2a\u89c2\u6d4b\u53d8\u91cf\u7684\u6240\u6709\u7236\u8282\u70b9\u90fd\u5df2\u77e5\u65f6\uff0c\u5c06\u8be5\u89c2\u6d4b\u53d8\u91cf\u8f6c\u5316\u4e3a\u6f5c\u5728\u7684\u539f\u56e0\u8282\u70b9\uff0c\u5373\u5c06\u8be5\u8282\u70b9\u4e0e\u5176\u6240\u6709\u7236\u8282\u70b9\u8fde\u63a5\u8d77\u6765\u5f62\u6210\u4e00\u4e2a\u65b0\u7684\u56e0\u679c\u6a21\u578b\uff0c\u8fd9\u4e2a\u6a21\u578b\u4e2d\u8be5\u8282\u70b9\u7684\u7236\u8282\u70b9\u4e4b\u95f4\u76f8\u4e92\u72ec\u7acb\u3002\u8fd9\u4e2a\u8fc7\u7a0b\u4e5f\u53ef\u4ee5\u770b\u4f5c\u662f\u5c06\u89c2\u6d4b\u53d8\u91cf\u201c\u9053\u5fb7\u5316\u201d\u4e3a\u6f5c\u5728\u7684\u539f\u56e0\u8282\u70b9\u3002<\/p>\n\n\n\n
                                                                                      \u9053\u5fb7\u5316\u7684\u76ee\u7684\u662f\u7b80\u5316\u56e0\u679c\u63a8\u65ad\u95ee\u9898\uff0c\u901a\u8fc7\u5c06\u4e00\u4e9b\u89c2\u6d4b\u53d8\u91cf\u8f6c\u5316\u4e3a\u539f\u56e0\u8282\u70b9\uff0c\u53ef\u4ee5\u66f4\u597d\u5730\u5206\u6790\u53d8\u91cf\u4e4b\u95f4\u7684\u56e0\u679c\u5173\u7cfb\uff0c\u4ece\u800c\u66f4\u51c6\u786e\u5730\u8fdb\u884c\u63a8\u65ad\u3002<\/p>\n\n\n\n
                                                                                      \u4e3e\u4e2a\u4f8b\u5b50\uff0c\u5047\u8bbe\u6211\u4eec\u6709\u4e00\u4e2a\u8d1d\u53f6\u65af\u7f51\u7edc\uff0c\u5176\u4e2dX\u548cY\u662f\u4e24\u4e2a\u89c2\u6d4b\u53d8\u91cf\uff0cA\u662f\u4e00\u4e2a\u672a\u77e5\u53d8\u91cf\u3002\u5982\u679c\u6211\u4eec\u60f3\u8981\u63a8\u65adA\u5bf9X\u7684\u5f71\u54cd\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06Y\u9053\u5fb7\u5316\u4e3a\u6f5c\u5728\u7684\u539f\u56e0\u8282\u70b9\uff0c\u7136\u540e\u5206\u6790A\u548cX\u5728Y\u7684\u4f5c\u7528\u4e0b\u7684\u5173\u7cfb\uff0c\u4ece\u800c\u5f97\u51faA\u5bf9X\u7684\u5f71\u54cd\u3002<\/p>\n\n\n\n
                                                                                      \u9053\u5fb7\u5316\u901a\u5e38\u7528\u4e8e\u56e0\u679c\u63a8\u65ad\u95ee\u9898\u4e2d\uff0c\u5b83\u5728\u673a\u5668\u5b66\u4e60\u548c\u4eba\u5de5\u667a\u80fd\u9886\u57df\u6709\u7740\u5e7f\u6cdb\u7684\u5e94\u7528\uff0c\u5982\u63a8\u8350\u7cfb\u7edf\u3001\u533b\u5b66\u8bca\u65ad\u3001\u81ea\u7136\u8bed\u8a00\u5904\u7406\u7b49\u3002<\/p>\n\n\n\n\n\n
                                                                                      \"gun<\/figure>\n\n\n\n
                                                                                      \u8bf7\u8be6\u7ec6\u89e3\u91ca\u8d1f\u5b9a<\/p>\n\n\n\n
                                                                                      \u5728\u7ebf\u6027\u4ee3\u6570\u548c\u77e9\u9635\u8bba\u4e2d\uff0c\u8d1f\u5b9a\u662f\u6307\u4e00\u4e2a\u5b9e\u5bf9\u79f0\u77e9\u9635 $A$ \u7684\u6240\u6709\u7279\u5f81\u503c\u90fd\u5c0f\u4e8e\u96f6\u7684\u6027\u8d28\u3002\u6362\u8a00\u4e4b\uff0c\u77e9\u9635 $A$ \u7684\u4efb\u610f\u4e00\u4e2a\u975e\u96f6\u5411\u91cf $x$ \u6ee1\u8db3 $x^T A x < 0$\u3002\u8fd9\u4e2a\u5b9a\u4e49\u53ef\u4ee5\u7528\u6765\u5224\u65ad\u77e9\u9635\u662f\u5426\u662f\u6b63\u5b9a\u3001\u534a\u6b63\u5b9a\u3001\u8d1f\u5b9a\u6216\u534a\u8d1f\u5b9a\u3002<\/p>\n\n\n\n
                                                                                      \u8d1f\u5b9a\u77e9\u9635\u5728\u5f88\u591a\u6570\u5b66\u548c\u5de5\u7a0b\u5e94\u7528\u4e2d\u90fd\u5f88\u91cd\u8981\uff0c\u4f8b\u5982\u5728\u4f18\u5316\u95ee\u9898\u4e2d\uff0c\u5f80\u5f80\u9700\u8981\u627e\u5230\u4e00\u4e2a\u8d1f\u5b9a\u7684\u77e9\u9635\u6765\u8fdb\u884c\u5206\u6790\u548c\u6c42\u89e3\uff1b\u5728\u7edf\u8ba1\u5b66\u4e2d\uff0c\u8d1f\u5b9a\u77e9\u9635\u88ab\u5e7f\u6cdb\u5e94\u7528\u4e8e\u534f\u65b9\u5dee\u77e9\u9635\u548c\u76f8\u5173\u77e9\u9635\u7684\u5b9a\u4e49\uff1b\u5728\u4fe1\u53f7\u5904\u7406\u4e2d\uff0c\u8d1f\u5b9a\u77e9\u9635\u7528\u4e8e\u63cf\u8ff0\u4fe1\u53f7\u7684\u529f\u7387\u8c31\u5bc6\u5ea6\u7b49\u6027\u8d28\u3002<\/p>\n\n\n\n
                                                                                      \u4e00\u4e9b\u5e38\u89c1\u7684\u6027\u8d28\u548c\u5224\u65ad\u65b9\u6cd5\u5982\u4e0b\uff1a<\/p>\n\n\n\n
                                                                                        \n
                                                                                      • \u4e00\u4e2a\u5b9e\u5bf9\u79f0\u77e9\u9635 $A$ \u662f\u8d1f\u5b9a\u7684\uff0c\u5f53\u4e14\u4ec5\u5f53 $A$ \u7684\u6240\u6709\u4e3b\u5b50\u5f0f\u90fd\u662f\u8d1f\u6570\u3002<\/li>\n\n\n\n
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                                                                                      • \u5982\u679c\u4e00\u4e2a\u5b9e\u5bf9\u79f0\u77e9\u9635 $A$ \u662f\u8d1f\u5b9a\u7684\uff0c\u5219\u5b83\u7684\u6bcf\u4e2a\u5947\u5f02\u503c\u90fd\u662f\u975e\u8d1f\u7684\uff0c\u4f46\u4e0d\u4e3a\u96f6\u3002<\/li>\n\n\n\n
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                                                                                        \"gun<\/figure>\n\n\n\n
                                                                                        \u8bf7\u8be6\u7ec6\u89e3\u91ca\u5206\u6bb5<\/p>\n\n\n\n
                                                                                        \u5206\u6bb5\uff08Piecewise\uff09\u51fd\u6570\u662f\u6307\u5728\u5b9a\u4e49\u57df\u4e0a\uff0c\u6309\u7167\u4e0d\u540c\u7684\u5b50\u533a\u95f4\u91c7\u7528\u4e0d\u540c\u7684\u51fd\u6570\u8868\u8fbe\u5f0f\u8fdb\u884c\u63cf\u8ff0\u7684\u4e00\u79cd\u51fd\u6570\u3002\u5206\u6bb5\u51fd\u6570\u53ef\u4ee5\u5206\u4e3a\u6709\u9650\u5206\u6bb5\u51fd\u6570\u548c\u65e0\u9650\u5206\u6bb5\u51fd\u6570\u4e24\u79cd\u3002<\/p>\n\n\n\n
                                                                                        \u6709\u9650\u5206\u6bb5\u51fd\u6570\u6307\u5728\u6709\u9650\u4e2a\u5b50\u533a\u95f4\u5185\uff0c\u91c7\u7528\u4e0d\u540c\u7684\u51fd\u6570\u8868\u8fbe\u5f0f\u63cf\u8ff0\u51fd\u6570\u3002\u4f8b\u5982\uff0c\u4e0b\u9762\u7684\u51fd\u6570 $f(x)$ \u5c31\u662f\u4e00\u4e2a\u6709\u9650\u5206\u6bb5\u51fd\u6570\uff1a<\/p>\n\n\n\n
                                                                                        \ufffd(\ufffd)={2\ufffd+1,\u2212\u221e<\ufffd<03\ufffd\u22121,0\u2264\ufffd<\u221ef<\/em>(x<\/em>)={2x<\/em>+1,3x<\/em>\u22121,\u200b\u2212\u221e<x<\/em><00\u2264x<\/em><\u221e\u200b<\/p>\n\n\n\n
                                                                                        \u65e0\u9650\u5206\u6bb5\u51fd\u6570\u6307\u5728\u5b9a\u4e49\u57df\u4e0a\uff0c\u6709\u65e0\u9650\u591a\u4e2a\u5b50\u533a\u95f4\uff0c\u6bcf\u4e2a\u5b50\u533a\u95f4\u91c7\u7528\u4e0d\u540c\u7684\u51fd\u6570\u8868\u8fbe\u5f0f\u63cf\u8ff0\u51fd\u6570\u3002\u4f8b\u5982\uff0c\u4e0b\u9762\u7684\u51fd\u6570 $f(x)$ \u5c31\u662f\u4e00\u4e2a\u65e0\u9650\u5206\u6bb5\u51fd\u6570\uff1a<\/p>\n\n\n\n
                                                                                        \ufffd(\ufffd)={\ufffd,\ufffd\u2208[0,1]2\u2212\ufffd,\ufffd\u2208(1,2]f<\/em>(x<\/em>)={x<\/em>,2\u2212x<\/em>,\u200bx<\/em>\u2208[0,1]x<\/em>\u2208(1,2]\u200b<\/p>\n\n\n\n
                                                                                        \u5206\u6bb5\u51fd\u6570\u5728\u6570\u5b66\u3001\u7269\u7406\u3001\u5de5\u7a0b\u7b49\u9886\u57df\u4e2d\u90fd\u6709\u5e7f\u6cdb\u7684\u5e94\u7528\uff0c\u4f8b\u5982\uff0c\u5728\u63a7\u5236\u7cfb\u7edf\u4e2d\uff0c\u6211\u4eec\u5e38\u5e38\u9700\u8981\u6839\u636e\u8f93\u5165\u91cf\u7684\u4e0d\u540c\u533a\u95f4\uff0c\u5bf9\u8f93\u51fa\u91cf\u8fdb\u884c\u4e0d\u540c\u7684\u63a7\u5236\uff1b\u5728\u4fe1\u53f7\u5904\u7406\u4e2d\uff0c\u6211\u4eec\u5e38\u5e38\u9700\u8981\u5c06\u4fe1\u53f7\u8fdb\u884c\u5206\u6bb5\u5904\u7406\uff0c\u4ee5\u8fbe\u5230\u66f4\u597d\u7684\u6548\u679c\u3002<\/p>\n\n\n\n\n\n
                                                                                        \"gun<\/figure>\n\n\n\n
                                                                                        \u8bf7\u8be6\u7ec6\u89e3\u91ca\u7f6e\u6362\u4e0d\u53d8\u6027<\/p>\n\n\n\n
                                                                                        \u7f6e\u6362\u4e0d\u53d8\u6027\u662f\u6307\u67d0\u4e2a\u7cfb\u7edf\u6216\u6a21\u578b\u5728\u8fdb\u884c\u7f6e\u6362\u64cd\u4f5c\u540e\uff0c\u5176\u6027\u8d28\u4e0d\u53d1\u751f\u53d8\u5316\u7684\u7279\u6027\u3002\u5728\u6570\u5b66\u548c\u7269\u7406\u5b66\u4e2d\uff0c\u7f6e\u6362\u4e0d\u53d8\u6027\u662f\u975e\u5e38\u91cd\u8981\u7684\u6982\u5ff5\uff0c\u4f8b\u5982\u5728\u5bf9\u79f0\u6027\u7814\u7a76\u3001\u5fae\u79ef\u5206\u3001\u4ee3\u6570\u5b66\u3001\u51e0\u4f55\u5b66\u7b49\u9886\u57df\u4e2d\u90fd\u6709\u5e94\u7528\u3002<\/p>\n\n\n\n
                                                                                        \u5728\u673a\u5668\u5b66\u4e60\u9886\u57df\u4e2d\uff0c\u7f6e\u6362\u4e0d\u53d8\u6027\u901a\u5e38\u6307\u8f93\u5165\u6570\u636e\u5728\u8fdb\u884c\u4e00\u4e9b\u7f6e\u6362\u64cd\u4f5c\u540e\uff0c\u6a21\u578b\u7684\u8f93\u51fa\u4e0d\u4f1a\u53d1\u751f\u6539\u53d8\u3002\u4f8b\u5982\u5728\u56fe\u50cf\u5206\u7c7b\u4efb\u52a1\u4e2d\uff0c\u5982\u679c\u4e00\u5f20\u56fe\u7247\u8fdb\u884c\u4e86\u5e73\u79fb\u3001\u65cb\u8f6c\u6216\u7f29\u653e\u7b49\u64cd\u4f5c\uff0c\u6a21\u578b\u5bf9\u4e8e\u8fd9\u5f20\u56fe\u7247\u7684\u5206\u7c7b\u7ed3\u679c\u5e94\u8be5\u4fdd\u6301\u4e0d\u53d8\uff0c\u8fd9\u5c31\u662f\u6a21\u578b\u7684\u7f6e\u6362\u4e0d\u53d8\u6027\u3002\u5728\u81ea\u7136\u8bed\u8a00\u5904\u7406\u9886\u57df\u4e2d\uff0c\u4e00\u4e2a\u53e5\u5b50\u7684\u8bcd\u5e8f\u53d1\u751f\u53d8\u5316\u540e\uff0c\u5176\u8bed\u4e49\u5e94\u8be5\u4fdd\u6301\u4e0d\u53d8\uff0c\u8fd9\u4e5f\u662f\u6a21\u578b\u7684\u7f6e\u6362\u4e0d\u53d8\u6027\u3002<\/p>\n\n\n\n
                                                                                        \u4e3a\u4e86\u5b9e\u73b0\u6a21\u578b\u7684\u7f6e\u6362\u4e0d\u53d8\u6027\uff0c\u901a\u5e38\u4f1a\u91c7\u7528\u4e00\u4e9b\u6280\u672f\uff0c\u4f8b\u5982\u5377\u79ef\u795e\u7ecf\u7f51\u7edc\u4e2d\u7684\u5377\u79ef\u64cd\u4f5c\u53ef\u4ee5\u5b9e\u73b0\u5e73\u79fb\u4e0d\u53d8\u6027\uff0c\u5faa\u73af\u795e\u7ecf\u7f51\u7edc\u4e2d\u7684\u5faa\u73af\u7ed3\u6784\u53ef\u4ee5\u5b9e\u73b0\u5e8f\u5217\u4e0d\u53d8\u6027\uff0c\u6c60\u5316\u64cd\u4f5c\u53ef\u4ee5\u5b9e\u73b0\u65cb\u8f6c\u4e0d\u53d8\u6027\u7b49\u3002<\/p>\n\n\n\n\n\n
                                                                                        \"gun<\/figure>\n\n\n\n
                                                                                        \u8bf7\u8be6\u7ec6\u89e3\u91ca\u6b63\u5219\u5316<\/p>\n\n\n\n
                                                                                        \u6b63\u5219\u5316\uff08Regularization\uff09\u662f\u673a\u5668\u5b66\u4e60\u4e2d\u7528\u4e8e\u9632\u6b62\u6a21\u578b\u8fc7\u5ea6\u62df\u5408\u7684\u4e00\u79cd\u65b9\u6cd5\u3002\u5f53\u8bad\u7ec3\u4e00\u4e2a\u6a21\u578b\u65f6\uff0c\u5982\u679c\u6a21\u578b\u8fc7\u4e8e\u590d\u6742\uff0c\u5c31\u4f1a\u5bfc\u81f4\u5bf9\u8bad\u7ec3\u6570\u636e\u7684\u8fc7\u5ea6\u62df\u5408\uff0c\u5bfc\u81f4\u6cdb\u5316\u6027\u80fd\u53d8\u5dee\u3002\u4e3a\u4e86\u907f\u514d\u8fd9\u79cd\u60c5\u51b5\uff0c\u6b63\u5219\u5316\u5c06\u6a21\u578b\u7684\u590d\u6742\u5ea6\u9650\u5236\u5728\u4e00\u5b9a\u8303\u56f4\u5185\uff0c\u4ece\u800c\u4f7f\u6a21\u578b\u66f4\u52a0\u7b80\u5355\uff0c\u66f4\u5bb9\u6613\u6cdb\u5316\u5230\u65b0\u6570\u636e\u3002<\/p>\n\n\n\n
                                                                                        \u5e38\u7528\u7684\u6b63\u5219\u5316\u65b9\u6cd5\u6709 L1 \u6b63\u5219\u5316\u548c L2 \u6b63\u5219\u5316\u3002L1 \u6b63\u5219\u5316\u901a\u8fc7\u5728\u76ee\u6807\u51fd\u6570\u4e2d\u6dfb\u52a0 L1 \u8303\u6570\uff08\u7edd\u5bf9\u503c\uff09\u7684\u60e9\u7f5a\u9879\uff0c\u53ef\u4ee5\u4f7f\u5f97\u90e8\u5206\u6743\u91cd\u53d8\u6210 0\uff0c\u5b9e\u73b0\u7279\u5f81\u9009\u62e9\u7684\u529f\u80fd\uff1bL2 \u6b63\u5219\u5316\u901a\u8fc7\u5728\u76ee\u6807\u51fd\u6570\u4e2d\u6dfb\u52a0 L2 \u8303\u6570\uff08\u5e73\u65b9\u548c\uff09\u7684\u60e9\u7f5a\u9879\uff0c\u53ef\u4ee5\u5c06\u6743\u91cd\u53c2\u6570\u538b\u7f29\u5230\u6bd4\u8f83\u5c0f\u7684\u8303\u56f4\u5185\uff0c\u9632\u6b62\u6743\u91cd\u53c2\u6570\u53d8\u5f97\u8fc7\u5927\u3002\u6b64\u5916\uff0c\u8fd8\u6709 Elastic Net \u6b63\u5219\u5316\u7b49\u65b9\u6cd5\uff0c\u53ef\u4ee5\u540c\u65f6\u4f7f\u7528 L1 \u548c L2 \u6b63\u5219\u5316\u3002<\/p>\n\n\n\n
                                                                                        \u6b63\u5219\u5316\u7684\u4e3b\u8981\u4f18\u70b9\u662f\u53ef\u4ee5\u964d\u4f4e\u6a21\u578b\u7684\u65b9\u5dee\uff0c\u51cf\u5c11\u5bf9\u8bad\u7ec3\u6570\u636e\u7684\u8fc7\u5ea6\u62df\u5408\uff0c\u63d0\u9ad8\u6cdb\u5316\u80fd\u529b\u3002\u7136\u800c\uff0c\u8fc7\u4e8e\u5f3a\u7684\u6b63\u5219\u5316\u4e5f\u53ef\u80fd\u4f1a\u5bfc\u81f4\u6b20\u62df\u5408\uff0c\u964d\u4f4e\u6a21\u578b\u7684\u51c6\u786e\u5ea6\u3002\u56e0\u6b64\uff0c\u9009\u62e9\u5408\u9002\u7684\u6b63\u5219\u5316\u65b9\u6cd5\u4ee5\u53ca\u6b63\u5219\u5316\u53c2\u6570\u662f\u5341\u5206\u91cd\u8981\u7684\u3002<\/p>\n\n\n\n\n\n
                                                                                        \"gun<\/figure>\n\n\n\n
                                                                                        \u8bf7\u8be6\u7ec6\u89e3\u91ca\u6b63\u5207\u4f20\u64ad<\/p>\n\n\n\n
                                                                                        \u6b63\u5207\u4f20\u64ad\uff08Tanh propagation\uff09\u662f\u4e00\u79cd\u795e\u7ecf\u7f51\u7edc\u4e2d\u5e38\u7528\u7684\u6fc0\u6d3b\u51fd\u6570\uff0c\u5b83\u662fsigmoid\u51fd\u6570\u7684\u53d8\u5f62\uff0c\u80fd\u591f\u5c06\u8f93\u5165\u7684\u503c\u57df\u9650\u5b9a\u5728[-1,1]\u8303\u56f4\u5185\u3002\u6b63\u5207\u4f20\u64ad\u51fd\u6570\u7684\u516c\u5f0f\u4e3a\uff1a<\/p>\n\n\n\n
                                                                                        \ufffd(\ufffd)=\ufffd\ufffd\u2212\ufffd\u2212\ufffd\ufffd\ufffd+\ufffd\u2212\ufffdf<\/em>(x<\/em>)=e<\/em>x<\/em>+e<\/em>\u2212x<\/em>e<\/em>x<\/em>\u2212e<\/em>\u2212x<\/em>\u200b<\/p>\n\n\n\n
                                                                                        \u6b63\u5207\u4f20\u64ad\u51fd\u6570\u5177\u6709\u4ee5\u4e0b\u7279\u70b9\uff1a<\/p>\n\n\n\n
                                                                                          \n
                                                                                        1. \u5b83\u7684\u8f93\u51fa\u503c\u57df\u662f[-1,1]\uff0c\u56e0\u6b64\u5177\u6709\u96f6\u5747\u503c\u3002<\/li>\n\n\n\n
                                                                                        2. \u5f53\u8f93\u5165\u7684\u7edd\u5bf9\u503c\u8f83\u5927\u65f6\uff0c\u8f93\u51fa\u7684\u503c\u63a5\u8fd1\u4e8e1\u6216-1\uff0c\u56e0\u6b64\u5177\u6709\u9971\u548c\u6027\uff0c\u53ef\u4ee5\u6291\u5236\u8fc7\u5927\u7684\u4fe1\u53f7\u3002<\/li>\n\n\n\n
                                                                                        3. \u6b63\u5207\u4f20\u64ad\u51fd\u6570\u662f\u4e00\u4e2a\u5947\u51fd\u6570\uff0c\u5373\u6ee1\u8db3 $f(-x)=-f(x)$\u3002<\/li>\n<\/ol>\n\n\n\n
                                                                                          \u5728\u795e\u7ecf\u7f51\u7edc\u4e2d\uff0c\u6b63\u5207\u4f20\u64ad\u51fd\u6570\u5e38\u7528\u4f5c\u4e2d\u95f4\u5c42\u7684\u6fc0\u6d3b\u51fd\u6570\uff0c\u53ef\u4ee5\u5c06\u8f93\u5165\u7684\u7ebf\u6027\u7ec4\u5408\u6620\u5c04\u5230\u975e\u7ebf\u6027\u7684\u8f93\u51fa\u7a7a\u95f4\uff0c\u589e\u52a0\u7f51\u7edc\u7684\u8868\u8fbe\u80fd\u529b\u3002\u540c\u65f6\uff0c\u7531\u4e8e\u5176\u5177\u6709\u96f6\u5747\u503c\u7684\u7279\u6027\uff0c\u4e5f\u6709\u52a9\u4e8e\u907f\u514d\u68af\u5ea6\u6d88\u5931\u95ee\u9898\u7684\u51fa\u73b0\u3002<\/p>\n\n\n\n
                                                                                          \u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u6b63\u5207\u4f20\u64ad\u51fd\u6570\u7684\u8ba1\u7b97\u5f00\u9500\u8f83\u5927\uff0c\u56e0\u6b64\u5728\u5b9e\u9645\u5e94\u7528\u4e2d\uff0c\u53ef\u80fd\u4f1a\u9009\u62e9\u4f7f\u7528\u5176\u5b83\u7684\u6fc0\u6d3b\u51fd\u6570\uff0c\u4f8b\u5982ReLU\u3001LeakyReLU\u7b49\u3002<\/p>\n\n\n\n\n\n
                                                                                          \"gun<\/figure>\n\n\n\n
                                                                                          \u8bf7\u8be6\u7ec6\u89e3\u91ca\u652f\u6301\u5411\u91cf\uff0c\u5e76\u7ed9\u51fa\u4ee3\u7801\u793a\u4f8b<\/p>\n\n\n\n
                                                                                          \u652f\u6301\u5411\u91cf\u662f\u4e00\u79cd\u7528\u4e8e\u5206\u7c7b\u548c\u56de\u5f52\u7684\u673a\u5668\u5b66\u4e60\u6a21\u578b\u3002\u5176\u4e3b\u8981\u601d\u60f3\u662f\u5c06\u6837\u672c\u6620\u5c04\u5230\u9ad8\u7ef4\u7a7a\u95f4\u4e2d\uff0c\u627e\u5230\u4e00\u4e2a\u8d85\u5e73\u9762\uff0c\u4f7f\u5f97\u4e24\u4e2a\u4e0d\u540c\u7684\u7c7b\u522b\u7684\u6837\u672c\u70b9\u80fd\u591f\u88ab\u5206\u9694\u5f00\u6765\uff0c\u5e76\u4e14\u6700\u5927\u5316\u6837\u672c\u70b9\u5230\u8d85\u5e73\u9762\u7684\u8ddd\u79bb\u3002<\/p>\n\n\n\n
                                                                                          \u5177\u4f53\u6765\u8bf4\uff0c\u652f\u6301\u5411\u91cf\u673a\u5206\u7c7b\u5668\u5c06\u8f93\u5165\u6570\u636e\u6620\u5c04\u5230\u4e00\u4e2a\u9ad8\u7ef4\u7a7a\u95f4\uff0c\u7136\u540e\u5728\u8fd9\u4e2a\u7a7a\u95f4\u4e2d\u627e\u5230\u4e00\u4e2a\u8d85\u5e73\u9762\uff0c\u5c06\u4e0d\u540c\u7684\u7c7b\u522b\u5206\u5f00\u3002\u901a\u5e38\u60c5\u51b5\u4e0b\uff0c\u9ad8\u7ef4\u7a7a\u95f4\u7684\u7ef4\u5ea6\u6bd4\u8f93\u5165\u6570\u636e\u7684\u7ef4\u5ea6\u9ad8\u5f97\u591a\uff0c\u8fd9\u4f7f\u5f97\u5728\u9ad8\u7ef4\u7a7a\u95f4\u4e2d\u627e\u5230\u5206\u79bb\u8d85\u5e73\u9762\u66f4\u5bb9\u6613\u3002\u652f\u6301\u5411\u91cf\u662f\u6307\u79bb\u5206\u79bb\u8d85\u5e73\u9762\u6700\u8fd1\u7684\u4e00\u4e9b\u70b9\uff0c\u5b83\u4eec\u51b3\u5b9a\u4e86\u5206\u79bb\u8d85\u5e73\u9762\u7684\u4f4d\u7f6e\u548c\u65b9\u5411\u3002<\/p>\n\n\n\n
                                                                                          \u4ee3\u7801\u793a\u4f8b\uff1a<\/p>\n\n\n\n
                                                                                          \u4e0b\u9762\u662f\u4f7f\u7528Python\u7684Scikit-learn\u5e93\u5b9e\u73b0\u652f\u6301\u5411\u91cf\u673a\u7684\u793a\u4f8b\u4ee3\u7801\uff1a<\/p>\n\n\n\n
                                                                                          pythonCopy codefrom sklearn import svm\nfrom sklearn.datasets import make_blobs\nimport matplotlib.pyplot as plt\n\n# \u751f\u6210\u6570\u636e\u96c6\nX, y = make_blobs(n_samples=100, centers=2, random_state=6)\n\n# \u8bad\u7ec3\u652f\u6301\u5411\u91cf\u673a\nclf = svm.SVC(kernel='linear', C=1000)\nclf.fit(X, y)\n\n# \u7ed8\u5236\u6570\u636e\u70b9\u548c\u51b3\u7b56\u8fb9\u754c\nplt.scatter(X[:, 0], X[:, 1], c=y, s=30, cmap=plt.cm.Paired)\nax = plt.gca()\nxlim = ax.get_xlim()\nylim = ax.get_ylim()\n\n# \u521b\u5efa\u7f51\u683c\u6765\u8bc4\u4f30\u6a21\u578b\nxx = np.linspace(xlim[0], xlim[1], 30)\nyy = np.linspace(ylim[0], ylim[1], 30)\nYY, XX = np.meshgrid(yy, xx)\nxy = np.vstack([XX.ravel(), YY.ravel()]).T\nZ = clf.decision_function(xy).reshape(XX.shape)\n\n# \u7ed8\u5236\u51b3\u7b56\u8fb9\u754c\u548c\u652f\u6301\u5411\u91cf\nax.contour(XX, YY, Z, colors='k', levels=[-1, 0, 1], alpha=0.5, linestyles=['--', '-', '--'])\nax.scatter(clf.support_vectors_[:, 0], clf.support_vectors_[:, 1], s=100, linewidth=1, facecolors='none', edgecolors='k')\nplt.show()\n<\/code><\/pre>\n\n\n\n
                                                                                          \u6b64\u793a\u4f8b\u751f\u6210\u4e24\u4e2a\u7c07\uff0c\u5e76\u4f7f\u7528\u7ebf\u6027\u652f\u6301\u5411\u91cf\u673a\u5c06\u5b83\u4eec\u5206\u5f00\u3002\u7ed8\u5236\u4e86\u6570\u636e\u70b9\u548c\u51b3\u7b56\u8fb9\u754c\uff0c\u5176\u4e2d\u652f\u6301\u5411\u91cf\u7528\u9ed1\u5706\u5708\u8868\u793a\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"
                                                                                          \u6211\u662f\u8981\u505a\u4eba\u5de5\u667a\u80fd\u4e2d\u7ea7\u6559\u80b2\u7684\u89c6\u9891\uff0c\u5185\u5bb9\u4e2d\u8981 … <\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"om_disable_all_campaigns":false,"_mi_skip_tracking":false,"footnotes":""},"categories":[23],"tags":[],"class_list":["post-888","post","type-post","status-publish","format-standard","hentry","category-material"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/edu.ujhb.org\/index.php?rest_route=\/wp\/v2\/posts\/888","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/edu.ujhb.org\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/edu.ujhb.org\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/edu.ujhb.org\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/edu.ujhb.org\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=888"}],"version-history":[{"count":1,"href":"https:\/\/edu.ujhb.org\/index.php?rest_route=\/wp\/v2\/posts\/888\/revisions"}],"predecessor-version":[{"id":889,"href":"https:\/\/edu.ujhb.org\/index.php?rest_route=\/wp\/v2\/posts\/888\/revisions\/889"}],"wp:attachment":[{"href":"https:\/\/edu.ujhb.org\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=888"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/edu.ujhb.org\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=888"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/edu.ujhb.org\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=888"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}