{"id":9326,"date":"2024-05-29T00:25:54","date_gmt":"2024-05-28T16:25:54","guid":{"rendered":"\/?p=9326"},"modified":"2024-05-29T01:26:47","modified_gmt":"2024-05-28T17:26:47","slug":"%e6%b7%b1%e5%ba%a6%e5%ad%a6%e4%b9%a0%e5%92%8c%e7%bb%8f%e5%85%b8%e7%bb%9f%e8%ae%a1%e5%ad%a6%e6%98%af%e4%b8%80%e5%9b%9e%e4%ba%8b%ef%bc%9f","status":"publish","type":"post","link":"\/?p=9326","title":{"rendered":"2.\u6df1\u5ea6\u5b66\u4e60\u548c\u7ecf\u5178\u7edf\u8ba1\u5b66\u662f\u4e00\u56de\u4e8b\uff1f"},"content":{"rendered":"<h2>\u6df1\u5ea6\u5b66\u4e60\u548c\u7ecf\u5178\u7edf\u8ba1\u5b66\u662f\u4e00\u56de\u4e8b\uff1f<\/h2>\n<p>\u6765\u6e90\uff1a\u673a\u5668\u4e4b\u5fc3<\/p>\n<p>\u672c\u6587<strong><em>\u7ea67000\u5b57<\/em><\/strong>\uff0c\u5efa\u8bae\u9605\u8bfb<strong>10+\u5206\u949f<\/strong><\/p>\n<p>\u672c\u6587\u4f1a\u89e3\u91ca\u4e3a\u4ec0\u4e48\u6df1\u5ea6\u5b66\u4e60\u7684\u57fa\u7840\u5176\u5b9e\u4e0d\u540c\u4e8e\u7edf\u8ba1\u5b66\uff0c\u751a\u81f3\u4e0d\u540c\u4e8e\u7ecf\u5178\u7684\u673a\u5668\u5b66\u4e60\u3002<\/p>\n<blockquote>\n<p>\u6df1\u5ea6\u5b66\u4e60\u548c\u7b80\u5355\u7684\u7edf\u8ba1\u5b66\u662f\u4e00\u56de\u4e8b\u5417\uff1f\u5f88\u591a\u4eba\u53ef\u80fd\u90fd\u6709\u8fd9\u4e2a\u7591\u95ee\uff0c\u6bd5\u7adf\u4e8c\u8005\u8fde\u672f\u8bed\u90fd\u6709\u5f88\u591a\u76f8\u4f3c\u7684\u5730\u65b9\u3002\u5728\u8fd9\u7bc7\u6587\u7ae0\u4e2d\uff0c\u7406\u8bba\u8ba1\u7b97\u673a\u79d1\u5b66\u5bb6\u3001\u54c8\u4f5b\u5927\u5b66\u77e5\u540d\u6559\u6388 Boaz Barak \u8be6\u7ec6\u6bd4\u8f83\u4e86\u6df1\u5ea6\u5b66\u4e60\u4e0e\u7ecf\u5178\u7edf\u8ba1\u5b66\u7684\u5dee\u5f02\uff0c\u8ba4\u4e3a\u201c\u5982\u679c\u7eaf\u7cb9\u4ece\u7edf\u8ba1\u5b66\u89d2\u5ea6\u8ba4\u8bc6\u6df1\u5ea6\u5b66\u4e60\uff0c\u5c31\u4f1a\u5ffd\u7565\u5176\u6210\u529f\u7684\u5173\u952e\u56e0\u7d20\u201d\u3002<\/p>\n<\/blockquote>\n<p><img src=\"https:\/\/lemon-guess.oss-cn-hangzhou.aliyuncs.com\/img\/640-1716822998378-93.webp\" alt=\"\u56fe\u7247\" \/><\/p>\n<p>\u56fe\u6e90\uff1a<a href=\"https:\/\/twitter.com\/YiMaTweets\/status\/1553913464183091200\">https:\/\/twitter.com\/YiMaTweets\/status\/1553913464183091200<\/a><\/p>\n<p>\u6df1\u5ea6\u5b66\u4e60\uff08\u6216\u4e00\u822c\u7684\u673a\u5668\u5b66\u4e60\uff09\u7ecf\u5e38\u88ab\u8ba4\u4e3a\u662f\u7b80\u5355\u7684\u7edf\u8ba1\u5b66\uff0c\u5373\u5b83\u4e0e\u7edf\u8ba1\u5b66\u5bb6\u7814\u7a76\u7684\u57fa\u672c\u662f\u76f8\u540c\u7684\u6982\u5ff5\uff0c\u4f46\u662f\u4f7f\u7528\u4e0e\u7edf\u8ba1\u5b66\u4e0d\u540c\u7684\u672f\u8bed\u6765\u63cf\u8ff0\u3002Rob Tibshirani \u66fe\u603b\u7ed3\u4e86\u4e0b\u9762\u8fd9\u4e2a\u6709\u8da3\u7684\u201c\u8bcd\u6c47\u8868\u201d\uff1a<\/p>\n<p><img src=\"https:\/\/lemon-guess.oss-cn-hangzhou.aliyuncs.com\/img\/640-1716822998378-94.webp\" alt=\"\u56fe\u7247\" \/><\/p>\n<p>\u8868\u4e2d\u7684\u67d0\u4e9b\u5185\u5bb9\u662f\u4e0d\u662f\u5f88\u80fd\u5f15\u8d77\u5171\u9e23\uff1f\u4e8b\u5b9e\u4e0a\u6240\u6709\u4ece\u4e8b\u673a\u5668\u5b66\u4e60\u7684\u4eba\u90fd\u6e05\u695a\uff0cTibshiriani 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\/><\/p>\n<p>\u5bf9\u6df1\u5ea6\u5b66\u4e60\u7684\u6070\u5f53\u8bc4\u4ef7\u4e0d\u662f\u5b83\u7528\u4e0d\u540c\u7684\u8bcd\u6765\u63cf\u8ff0\u65e7\u7684\u7edf\u8ba1\u672f\u8bed\uff0c\u800c\u662f\u5b83\u7528\u8fd9\u4e9b\u672f\u8bed\u6765\u63cf\u8ff0\u5b8c\u5168\u4e0d\u540c\u7684\u8fc7\u7a0b\u3002<\/p>\n<p>\u672c\u6587\u4f1a\u89e3\u91ca\u4e3a\u4ec0\u4e48\u6df1\u5ea6\u5b66\u4e60\u7684\u57fa\u7840\u5176\u5b9e\u4e0d\u540c\u4e8e\u7edf\u8ba1\u5b66\uff0c\u751a\u81f3\u4e0d\u540c\u4e8e\u7ecf\u5178\u7684\u673a\u5668\u5b66\u4e60\u3002\u672c\u6587\u9996\u5148\u8ba8\u8bba\u6a21\u578b\u62df\u5408\u6570\u636e\u65f6\u7684\u300c\u89e3\u91ca\uff08explanation\uff09\u300d\u4efb\u52a1\u548c\u300c\u9884\u6d4b\uff08prediction\uff09\u300d\u4efb\u52a1\u4e4b\u95f4\u7684\u5dee\u5f02\u3002\u63a5\u7740\u8ba8\u8bba\u5b66\u4e60\u8fc7\u7a0b\u7684\u4e24\u4e2a\u573a\u666f\uff1a<\/p>\n<ol>\n<li>\u4f7f\u7528\u7ecf\u9a8c\u98ce\u9669\u6700\u5c0f\u5316\u62df\u5408\u7edf\u8ba1\u6a21\u578b; 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src=\"https:\/\/lemon-guess.oss-cn-hangzhou.aliyuncs.com\/img\/640-1716822998378-96.webp\" alt=\"\u56fe\u7247\" \/><\/p>\n<p>\u6258\u52d2\u5bc6\u548c\u54e5\u767d\u5c3c\u7684\u6a21\u578b\u90fd\u662f\u65e0\u4e0e\u4f26\u6bd4\u7684\u3002\u5982\u679c\u6211\u4eec\u60f3\u901a\u8fc7 \u201c\u9ed1\u76d2\u201d 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\u201c\u8fd0\u52a8\u90e8\u5206\u201d \u6781\u5c11\u3002\u591a\u5e74\u6765\uff0c\u5929\u6587\u5b66\u4e00\u76f4\u662f\u53d1\u5c55\u7edf\u8ba1\u6280\u672f\u7684\u7075\u611f\u6765\u6e90\u3002\u9ad8\u65af\u548c\u52d2\u8ba9\u5fb7\u5206\u522b\u72ec\u7acb\u5730\u5728 1800 \u5e74\u5de6\u53f3\u53d1\u660e\u4e86\u6700\u5c0f\u4e8c\u4e58\u56de\u5f52\uff0c\u4ee5\u9884\u6d4b\u5c0f\u884c\u661f\u548c\u5176\u4ed6\u5929\u4f53\u7684\u8f68\u9053\u30021847 \u5e74\uff0c\u67ef\u897f\u53d1\u660e\u4e86\u68af\u5ea6\u4e0b\u964d\u6cd5\uff0c\u8fd9\u4e5f\u662f\u7531\u5929\u6587\u9884\u6d4b\u63a8\u52a8\u7684\u3002<\/p>\n<p>\u5728\u7269\u7406\u5b66\u4e2d\uff0c\u6709\u65f6\u5b66\u8005\u4eec\u53ef\u4ee5\u638c\u63e1\u5168\u90e8\u7ec6\u8282\uff0c\u4ece\u800c\u627e\u5230 \u201c\u6b63\u786e\u201d \u7684\u7406\u8bba\uff0c\u628a\u9884\u6d4b\u51c6\u786e\u6027\u505a\u5230\u6700\u4f18\uff0c\u5e76\u4e14\u5bf9\u6570\u636e\u505a\u51fa\u6700\u597d\u7684\u89e3\u91ca\u3002\u8fd9\u4e9b\u90fd\u5728\u5965\u5361\u59c6\u5243\u5200\u4e4b\u7c7b\u7684\u89c2\u70b9\u8303\u7574\u5185\uff0c\u53ef\u4ee5\u8ba4\u4e3a\u662f\u5047\u8bbe\u7b80\u5355\u6027\u3001\u9884\u6d4b\u80fd\u529b\u548c\u89e3\u91ca\u6027\u90fd\u76f8\u4e92\u548c\u8c10\u4e00\u81f4\u7684\u3002<\/p>\n<p>\u7136\u800c\uff0c\u5728\u8bb8\u591a\u5176\u5b83\u9886\u57df\uff0c\u89e3\u91ca\u548c\u9884\u6d4b\u8fd9\u4e24\u4e2a\u76ee\u6807\u4e4b\u95f4\u7684\u5173\u7cfb\u5374\u6ca1\u6709\u90a3\u4e48\u548c\u8c10\u3002\u5982\u679c\u53ea\u60f3\u9884\u6d4b\u89c2\u5bdf\u7ed3\u679c\uff0c\u901a\u8fc7 \u201c\u9ed1\u76d2\u201d \u53ef\u80fd\u662f\u6700\u597d\u7684\u3002\u53e6\u4e00\u65b9\u9762\uff0c\u5982\u679c\u60f3\u83b7\u5f97\u89e3\u91ca\u6027\u7684\u4fe1\u606f\uff0c\u5982\u56e0\u679c\u6a21\u578b\u3001\u901a\u7528\u539f\u5219\u6216\u91cd\u8981\u7279\u5f81\uff0c\u90a3\u4e48\u53ef\u4ee5\u7406\u89e3\u548c\u89e3\u91ca\u7684\u6a21\u578b\u53ef\u80fd\u8d8a\u7b80\u5355\u8d8a\u597d\u3002<\/p>\n<p>\u6a21\u578b\u7684\u6b63\u786e\u9009\u62e9\u4e0e\u5426\u53d6\u51b3\u4e8e\u5176\u7528\u9014\u3002\u4f8b\u5982\uff0c\u8003\u8651\u4e00\u4e2a\u5305\u542b\u8bb8\u591a\u4e2a\u4f53\u7684\u9057\u4f20\u8868\u8fbe\u548c\u8868\u578b\uff08\u4f8b\u5982\u67d0\u4e9b\u75be\u75c5\uff09\u7684\u6570\u636e\u96c6\uff0c\u5982\u679c\u76ee\u6807\u662f\u9884\u6d4b\u4e00\u4e2a\u4eba\u751f\u75c5\u7684\u51e0\u7387\uff0c\u90a3\u4e48\u65e0\u8bba\u5b83\u6709\u591a\u590d\u6742\u6216\u4f9d\u8d56\u4e8e\u591a\u5c11\u4e2a\u57fa\u56e0\uff0c\u90fd\u8981\u4f7f\u7528\u9002\u914d\u8be5\u4efb\u52a1\u7684\u6700\u4f73\u9884\u6d4b\u6a21\u578b\u3002\u76f8\u53cd\uff0c\u5982\u679c\u76ee\u7684\u662f\u8bc6\u522b\u4e00\u4e9b\u57fa\u56e0\uff0c\u4ee5\u4fbf\u8fdb\u884c\u8fdb\u4e00\u6b65\u7814\u7a76\uff0c\u90a3\u4e48\u4e00\u4e2a\u590d\u6742\u7684\u975e\u5e38\u7cbe\u786e\u7684 \u201c\u9ed1\u76d2\u201d \u7684\u7528\u5904\u662f\u6709\u9650\u7684\u3002<\/p>\n<p>\u7edf\u8ba1\u5b66\u5bb6 Leo Breiman \u5728 2001 \u5e74\u5173\u4e8e\u7edf\u8ba1\u5efa\u6a21\u7684\u4e24\u79cd\u6587\u5316\u7684\u8457\u540d\u6587\u7ae0\u4e2d\u9610\u8ff0\u4e86\u8fd9\u4e00\u70b9\u3002\u7b2c\u4e00\u79cd\u662f\u201c\u6570\u636e\u5efa\u6a21\u6587\u5316\u201d\uff0c\u4fa7\u91cd\u4e8e\u80fd\u89e3\u91ca\u6570\u636e\u7684\u7b80\u5355\u751f\u6210\u6a21\u578b\u3002\u7b2c\u4e8c\u79cd\u662f\u201c\u7b97\u6cd5\u5efa\u6a21\u6587\u5316\u201d\uff0c\u5bf9\u6570\u636e\u7684\u751f\u6210\u65b9\u5f0f\u4e0d\u53ef\u77e5\uff0c\u4fa7\u91cd\u4e8e\u5bfb\u627e\u80fd\u591f\u9884\u6d4b\u6570\u636e\u7684\u6a21\u578b\uff0c\u65e0\u8bba\u5176\u591a\u4e48\u590d\u6742\u3002<\/p>\n<p><img src=\"https:\/\/lemon-guess.oss-cn-hangzhou.aliyuncs.com\/img\/640-1716822998378-97.webp\" alt=\"\u56fe\u7247\" \/><\/p>\n<p>\u6587\u7ae0\u94fe\u63a5\uff1a<a href=\"https:\/\/projecteuclid.org\/journals\/statistical-science\/volume-16\/issue-3\/Statistical-Modeling--The-Two-Cultures-with-comments-and-a\/10.1214\/ss\/1009213726.full\">https:\/\/projecteuclid.org\/journals\/statistical-science\/volume-16\/issue-3\/Statistical-Modeling--The-Two-Cultures-with-comments-and-a\/10.1214\/ss\/1009213726.full<\/a><\/p>\n<p>Breiman \u8ba4\u4e3a\uff0c\u7edf\u8ba1\u5b66\u8fc7\u4e8e\u53d7\u7b2c\u4e00\u79cd\u6587\u5316\u7684\u652f\u914d\uff0c\u8fd9\u79cd\u5173\u6ce8\u9020\u6210\u4e24\u79cd\u95ee\u9898\uff1a<\/p>\n<ul>\n<li>\u5bfc\u81f4\u4e86\u4e0d\u76f8\u5173\u7684\u7406\u8bba\u548c\u53ef\u7591\u7684\u79d1\u5b66\u7ed3\u8bba\u3002<\/li>\n<li>\u963b\u6b62\u4e86\u7edf\u8ba1\u5b66\u5bb6\u7814\u7a76\u4ee4\u4eba\u5174\u594b\u7684\u65b0\u95ee\u9898\u3002<\/li>\n<\/ul>\n<p>Breiman \u7684\u8bba\u6587\u4e00\u51fa\uff0c\u5c31\u5f15\u8d77\u4e86\u4e00\u4e9b\u4e89\u8bae\u3002\u540c\u4e3a\u7edf\u8ba1\u5b66\u5bb6\u7684 Brad Efron \u56de\u5e94\u8bf4\uff0c\u867d\u7136\u4ed6\u540c\u610f\u4e00\u4e9b\u89c2\u70b9\uff0c\u4f46\u4ed6\u4e5f\u5f3a\u8c03\uff0cBreiman \u7684\u8bba\u70b9\u4f3c\u4e4e\u662f\u53cd\u5bf9\u8282\u4fed\u548c\u79d1\u5b66\u89c1\u89e3\uff0c\u652f\u6301\u82b1\u5927\u529b\u6c14\u5236\u9020\u590d\u6742\u7684\u201c\u9ed1\u76d2\u201d\u3002\u4f46\u5728\u6700\u8fd1\u7684\u4e00\u7bc7\u6587\u7ae0\u4e2d\uff0cEfron \u6452\u5f03\u4e86\u4e4b\u524d\u7684\u89c2\u70b9\uff0c\u627f\u8ba4 Breima \u66f4\u6709\u5148\u89c1\u4e4b\u660e\uff0c\u56e0\u4e3a\u201c21 \u4e16\u7eaa\u7edf\u8ba1\u5b66\u7684\u7126\u70b9\u90fd\u805a\u7126\u5728\u9884\u6d4b\u7b97\u6cd5\u4e0a\uff0c\u5728\u5f88\u5927\u7a0b\u5ea6\u4e0a\u6cbf\u7740 Breiman \u63d0\u51fa\u7684\u8def\u7ebf\u6f14\u8fdb\u201d\u3002<\/p>\n<h2><strong>2\u3001\u7ecf\u5178\u548c\u73b0\u4ee3\u9884\u6d4b\u6a21\u578b<\/strong><\/h2>\n<p>\u673a\u5668\u5b66\u4e60\uff0c\u65e0\u8bba\u662f\u4e0d\u662f\u6df1\u5ea6\u5b66\u4e60\uff0c\u90fd\u6cbf\u7740 Breiman \u7684\u7b2c\u4e8c\u79cd\u89c2\u70b9\u6f14\u8fdb\uff0c\u5373\u4ee5\u9884\u6d4b\u4e3a\u91cd\u70b9\u3002\u8fd9\u79cd\u6587\u5316\u6709\u7740\u60a0\u4e45\u7684\u5386\u53f2\u3002\u4f8b\u5982\uff0cDuda \u548c Hart \u5728 1973 \u5e74\u51fa\u7248\u7684\u6559\u79d1\u4e66\u548c Highleyman 1962 \u5e74\u7684\u8bba\u6587\u5c31\u5199\u5230\u4e86\u4e0b\u56fe\u4e2d\u7684\u5185\u5bb9\uff0c\u8fd9\u5bf9\u4e8e\u4eca\u5929\u7684\u6df1\u5ea6\u5b66\u4e60\u7814\u7a76\u8005\u6765\u8bf4\u662f\u975e\u5e38\u5bb9\u6613\u7406\u89e3\u7684\uff1a<\/p>\n<p><img src=\"https:\/\/lemon-guess.oss-cn-hangzhou.aliyuncs.com\/img\/640-1716822998378-98.webp\" alt=\"\u56fe\u7247\" \/><\/p>\n<p>Duda \u548c Hart \u7684\u6559\u79d1\u4e66\u300aPattern classification and scene analysis\u300b\u548c Highleyman 1962 \u5e74\u7684\u8bba\u6587\u300aThe Design and Analysis of Pattern Recognition Experiments\u300b\u4e2d\u7684\u7247\u6bb5\u3002<\/p>\n<p>\u7c7b\u4f3c\u5730\uff0c\u4e0b\u56fe\u4e2d\u7684 Highleyman \u7684\u624b\u5199\u5b57\u7b26\u6570\u636e\u96c6\u548c\u7528\u4e8e\u62df\u5408\u5b83\u7684\u67b6\u6784 Chow\uff081962\uff09\uff08\u51c6\u786e\u7387\u7ea6\u4e3a 58%\uff09\u4e5f\u4f1a\u5f15\u8d77\u5f88\u591a\u4eba\u7684\u5171\u9e23\u3002<\/p>\n<p><img src=\"https:\/\/lemon-guess.oss-cn-hangzhou.aliyuncs.com\/img\/640-1716822998378-99.webp\" alt=\"\u56fe\u7247\" \/><\/p>\n<h2><strong>3\u3001\u4e3a\u4ec0\u4e48\u6df1\u5ea6\u5b66\u4e60\u4e0e\u4f17\u4e0d\u540c\uff1f<\/strong><\/h2>\n<p>1992 \u5e74\uff0cGeman\u3001Bienenstock \u548c Doursat \u5199\u4e86\u4e00\u7bc7\u5173\u4e8e\u795e\u7ecf\u7f51\u7edc\u7684\u60b2\u89c2\u6587\u7ae0\uff0c\u8ba4\u4e3a \u201c\u5f53\u524d\u7684\u524d\u9988\u795e\u7ecf\u7f51\u7edc\u5728\u5f88\u5927\u7a0b\u5ea6\u4e0a\u4e0d\u8db3\u4ee5\u89e3\u51b3\u673a\u5668\u611f\u77e5\u548c\u673a\u5668\u5b66\u4e60\u4e2d\u7684\u96be\u9898\u201d\u3002\u5177\u4f53\u6765\u8bf4\uff0c\u4ed6\u4eec\u8ba4\u4e3a\u901a\u7528\u795e\u7ecf\u7f51\u7edc\u5728\u5904\u7406\u56f0\u96be\u4efb\u52a1\u65b9\u9762\u4e0d\u4f1a\u6210\u529f\uff0c\u800c\u5b83\u4eec\u6210\u529f\u7684\u552f\u4e00\u9014\u5f84\u662f\u901a\u8fc7\u4eba\u5de5\u8bbe\u8ba1\u7684\u7279\u5f81\u3002\u7528\u4ed6\u4eec\u7684\u8bdd\u8bf4\uff1a\u201c\u91cd\u8981\u5c5e\u6027\u5fc5\u987b\u662f\u5185\u7f6e\u7684\u6216\u201c\u786c\u8fde\u63a5\u7684\u201d\u2026\u2026 \u800c\u4e0d\u662f\u4ee5\u4efb\u4f55\u7edf\u8ba1\u610f\u4e49\u4e0a\u7684\u65b9\u5f0f\u5b66\u4e60\u3002\u201d \u73b0\u5728\u770b\u6765 Geman \u7b49\u4eba\u5b8c\u5168\u9519\u4e86\uff0c\u4f46\u66f4\u6709\u610f\u601d\u7684\u662f\u4e86\u89e3\u4ed6\u4eec\u4e3a\u4ec0\u4e48\u9519\u4e86\u3002<\/p>\n<p>\u6df1\u5ea6\u5b66\u4e60\u786e\u5b9e\u4e0d\u540c\u4e8e\u5176\u5b83\u5b66\u4e60\u65b9\u6cd5\u3002\u867d\u7136\u6df1\u5ea6\u5b66\u4e60\u4f3c\u4e4e\u53ea\u662f\u9884\u6d4b\uff0c\u5c31\u50cf\u6700\u8fd1\u90bb\u6216\u968f\u673a\u68ee\u6797\u4e00\u6837\uff0c\u4f46\u5b83\u53ef\u80fd\u6709\u66f4\u591a\u7684\u590d\u6742\u53c2\u6570\u3002\u8fd9\u770b\u8d77\u6765\u4f3c\u4e4e\u53ea\u662f\u91cf\u7684\u5dee\u5f02\uff0c\u800c\u4e0d\u662f\u8d28\u7684\u5dee\u5f02\u3002\u4f46\u5728\u7269\u7406\u5b66\u4e2d\uff0c\u4e00\u65e6\u5c3a\u5ea6\u53d8\u5316\u4e86\u51e0\u4e2a\u6570\u91cf\u7ea7\uff0c\u901a\u5e38\u5c31\u9700\u8981\u5b8c\u5168\u4e0d\u540c\u7684\u7406\u8bba\uff0c\u6df1\u5ea6\u5b66\u4e60\u4e5f\u662f\u5982\u6b64\u3002\u6df1\u5ea6\u5b66\u4e60\u4e0e\u7ecf\u5178\u6a21\u578b\uff08\u53c2\u6570\u5316\u6216\u975e\u53c2\u6570\u5316\uff09\u7684\u57fa\u7840\u8fc7\u7a0b\u5b8c\u5168\u4e0d\u540c\uff0c\u867d\u7136\u5b83\u4eec\u7684\u6570\u5b66\u65b9\u7a0b\uff08\u548c Python \u4ee3\u7801\uff09\u5728\u66f4\u9ad8\u5c42\u6b21\u4e0a\u6765\u770b\u662f\u76f8\u540c\u7684\u3002<\/p>\n<p>\u4e3a\u4e86\u8bf4\u660e\u8fd9\u4e00\u70b9\uff0c\u4e0b\u9762\u8003\u8651\u4e24\u4e2a\u4e0d\u540c\u7684\u573a\u666f\uff1a\u62df\u5408\u7edf\u8ba1\u6a21\u578b\u548c\u5411\u5b66\u751f\u6559\u6388\u6570\u5b66\u3002<\/p>\n<h3><strong>\u573a\u666f A\uff1a\u62df\u5408\u4e00\u4e2a\u7edf\u8ba1\u6a21\u578b<\/strong><\/h3>\n<p>\u901a\u8fc7\u6570\u636e\u53bb\u62df\u5408\u4e00\u4e2a\u7edf\u8ba1\u6a21\u578b\u7684\u5178\u578b\u6b65\u9aa4\u5982\u4e0b\uff1a<\/p>\n<p>1.\u8fd9\u91cc\u6709\u4e00\u4e9b\u6570\u636e$X,y$($X$\u662f$n*p$\u7684\u77e9\u9635\uff1b$y$\u662f$n$\u7ef4\u5411\u91cf\uff0c\u5373\u7c7b\u522b\u6807\u7b7e\u3002\u628a\u6570\u636e\u8ba4\u4e3a\u662f\u6765\u81ea\u67d0\u4e2a\u6709\u7ed3\u6784\u4e14\u5305\u542b\u566a\u58f0\u7684\u6a21\u578b\uff0c\u5c31\u662f\u8981\u53bb\u62df\u5408\u7684\u6a21\u578b)<\/p>\n<p>2.\u4f7f\u7528\u4e0a\u9762\u7684\u6570\u636e\u62df\u5408\u4e00\u4e2a\u6a21\u578b $\\hat{f}$\uff0c\u5e76\u7528\u4f18\u5316\u7b97\u6cd5\u6765\u6700\u5c0f\u5316\u7ecf\u9a8c\u98ce\u9669\u3002\u5c31\u662f\u8bf4\u901a\u8fc7\u4f18\u5316\u7b97\u6cd5\u627e\u5230\u8fd9\u6837\u7684\uff0c\u4f7f\u5f97 $L(\\hat{f})+R(\\hat{f})$\u6700\u5c0f\uff0c$L(\u00b7)$\u4ee3\u8868\u635f\u5931\uff08\u8868\u660e\u9884\u6d4b\u503c\u6709\u591a\u63a5\u8fd1\u771f\u5b9e\u503c\uff09\uff0c $R(\\hat{f})$\u662f\u53ef\u9009\u7684\u6b63\u5219\u5316\u9879\u3002<\/p>\n<ol start=\"3\">\n<li>\u6a21\u578b\u7684\u603b\u4f53\u635f\u5931\u8d8a\u5c0f\u8d8a\u597d\uff0c\u5373\u6cdb\u5316\u8bef\u5dee$\\mathbb{E}|L(\\hat{f}(x),y)$\u7684\u503c\u76f8\u5bf9\u6700\u5c0f\u3002<\/li>\n<\/ol>\n<p><img src=\"https:\/\/lemon-guess.oss-cn-hangzhou.aliyuncs.com\/img\/640-1716822998379-110.webp\" alt=\"\u56fe\u7247\" \/><\/p>\n<p>Effron \u4ece\u5305\u542b\u566a\u58f0\u7684\u89c2\u6d4b\u4e2d\u6062\u590d\u725b\u987f\u7b2c\u4e00\u5b9a\u5f8b\u7684\u5c55\u793a\u56fe<\/p>\n<p>\u8fd9\u4e2a\u975e\u5e38\u901a\u7528\u7684\u8303\u4f8b\u5176\u5b9e\u5305\u542b\u8bb8\u591a\u5185\u5bb9\uff0c\u5982\u6700\u5c0f\u4e8c\u4e58\u7ebf\u6027\u56de\u5f52\u3001\u6700\u8fd1\u90bb\u3001\u795e\u7ecf\u7f51\u7edc\u8bad\u7ec3\u7b49\u7b49\u3002\u5728\u7ecf\u5178\u7edf\u8ba1\u573a\u666f\u4e2d\uff0c\u6211\u4eec\u901a\u5e38\u4f1a\u78b0\u5230\u4e0b\u9762\u7684\u60c5\u51b5\uff1a<\/p>\n<p>\u6743\u8861\uff1a\u5047\u8bbe$\\mathcal{F}$\u662f\u7ecf\u8fc7\u4f18\u5316\u7684\u6a21\u578b\u96c6\u5408(\u5982\u679c\u51fd\u6570\u662f\u975e\u51f8\u7684\u6216\u5305\u542b\u6b63\u5219\u5316\u9879\uff0c\u7cbe\u5fc3\u9009\u62e9\u7b97\u6cd5\u548c\u6b63\u5219\u5316\uff0c\u53ef\u5f97\u5230\u6a21\u578b\u96c6$\\mathcal{F}$\u3002\u7684$\\mathcal{F}$\u504f\u5dee\u662f\u5143\u7d20\u6240$\\hat{f}\\in\\mathcal{F}$\u80fd\u8fbe\u5230\u7684\u6700\u63a5\u8fd1\u771f\u503c\u7684\u8fd1\u4f3c\u503c\u3002\u96c6\u5408$\\mathcal{F}$\u8d8a\u5927\uff0c\u504f\u5dee\u8d8a\u5c0f\uff0c\u5e76\u4e14\u53ef\u80fd\u4e3a 0(\u5982\u679c$f_0\\in\\mathcal{F}$)\u3002<\/p>\n<p>\u7136\u800c\uff0c$\\mathcal{F}$\u8d8a\u5927\uff0c\u9700\u8981\u7f29\u5c0f\u5176\u6210\u5458\u8303\u56f4\u7684\u6837\u672c\u8d8a\u591a\uff0c\u56e0\u6b64\u7b97\u6cd5\u8f93\u51fa\u6a21\u578b\u7684\u65b9\u5dee\u8d8a\u5927\u3002\u603b\u4f53\u6cdb\u5316\u8bef\u5dee\u662f\u504f\u5dee\u548c\u65b9\u5dee\u7684\u603b\u548c\u3002\u56e0\u6b64\uff0c\u7edf\u8ba1\u5b66\u4e60\u901a\u5e38\u662f Bias-Variance \u6743\u8861\uff0c\u6b63\u786e\u7684\u6a21\u578b\u590d\u6742\u5ea6\u662f\u5c06\u603b\u4f53\u8bef\u5dee\u964d\u81f3\u6700\u4f4e\u3002\u4e8b\u5b9e\u4e0a\uff0cGeman \u7b49\u4eba\u8bc1\u660e\u4e86\u5176\u5bf9\u795e\u7ecf\u7f51\u7edc\u7684\u60b2\u89c2\u6001\u5ea6\uff0c\u4ed6\u4eec\u8ba4\u4e3a\uff1aBias-Variance \u56f0\u5883\u9020\u6210\u7684\u57fa\u672c\u9650\u5236\u9002\u7528\u4e8e\u6240\u6709\u975e\u53c2\u6570\u63a8\u7406\u6a21\u578b\uff0c\u5305\u62ec\u795e\u7ecf\u7f51\u7edc\u3002<\/p>\n<p>\u201c\u591a\u591a\u76ca\u5584\u201d\u5e76\u4e0d\u603b\u662f\u6210\u7acb\uff1a\u5728\u7edf\u8ba1\u5b66\u4e60\u4e2d\uff0c\u66f4\u591a\u7684\u7279\u5f81\u6216\u6570\u636e\u5e76\u4e0d\u4e00\u5b9a\u4f1a\u63d0\u9ad8\u6027\u80fd\u3002\u4f8b\u5982\uff0c\u4ece\u5305\u542b\u8bb8\u591a\u4e0d\u76f8\u5173\u7279\u5f81\u7684\u6570\u636e\u4e2d\u5b66\u4e60\u662f\u5f88\u96be\u7684\u3002\u7c7b\u4f3c\u5730\uff0c\u4ece\u6df7\u5408\u6a21\u578b\u4e2d\u5b66\u4e60\uff0c\u5176\u4e2d\u6570\u636e\u6765\u81ea\u4e24\u4e2a\u5206\u5e03\u4e2d\u7684\u4e00\u4e2a\uff08\u5982$y=f_0(x)$\u548c$y=f_1(x)$)\uff0c\u6bd4\u72ec\u7acb\u5b66\u4e60\u6bcf\u4e2a\u5206\u5e03\u66f4\u96be\u3002<\/p>\n<p>\u6536\u76ca\u9012\u51cf\uff1a\u5728\u5f88\u591a\u60c5\u51b5\u4e2d\uff0c\u5c06\u9884\u6d4b\u566a\u58f0\u964d\u4f4e\u5230\u6c34\u5e73$\\epsilon$\u6240\u9700\u7684\u6570\u636e\u70b9\u6570\u91cf\u4e0e\u53c2\u6570$k$\u548c$\\epsilon$\u662f\u6709\u5173\u7684\uff0c\u5373\u6570\u636e\u70b9\u6570\u91cf\u7ea6\u7b49\u4e8e$k\/\\epsilon^2$\u3002\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u9700\u8981\u5927\u7ea6 k \u4e2a\u6837\u672c\u624d\u80fd\u542f\u52a8\uff0c\u4f46\u4e00\u65e6\u8fd9\u6837\u505a\uff0c\u5c31\u9762\u4e34\u7740\u56de\u62a5\u9012\u51cf\u7684\u60c5\u51b5\uff0c\u5373\u5982\u679c\u9700\u8981$n$\u4e2a\u70b9\u624d\u80fd\u8fbe\u5230 90% \u7684\u51c6\u786e\u7387\uff0c\u5219\u9700\u8981\u5927\u7ea6\u989d\u5916\u7684$3n$\u4e2a\u70b9\u6765\u5c06\u51c6\u786e\u7387\u63d0\u9ad8\u5230 95%\u3002\u4e00\u822c\u6765\u8bf4\uff0c\u968f\u7740\u8d44\u6e90\u589e\u52a0\uff08\u65e0\u8bba\u662f\u6570\u636e\u3001\u6a21\u578b\u590d\u6742\u5ea6\u8fd8\u662f\u8ba1\u7b97)\uff0c\u4eba\u4eec\u5e0c\u671b\u83b7\u5f97\u8d8a\u6765\u8d8a\u7cbe\u7ec6\u7684\u533a\u5206\uff0c\u800c\u4e0d\u662f\u89e3\u9501\u7279\u5b9a\u7684\u65b0\u529f\u80fd\u3002<\/p>\n<p>\u5bf9\u635f\u5931\u3001\u6570\u636e\u7684\u4e25\u91cd\u4f9d\u8d56\u6027\uff1a\u5f53\u5c06\u6a21\u578b\u62df\u5408\u5230\u9ad8\u7ef4\u6570\u636e\u65f6\uff0c\u4efb\u4f55\u5c0f\u7ec6\u8282\u90fd\u53ef\u80fd\u4f1a\u4ea7\u751f\u5f88\u5927\u7684\u5dee\u5f02\u3002L1 \u6216 L2 \u6b63\u5219\u5316\u5668\u7b49\u9009\u62e9\u5f88\u91cd\u8981\uff0c\u66f4\u4e0d\u7528\u8bf4\u4f7f\u7528\u5b8c\u5168\u4e0d\u540c\u7684\u6570\u636e\u96c6\u3002\u4e0d\u540c\u6570\u91cf\u7684\u9ad8\u7ef4\u4f18\u5316\u5668\u76f8\u4e92\u4e4b\u95f4\u4e5f\u975e\u5e38\u4e0d\u540c\u3002<\/p>\n<p>\u6570\u636e\u662f\u76f8\u5bf9 \u201c\u5355\u7eaf\u201d \u7684\uff1a\u901a\u5e38\u4f1a\u5047\u8bbe\u6570\u636e\u662f\u72ec\u7acb\u4e8e\u67d0\u4e9b\u5206\u5e03\u8fdb\u884c\u91c7\u6837\u7684\u3002\u867d\u7136\u9760\u8fd1\u51b3\u7b56\u8fb9\u754c\u7684\u70b9\u5f88\u96be\u5206\u7c7b\uff0c\u4f46\u8003\u8651\u5230\u9ad8\u7ef4\u5ea6\u4e0a\u6d4b\u91cf\u96c6\u4e2d\u73b0\u8c61\uff0c\u53ef\u4ee5\u8ba4\u4e3a\u5927\u591a\u6570\u70b9\u7684\u8ddd\u79bb\u90fd\u662f\u76f8\u8fd1\u7684\u3002\u56e0\u6b64\u5728\u7ecf\u5178\u7684\u6570\u636e\u5206\u5e03\u4e2d\uff0c\u6570\u636e\u70b9\u95f4\u7684\u8ddd\u79bb\u5dee\u5f02\u662f\u4e0d\u5927\u7684\u3002\u7136\u800c\uff0c\u6df7\u5408\u6a21\u578b\u53ef\u4ee5\u663e\u793a\u8fd9\u79cd\u5dee\u5f02\uff0c\u56e0\u6b64\uff0c\u4e0e\u4e0a\u8ff0\u5176\u4ed6\u95ee\u9898\u4e0d\u540c\uff0c\u8fd9\u79cd\u5dee\u5f02\u5728\u7edf\u8ba1\u4e2d\u5f88\u5e38\u89c1\u3002<\/p>\n<h3><strong>\u573a\u666f B\uff1a\u5b66\u4e60\u6570\u5b66<\/strong><\/h3>\n<p>\u5728\u8fd9\u4e2a\u573a\u666f\u4e2d\uff0c\u6211\u4eec\u5047\u8bbe\u4f60\u60f3\u901a\u8fc7\u4e00\u4e9b\u8bf4\u660e\u548c\u7ec3\u4e60\u6765\u6559\u5b66\u751f\u6570\u5b66\uff08\u5982\u8ba1\u7b97\u5bfc\u6570\uff09\u3002\u8fd9\u4e2a\u573a\u666f\u867d\u7136\u6ca1\u6709\u6b63\u5f0f\u5b9a\u4e49\uff0c\u4f46\u6709\u4e00\u4e9b\u5b9a\u6027\u7279\u5f81\uff1a<\/p>\n<p><img src=\"https:\/\/lemon-guess.oss-cn-hangzhou.aliyuncs.com\/img\/640-1716822998379-124.webp\" alt=\"\u56fe\u7247\" \/><\/p>\n<p>\u5b66\u4e60\u4e00\u9879\u6280\u80fd\uff0c\u800c\u4e0d\u662f\u53bb\u8fd1\u4f3c\u4e00\u4e2a\u7edf\u8ba1\u5206\u5e03\uff1a\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u5b66\u751f\u5b66\u4e60\u7684\u662f\u4e00\u79cd\u6280\u80fd\uff0c\u800c\u4e0d\u662f\u67d0\u4e2a\u91cf\u7684\u4f30\u8ba1 \/ \u9884\u6d4b\u3002\u5177\u4f53\u6765\u8bf4\uff0c\u5373\u4f7f\u5c06\u7ec3\u4e60\u6620\u5c04\u5230\u89e3\u7684\u51fd\u6570\u4e0d\u80fd\u88ab\u7528\u4f5c\u89e3\u51b3\u67d0\u4e9b\u672a\u77e5\u4efb\u52a1\u7684\u201c\u9ed1\u76d2\u201d\uff0c\u4f46\u5b66\u751f\u5728\u89e3\u51b3\u8fd9\u4e9b\u95ee\u9898\u65f6\u5f62\u6210\u7684\u601d\u7ef4\u6a21\u5f0f\u4ecd\u7136\u5bf9\u672a\u77e5\u4efb\u52a1\u662f\u6709\u7528\u7684\u3002<\/p>\n<p>\u591a\u591a\u76ca\u5584\uff1a\u4e00\u822c\u6765\u8bf4\uff0c\u505a\u9898\u8d8a\u591a\u3001\u9898\u578b\u6d89\u730e\u8d8a\u5e7f\u7684\u5b66\u751f\u8868\u73b0\u8d8a\u597d\u3002\u540c\u65f6\u505a\u4e00\u4e9b\u5fae\u79ef\u5206\u9898\u548c\u4ee3\u6570\u9898\uff0c\u4e0d\u4f1a\u5bfc\u81f4\u5b66\u751f\u7684\u5fae\u79ef\u5206\u6210\u7ee9\u4e0b\u964d\uff0c\u76f8\u53cd\u53ef\u80fd\u5e2e\u52a9\u5176\u5fae\u79ef\u5206\u6210\u7ee9\u63d0\u5347\u3002<\/p>\n<p>\u4ece\u63d0\u5347\u80fd\u529b\u5230\u81ea\u52a8\u5316\u8868\u793a\uff1a\u867d\u7136\u5728\u67d0\u4e9b\u60c5\u51b5\u4e0b\uff0c\u89e3\u51b3\u95ee\u9898\u7684\u56de\u62a5\u4e5f\u4f1a\u9012\u51cf\uff0c\u4f46\u5b66\u751f\u7684\u5b66\u4e60\u4f1a\u7ecf\u5386\u51e0\u4e2a\u9636\u6bb5\u3002\u6709\u4e00\u4e2a\u9636\u6bb5\uff0c\u89e3\u51b3\u4e00\u4e9b\u95ee\u9898\u6709\u52a9\u4e8e\u7406\u89e3\u6982\u5ff5\u5e76\u89e3\u9501\u65b0\u7684\u80fd\u529b\u3002\u6b64\u5916\uff0c\u5f53\u5b66\u751f\u91cd\u590d\u67d0\u4e00\u7279\u5b9a\u7c7b\u578b\u7684\u95ee\u9898\u65f6\uff0c\u4ed6\u4eec\u89c1\u5230\u540c\u7c7b\u95ee\u9898\u5c31\u4f1a\u5f62\u6210\u81ea\u52a8\u5316\u7684\u89e3\u9898\u6d41\u7a0b\uff0c\u4ece\u4e4b\u524d\u7684\u80fd\u529b\u63d0\u5347\u8f6c\u53d8\u4e3a\u81ea\u52a8\u5316\u89e3\u9898\u3002<\/p>\n<p>\u8868\u73b0\u72ec\u7acb\u4e8e\u6570\u636e\u548c\u635f\u5931\uff1a\u6559\u6388\u6570\u5b66\u6982\u5ff5\u7684\u65b9\u6cd5\u4e0d\u6b62\u4e00\u79cd\u3002\u4f7f\u7528\u4e0d\u540c\u4e66\u3001\u6559\u80b2\u65b9\u6cd5\u6216\u8bc4\u5206\u7cfb\u7edf\u5b66\u4e60\u7684\u5b66\u751f\u6700\u7ec8\u53ef\u4ee5\u5b66\u4e60\u5230\u76f8\u540c\u7684\u5185\u5bb9\u4ee5\u53ca\u76f8\u4f3c\u7684\u6570\u5b66\u80fd\u529b\u3002<\/p>\n<p>\u6709\u4e9b\u95ee\u9898\u66f4\u56f0\u96be\uff1a\u5728\u6570\u5b66\u7ec3\u4e60\u4e2d\uff0c\u6211\u4eec\u7ecf\u5e38\u770b\u5230\u4e0d\u540c\u5b66\u751f\u89e3\u51b3\u540c\u4e00\u95ee\u9898\u7684\u65b9\u5f0f\u4e4b\u95f4\u5b58\u5728\u7740\u5f88\u5f3a\u7684\u76f8\u5173\u6027\u3002\u5bf9\u4e8e\u4e00\u4e2a\u95ee\u9898\u6765\u8bf4\uff0c\u4f3c\u4e4e\u786e\u5b9e\u5b58\u5728\u4e00\u4e2a\u56fa\u6709\u7684\u96be\u5ea6\u6c34\u5e73\uff0c\u4ee5\u53ca\u4e00\u4e2a\u5bf9\u5b66\u4e60\u6700\u6709\u5229\u7684\u81ea\u7136\u96be\u5ea6\u9012\u8fdb\u3002<\/p>\n<h2><strong>4\u3001\u6df1\u5ea6\u5b66\u4e60\u66f4\u50cf\u662f\u7edf\u8ba1\u4f30\u8ba1\u8fd8\u662f\u5b66\u751f\u5b66\u4e60\u6280\u80fd\uff1f<\/strong><\/h2>\n<p>\u4e0a\u9762\u4e24\u4e2a\u573a\u666f\u7684\u6bd4\u55bb\u4e2d\uff0c\u54ea\u4e00\u4e2a\u7528\u6765\u63cf\u8ff0\u73b0\u4ee3\u6df1\u5ea6\u5b66\u4e60\u66f4\u6070\u5f53\uff1f\u5177\u4f53\u6765\u8bf4\uff0c\u5b83\u6210\u529f\u7684\u539f\u56e0\u662f\u4ec0\u4e48\uff1f\u7edf\u8ba1\u6a21\u578b\u62df\u5408\u53ef\u4ee5\u5f88\u597d\u5730\u4f7f\u7528\u6570\u5b66\u548c\u4ee3\u7801\u6765\u8868\u8fbe\u3002\u5b9e\u9645\u4e0a\uff0c\u89c4\u8303\u7684 Pytorch \u8bad\u7ec3\u5faa\u73af\u901a\u8fc7\u7ecf\u9a8c\u98ce\u9669\u6700\u5c0f\u5316\u8bad\u7ec3\u6df1\u5ea6\u7f51\u7edc\uff1a<\/p>\n<p><img src=\"https:\/\/lemon-guess.oss-cn-hangzhou.aliyuncs.com\/img\/640-1716822998379-125.webp\" alt=\"\u56fe\u7247\" \/><\/p>\n<p>\u5728\u66f4\u6df1\u7684\u5c42\u6b21\u4e0a\uff0c\u8fd9\u4e24\u79cd\u573a\u666f\u4e4b\u95f4\u7684\u5173\u7cfb\u5e76\u4e0d\u6e05\u695a\u3002\u4e3a\u4e86\u66f4\u5177\u4f53\uff0c\u8fd9\u91cc\u4ee5\u4e00\u4e2a\u7279\u5b9a\u7684\u5b66\u4e60\u4efb\u52a1\u4e3a\u4f8b\u3002\u8003\u8651\u4f7f\u7528 \u201c\u81ea\u76d1\u7763\u5b66\u4e60 + \u7ebf\u6027\u63a2\u6d4b\u201d \u65b9\u6cd5\u8bad\u7ec3\u7684\u5206\u7c7b\u7b97\u6cd5\u3002\u5177\u4f53\u7b97\u6cd5\u8bad\u7ec3\u5982\u4e0b\uff1a<\/p>\n<ol>\n<li>\n<p>\u5047\u8bbe\u6570\u636e\u662f\u4e00\u4e2a\u5e8f\u5217${{(x_i,y_i)}},{i=1...n}$\uff0c\u5176\u4e2d$x_i\\in\\mathbb{R}^p$\u662f\u67d0\u4e2a\u6570\u636e\u70b9\uff08\u6bd4\u5982\u4e00\u5f20\u56fe\u7247\uff09\uff0c$y_i$\u662f\u6807\u7b7e\u3002<\/p>\n<\/li>\n<li>\n<p>\u9996\u5148\u5f97\u5230\u8868\u793a\u51fd\u6570$\\hat{r}:\\mathbb{R}^p\\rightarrow\\mathbb{R}^d$\u7684\u6df1\u5ea6\u795e\u7ecf\u7f51\u7edc\u3002\u901a\u8fc7\u6700\u5c0f\u5316\u67d0\u79cd\u7c7b\u578b\u7684\u81ea\u76d1\u7763\u635f\u5931\u51fd\u6570\uff0c\u4ec5\u4f7f\u7528\u6570\u636e\u70b9${x_1,...,x_n}$\u800c\u4e0d\u4f7f\u7528\u6807\u7b7e\u6765\u8bad\u7ec3\u8be5\u51fd\u6570\u3002\u8fd9\u79cd\u635f\u5931\u51fd\u6570\u7684\u4f8b\u5b50\u662f\u91cd\u5efa\uff08\u7528\u5176\u5b83\u8f93\u5165\u6062\u590d\u8f93\u5165\uff09\u6216\u5bf9\u6bd4\u5b66\u4e60\uff08\u6838\u5fc3\u601d\u60f3\u662f\u6b63\u6837\u672c\u548c\u8d1f\u6837\u672c\u5728\u7279\u5f81\u7a7a\u95f4\u5bf9\u6bd4\uff0c\u5b66\u4e60\u6837\u672c\u7684\u7279\u5f81\u8868\u793a)\u3002<\/p>\n<\/li>\n<li>\n<p>\u4f7f\u7528\u5b8c\u6574\u7684\u6807\u8bb0\u6570\u636e${(x_i,y_i)}$\u62df\u5408\u7ebf\u6027\u5206\u7c7b\u5668$\\hat{\\ell}:\\hat{r}:\\mathbb{R}^d\\rightarrow\\mathbb{R}^c$\uff08$C$\u662f\u7c7b\u6570\uff09\uff0c\u4ee5\u6700\u5c0f\u5316\u4ea4\u53c9\u71b5\u635f\u5931\u3002\u6211\u4eec\u7684\u6700\u7ec8\u5206\u7c7b\u5668\u662f:$x\\rightarrow \\mathbf{arg}\\;\\mathbf{max}\\hat{\\ell}(\\hat{r}(x))$<\/p>\n<\/li>\n<\/ol>\n<p>\u6b65\u9aa4 3 \u4ec5\u9002\u7528\u4e8e\u7ebf\u6027\u5206\u7c7b\u5668\uff0c\u56e0\u6b64 \u201c\u9b54\u672f\u201d \u53d1\u751f\u5728\u6b65\u9aa4 2 \u4e2d\uff08\u6df1\u5ea6\u7f51\u7edc\u7684\u81ea\u76d1\u7763\u5b66\u4e60\uff09\u3002\u5728\u81ea\u76d1\u7763\u5b66\u4e60\u4e2d\u6709\u4e9b\u91cd\u8981\u5c5e\u6027\uff1a<\/p>\n<p><strong>\u5b66\u4e60\u4e00\u9879\u6280\u80fd\u800c\u4e0d\u662f\u53bb\u8fd1\u4f3c\u4e00\u4e2a\u51fd\u6570\uff1a<\/strong>\u81ea\u76d1\u7763\u5b66\u4e60\u4e0d\u662f\u903c\u8fd1\u51fd\u6570\uff0c\u800c\u662f\u5b66\u4e60\u53ef\u7528\u4e8e\u5404\u79cd\u4e0b\u6e38\u4efb\u52a1\u7684\u8868\u793a(\u8fd9\u662f\u81ea\u7136\u8bed\u8a00\u5904\u7406\u7684\u4e3b\u5bfc\u8303\u5f0f)\u3002\u901a\u8fc7\u7ebf\u6027\u63a2\u6d4b\u3001\u5fae\u8c03\u6216\u6fc0\u52b1\u83b7\u5f97\u4e0b\u6e38\u4efb\u52a1\u662f\u6b21\u8981\u7684\u3002<\/p>\n<p><strong>\u591a\u591a\u76ca\u5584\uff1a<\/strong>\u5728\u81ea\u76d1\u7763\u5b66\u4e60\u4e2d\uff0c\u8868\u793a\u8d28\u91cf\u968f\u7740\u6570\u636e\u91cf\u7684\u589e\u52a0\u800c\u63d0\u9ad8\uff0c\u4e0d\u4f1a\u56e0\u4e3a\u6df7\u5408\u4e86\u51e0\u4e2a\u6765\u6e90\u7684\u6570\u636e\u800c\u53d8\u7cdf\u3002\u4e8b\u5b9e\u4e0a\uff0c\u6570\u636e\u8d8a\u591a\u6837\u5316\u8d8a\u597d\u3002<\/p>\n<p><img src=\"https:\/\/lemon-guess.oss-cn-hangzhou.aliyuncs.com\/img\/640-1716822998380-135.webp\" alt=\"\u56fe\u7247\" \/><\/p>\n<p>Google PaLM \u6a21\u578b\u7684\u6570\u636e\u96c6\u3002<\/p>\n<p><strong>\u89e3\u9501\u65b0\u80fd\u529b\uff1a<\/strong>\u968f\u7740\u8d44\u6e90\uff08\u6570\u636e\u3001\u8ba1\u7b97\u3001\u6a21\u578b\u5927\u5c0f\uff09\u6295\u5165\u7684\u589e\u52a0\uff0c\u6df1\u5ea6\u5b66\u4e60\u6a21\u578b\u4e5f\u5728\u4e0d\u8fde\u7eed\u5730\u6539\u8fdb\u3002\u5728\u4e00\u4e9b\u7ec4\u5408\u73af\u5883\u4e2d\u4e5f\u8bc1\u660e\u4e86\u8fd9\u4e00\u70b9\u3002<\/p>\n<p><img src=\"https:\/\/lemon-guess.oss-cn-hangzhou.aliyuncs.com\/img\/640-1716822998380-135.webp\" alt=\"\u56fe\u7247\" \/><\/p>\n<p>\u968f\u7740\u6a21\u578b\u89c4\u6a21\u7684\u589e\u52a0\uff0cPaLM \u5728\u57fa\u51c6\u6d4b\u8bd5\u4e2d\u663e\u793a\u51fa\u4e0d\u8fde\u7eed\u7684\u6539\u8fdb\uff0c\u5e76\u4e14\u89e3\u9501\u4ee4\u4eba\u60ca\u8bb6\u7684\u529f\u80fd\uff0c\u6bd4\u5982\u89e3\u91ca\u7b11\u8bdd\u4e3a\u4ec0\u4e48\u597d\u7b11\u3002<\/p>\n<p><strong>\u6027\u80fd\u51e0\u4e4e\u4e0e\u635f\u5931\u6216\u6570\u636e\u65e0\u5173\uff1a<\/strong>\u5b58\u5728\u591a\u4e2a\u81ea\u76d1\u7763\u635f\u5931\uff0c\u56fe\u50cf\u7814\u7a76\u4e2d\u5176\u5b9e\u4f7f\u7528\u4e86\u591a\u79cd\u5bf9\u6bd4\u548c\u91cd\u5efa\u635f\u5931\uff0c\u8bed\u8a00\u6a21\u578b\u4f7f\u7528\u5355\u8fb9\u91cd\u5efa\uff08\u9884\u6d4b\u4e0b\u4e00\u4e2a token\uff09\u6216\u4f7f\u7528 mask \u6a21\u578b\uff0c\u9884\u6d4b\u6765\u81ea\u5de6\u53f3 token \u7684 mask \u8f93\u5165\u3002\u4e5f\u53ef\u4ee5\u4f7f\u7528\u7a0d\u5fae\u4e0d\u540c\u7684\u6570\u636e\u96c6\u3002\u8fd9\u4e9b\u53ef\u80fd\u4f1a\u5f71\u54cd\u6548\u7387\uff0c\u4f46\u53ea\u8981\u505a\u51fa \u201c\u5408\u7406\u201d \u7684\u9009\u62e9\uff0c\u901a\u5e38\u539f\u59cb\u8d44\u6e90\u6bd4\u4f7f\u7528\u7684\u7279\u5b9a\u635f\u5931\u6216\u6570\u636e\u96c6\u66f4\u80fd\u63d0\u5347\u9884\u6d4b\u6027\u80fd\u3002<\/p>\n<p><strong>\u6709\u4e9b\u60c5\u51b5\u6bd4\u5176\u4ed6\u60c5\u51b5\u66f4\u56f0\u96be\uff1a<\/strong>\u8fd9\u4e00\u70b9\u5e76\u4e0d\u7279\u5b9a\u4e8e\u81ea\u76d1\u7763\u5b66\u4e60\u3002\u6570\u636e\u70b9\u4f3c\u4e4e\u6709\u4e00\u4e9b\u56fa\u6709\u7684 \u201c\u96be\u5ea6\u7ea7\u522b\u201d\u3002\u4e8b\u5b9e\u4e0a\uff0c\u4e0d\u540c\u7684\u5b66\u4e60\u7b97\u6cd5\u5177\u6709\u4e0d\u540c\u7684\u201c\u6280\u80fd\u6c34\u5e73\u201d\uff0c\u4e0d\u540c\u7684\u6570\u636e dian \u5177\u6709\u4e0d\u540c\u7684\u201d \u96be\u5ea6\u6c34\u5e73\u201c\uff08\u5206\u7c7b\u5668$f$\u6b63\u786e\u5206\u7c7b\u70b9$x$\u7684\u6982\u7387\u968f$f$\u7684\u6280\u80fd\u800c\u5355\u8c03\u63d0\u5347\uff0c\u968f$x$\u96be\u5ea6\u5355\u8c03\u964d\u4f4e)\u3002<\/p>\n<p>\u201c\u6280\u80fd\u4e0e\u96be\u5ea6\uff08skill vs. difficulty\uff09\u201d\u8303\u5f0f\u662f\u5bf9 Recht \u7b49\u4eba\u548c Miller \u7b49\u4eba\u53d1\u73b0\u7684 \u201caccuracy on the line\u201d \u73b0\u8c61\u7684\u6700\u6e05\u6670\u89e3\u91ca\u3002Kaplen\u3001Ghosh\u3001Garg \u548c Nakkiran \u7684\u8bba\u6587\u8fd8\u5c55\u793a\u4e86\u6570\u636e\u96c6\u4e2d\u7684\u4e0d\u540c\u8f93\u5165\u5982\u4f55\u5177\u6709\u56fa\u6709\u7684\u201c\u96be\u5ea6\u5256\u9762\u201d\uff0c\u5bf9\u4e8e\u4e0d\u540c\u7684\u6a21\u578b\u65cf\uff0c\u8be5\u5256\u9762\u901a\u5e38\u662f\u7a33\u5065\u7684\u3002<\/p>\n<p><img src=\"https:\/\/lemon-guess.oss-cn-hangzhou.aliyuncs.com\/img\/640-1716822998380-140.webp\" alt=\"\u56fe\u7247\" \/><\/p>\n<p>CIFAR-10 \u4e0a\u8bad\u7ec3\u5e76\u5728 CINIC-10 \u4e0a\u6d4b\u8bd5\u7684\u5206\u7c7b\u5668\u7684 accuracy on the line \u73b0\u8c61\u3002\u56fe\u6e90\uff1a<a href=\"https:\/\/millerjohnp-linearfits-app-app-ryiwcq.streamlitapp.com\/\">https:\/\/millerjohnp-linearfits-app-app-ryiwcq.streamlitapp.com\/<\/a><\/p>\n<p><img src=\"https:\/\/lemon-guess.oss-cn-hangzhou.aliyuncs.com\/img\/640-1716822998380-141.webp\" alt=\"\u56fe\u7247\" \/><\/p>\n<p>\u9876\u90e8\u7684\u56fe\u63cf\u8ff0\u4e86\u6700\u53ef\u80fd\u7c7b\u522b\u7684\u4e0d\u540c softmax \u6982\u7387\uff0c\u4f5c\u4e3a\u67d0\u4e2a\u7c7b\u522b\u5206\u7c7b\u5668\u7684\u5168\u5c40\u7cbe\u5ea6\u7684\u51fd\u6570\uff0c\u8be5\u7c7b\u522b\u7531\u8bad\u7ec3\u65f6\u95f4\u7d22\u5f15\u3002\u5e95\u90e8\u7684\u997c\u56fe\u663e\u793a\u4e86\u4e0d\u540c\u6570\u636e\u96c6\u5206\u89e3\u4e3a\u4e0d\u540c\u7c7b\u578b\u7684\u70b9(\u6ce8\u610f\uff0c\u8fd9\u79cd\u5206\u89e3\u5bf9\u4e8e\u4e0d\u540c\u7684\u795e\u7ecf\u7ed3\u6784\u662f\u76f8\u4f3c\u7684)\u3002<\/p>\n<p><strong>\u8bad\u7ec3\u5c31\u662f\u6559\u5b66\uff1a<\/strong>\u73b0\u4ee3\u5927\u6a21\u578b\u7684\u8bad\u7ec3\u4f3c\u4e4e\u66f4\u50cf\u662f\u6559\u5b66\u751f\uff0c\u800c\u4e0d\u662f\u8ba9\u6a21\u578b\u62df\u5408\u6570\u636e\uff0c\u5f53\u5b66\u751f\u4e0d\u61c2\u6216\u611f\u5230\u75b2\u5026\u65f6\uff0c\u5c31 \u201c\u4f11\u606f\u201d \u6216\u5c1d\u8bd5\u4e0d\u540c\u7684\u65b9\u6cd5\uff08\u8bad\u7ec3\u5dee\u5f02\uff09\u3002Meta \u7684\u5927\u6a21\u578b\u8bad\u7ec3\u65e5\u5fd7\u5f88\u6709\u542f\u53d1\u6027\u2014\u2014\u9664\u4e86\u786c\u4ef6\u95ee\u9898\u5916\uff0c\u6211\u4eec\u8fd8\u53ef\u4ee5\u770b\u5230\u5e72\u9884\u63aa\u65bd\uff0c\u4f8b\u5982\u5728\u8bad\u7ec3\u8fc7\u7a0b\u4e2d\u5207\u6362\u4e0d\u540c\u7684\u4f18\u5316\u7b97\u6cd5\uff0c\u751a\u81f3\u8003\u8651 \u201chot swapping\u201d \u6fc0\u6d3b\u51fd\u6570\uff08GELU to RELU\uff09\u3002\u5982\u679c\u5c06\u6a21\u578b\u8bad\u7ec3\u89c6\u4e3a\u62df\u5408\u6570\u636e\uff0c\u800c\u4e0d\u662f\u5b66\u4e60\u8868\u793a\uff0c\u5219\u540e\u8005\u6ca1\u6709\u591a\u5927\u610f\u4e49\u3002<\/p>\n<p><img src=\"https:\/\/lemon-guess.oss-cn-hangzhou.aliyuncs.com\/img\/640-1716822998380-142.webp\" alt=\"\u56fe\u7247\" \/><\/p>\n<p>Meta \u8bad\u7ec3\u65e5\u5fd7\u6458\u5f55<\/p>\n<p><strong>4.1\uff09\u4f46\u662f\u76d1\u7763\u5b66\u4e60\u600e\u6837\u5462\uff1f<\/strong><\/p>\n<p>\u524d\u9762\u8ba8\u8bba\u4e86\u81ea\u76d1\u7763\u5b66\u4e60\uff0c\u4f46\u6df1\u5ea6\u5b66\u4e60\u7684\u5178\u578b\u4f8b\u5b50\uff0c\u4ecd\u7136\u662f\u76d1\u7763\u5b66\u4e60\u3002\u6bd5\u7adf\uff0c\u6df1\u5ea6\u5b66\u4e60\u7684 \u201cImageNet \u65f6\u523b\u201d \u6765\u81ea ImageNet\u3002\u90a3\u4e48\u4e0a\u9762\u6240\u8ba8\u8bba\u7684\u662f\u5426\u4ecd\u7136\u9002\u7528\u4e8e\u8fd9\u4e2a\u8bbe\u5b9a\uff1f<\/p>\n<p>\u9996\u5148\uff0c\u6709\u76d1\u7763\u7684\u5927\u89c4\u6a21\u6df1\u5ea6\u5b66\u4e60\u7684\u51fa\u73b0\u5728\u67d0\u79cd\u7a0b\u5ea6\u4e0a\u662f\u4e2a\u5076\u7136\uff0c\u8fd9\u5f97\u76ca\u4e8e\u5927\u578b\u9ad8\u8d28\u91cf\u6807\u8bb0\u6570\u636e\u96c6\uff08\u5373 ImageNet\uff09\u7684\u53ef\u7528\u6027\u3002\u5982\u679c\u4f60\u60f3\u8c61\u529b\u4e30\u5bcc\uff0c\u53ef\u4ee5\u60f3\u8c61\u53e6\u4e00\u79cd\u5386\u53f2\uff0c\u5373\u6df1\u5ea6\u5b66\u4e60\u9996\u5148\u5f00\u59cb\u901a\u8fc7\u65e0\u76d1\u7763\u5b66\u4e60\u5728\u81ea\u7136\u8bed\u8a00\u5904\u7406\u65b9\u9762\u53d6\u5f97\u7a81\u7834\u6027\u8fdb\u5c55\uff0c\u7136\u540e\u624d\u8f6c\u79fb\u5230\u89c6\u89c9\u548c\u76d1\u7763\u5b66\u4e60\u4e2d\u3002<\/p>\n<p>\u5176\u6b21\uff0c\u6709\u8bc1\u636e\u8868\u660e\uff0c\u5c3d\u7ba1\u4f7f\u7528\u5b8c\u5168\u4e0d\u540c\u7684\u635f\u5931\u51fd\u6570\uff0c\u4f46\u76d1\u7763\u5b66\u4e60\u548c\u81ea\u76d1\u7763\u5b66\u4e60\u5728\u201d\u5185\u90e8\u201c\u7684\u884c\u4e3a\u5176\u5b9e\u662f\u76f8\u4f3c\u7684\u3002\u4e24\u8005\u901a\u5e38\u90fd\u80fd\u8fbe\u5230\u76f8\u540c\u7684\u6027\u80fd\u3002\u5177\u4f53\u5730\uff0c\u5bf9\u4e8e\u6bcf\u4e00\u4e2a$k\\in{1..d-1}$\uff0c\u4eba\u4eec\u53ef\u4ee5\u5c06\u901a\u8fc7\u81ea\u76d1\u7763\u8bad\u7ec3\u7684\u6df1\u5ea6\u4e3a d \u7684\u6a21\u578b\u7684\u524d k \u5c42\u4e0e\u76d1\u7763\u6a21\u578b\u7684\u6700\u540e d-k \u5c42\u5408\u5728\u4e00\u8d77\uff0c\u800c\u6027\u80fd\u635f\u5931\u5f88\u5c0f\u3002<\/p>\n<p><img src=\"https:\/\/lemon-guess.oss-cn-hangzhou.aliyuncs.com\/img\/640-1716822998380-144.webp\" alt=\"\u56fe\u7247\" \/><\/p>\n<p>SimCLR v2 \u8bba\u6587\u7684\u8868\u683c\u3002\u8bf7\u6ce8\u610f\u76d1\u7763\u5b66\u4e60\u3001\u5fae\u8c03\uff08100%\uff09\u81ea\u76d1\u7763\u548c\u81ea\u76d1\u7763 + \u7ebf\u6027\u63a2\u6d4b\u4e4b\u95f4\u5728\u6027\u80fd\u4e0a\u7684\u4e00\u822c\u76f8\u4f3c\u6027\uff08\u56fe\u6e90\uff1a<a href=\"https:\/\/arxiv.org\/abs\/2006.10029\">https:\/\/arxiv.org\/abs\/2006.10029<\/a>\uff09<\/p>\n<p><img src=\"https:\/\/lemon-guess.oss-cn-hangzhou.aliyuncs.com\/img\/640-1716822998380-145.webp\" alt=\"\u56fe\u7247\" \/><\/p>\n<p>\u62fc\u63a5\u81ea\u76d1\u7763\u6a21\u578b\u548c Bansal \u7b49\u4eba\u7684\u76d1\u7763\u6a21\u578b\uff08<a href=\"https:\/\/arxiv.org\/abs\/2106.07682\uff09\u3002\u5de6\uff1a\u5982\u679c\u81ea\u76d1\u7763\u6a21\u578b\u7684\u51c6\u786e\u7387\uff08\u6bd4\u5982\uff09\u6bd4\u76d1\u7763\u6a21\u578b\u4f4e\">https:\/\/arxiv.org\/abs\/2106.07682\uff09\u3002\u5de6\uff1a\u5982\u679c\u81ea\u76d1\u7763\u6a21\u578b\u7684\u51c6\u786e\u7387\uff08\u6bd4\u5982\uff09\u6bd4\u76d1\u7763\u6a21\u578b\u4f4e<\/a> 3%\uff0c\u5219\u5f53\u5c42\u7684 p \u90e8\u5206\u6765\u81ea\u81ea\u76d1\u7763\u6a21\u578b\u65f6\uff0c\u5b8c\u5168\u517c\u5bb9\u7684\u8868\u793a\u5c06\u5bfc\u81f4\u62fc\u63a5\u60e9\u7f5a\u4e3a p 3%\u3002\u5982\u679c\u6a21\u578b\u5b8c\u5168\u4e0d\u517c\u5bb9\uff0c\u90a3\u4e48\u6211\u4eec\u9884\u8ba1\u968f\u7740\u5408\u5e76\u66f4\u591a\u6a21\u578b\uff0c\u51c6\u786e\u7387\u4f1a\u6025\u5267\u4e0b\u964d\u3002\u53f3\uff1a\u5408\u5e76\u4e0d\u540c\u81ea\u76d1\u7763\u6a21\u578b\u7684\u5b9e\u9645\u7ed3\u679c\u3002<\/p>\n<p>\u81ea\u76d1\u7763 + \u7b80\u5355\u6a21\u578b\u7684\u4f18\u52bf\u5728\u4e8e\uff0c\u5b83\u4eec\u53ef\u4ee5\u5c06\u7279\u5f81\u5b66\u4e60\u6216 \u201c\u6df1\u5ea6\u5b66\u4e60\u9b54\u6cd5\u201d\uff08\u7531\u6df1\u5ea6\u8868\u793a\u51fd\u6570\u5b8c\u6210\uff09\u4e0e\u7edf\u8ba1\u6a21\u578b\u62df\u5408\uff08\u7531\u7ebf\u6027\u6216\u5176\u4ed6\u201c\u7b80\u5355\u201d \u5206\u7c7b\u5668\u5728\u6b64\u8868\u793a\u4e4b\u4e0a\u5b8c\u6210\uff09\u5206\u79bb\u3002<\/p>\n<p>\u6700\u540e\uff0c\u867d\u7136\u8fd9\u66f4\u50cf\u662f\u4e00\u79cd\u63a8\u6d4b\uff0c\u4f46\u4e8b\u5b9e\u4e0a \u201c\u5143\u5b66\u4e60\u201d \u4f3c\u4e4e\u5f80\u5f80\u7b49\u540c\u4e8e\u5b66\u4e60\u8868\u5f81\uff08\u53c2\u89c1\uff1a<a href=\"https:\/\/arxiv.org\/abs\/1909.09157\uff0chttps:\/\/arxiv.org\/abs\/2206.03271\uff09\uff0c\u8fd9\u53ef\u4ee5\u88ab\u89c6\u4e3a\u53e6\u4e00\u4e2a\u8bc1\u636e\uff0c\u8bc1\u660e\u8fd9\u5728\u5f88\u5927\u7a0b\u5ea6\u4e0a\u662f\u5728\u8fdb\u884c\u7684\uff0c\u800c\u4e0d\u7ba1\u6a21\u578b\u4f18\u5316\u7684\u76ee\u6807\u662f\u4ec0\u4e48\">https:\/\/arxiv.org\/abs\/1909.09157\uff0chttps:\/\/arxiv.org\/abs\/2206.03271\uff09\uff0c\u8fd9\u53ef\u4ee5\u88ab\u89c6\u4e3a\u53e6\u4e00\u4e2a\u8bc1\u636e\uff0c\u8bc1\u660e\u8fd9\u5728\u5f88\u5927\u7a0b\u5ea6\u4e0a\u662f\u5728\u8fdb\u884c\u7684\uff0c\u800c\u4e0d\u7ba1\u6a21\u578b\u4f18\u5316\u7684\u76ee\u6807\u662f\u4ec0\u4e48<\/a>\u3002<\/p>\n<p><strong>4.2\uff09\u8fc7\u5ea6\u53c2\u6570\u5316\u600e\u4e48\u529e\uff1f<\/strong><\/p>\n<p>\u672c\u6587\u8df3\u8fc7\u4e86\u88ab\u8ba4\u4e3a\u662f\u7edf\u8ba1\u5b66\u4e60\u6a21\u578b\u548c\u6df1\u5ea6\u5b66\u4e60\u5728\u5b9e\u8df5\u4e2d\u5b58\u5728\u5dee\u5f02\u7684\u5178\u578b\u4f8b\u5b50\uff1a\u7f3a\u4e4f \u201cBias-Variance \u6743\u8861\u201d \u4ee5\u53ca\u8fc7\u5ea6\u53c2\u6570\u5316\u6a21\u578b\u7684\u826f\u597d\u6cdb\u5316\u80fd\u529b\u3002<\/p>\n<p>\u4e3a\u4ec0\u4e48\u8981\u8df3\u8fc7\uff1f\u6709\u4e24\u4e2a\u539f\u56e0\uff1a<\/p>\n<ul>\n<li>\u9996\u5148\uff0c\u5982\u679c\u76d1\u7763\u5b66\u4e60\u786e\u5b9e\u7b49\u4e8e\u81ea\u76d1\u7763 + \u7b80\u5355\u5b66\u4e60\uff0c\u90a3\u4e48\u8fd9\u53ef\u80fd\u89e3\u91ca\u4e86\u5b83\u7684\u6cdb\u5316\u80fd\u529b\u3002<\/li>\n<li>\u5176\u6b21\uff0c\u8fc7\u5ea6\u53c2\u6570\u5316\u5e76\u4e0d\u662f\u6df1\u5ea6\u5b66\u4e60\u6210\u529f\u7684\u5173\u952e\u3002\u6df1\u5ea6\u7f51\u7edc\u4e4b\u6240\u4ee5\u7279\u522b\uff0c\u5e76\u4e0d\u662f\u56e0\u4e3a\u5b83\u4eec\u4e0e\u6837\u672c\u6570\u91cf\u76f8\u6bd4\u5927\uff0c\u800c\u662f\u56e0\u4e3a\u5b83\u4eec\u5728\u7edd\u5bf9\u503c\u4e0a\u5927\u3002\u4e8b\u5b9e\u4e0a\uff0c\u901a\u5e38\u5728\u65e0\u76d1\u7763 \/ \u81ea\u76d1\u7763\u5b66\u4e60\u4e2d\uff0c\u6a21\u578b\u4e0d\u4f1a\u8fc7\u5ea6\u53c2\u6570\u5316\u3002\u5373\u4f7f\u5bf9\u4e8e\u975e\u5e38\u5927\u7684\u8bed\u8a00\u6a21\u578b\uff0c\u5b83\u4eec\u7684\u6570\u636e\u96c6\u4e5f\u66f4\u5927\u3002<\/li>\n<\/ul>\n<p><img src=\"https:\/\/lemon-guess.oss-cn-hangzhou.aliyuncs.com\/img\/640-1716822998380-146.webp\" alt=\"\u56fe\u7247\" \/><\/p>\n<p>Nakkiran-Neyshabur-Sadghi\u201cdeep bootstrap\u201d\u8bba\u6587\u8868\u660e\uff0c\u73b0\u4ee3\u67b6\u6784\u5728 \u201c\u8fc7\u5ea6\u53c2\u6570\u5316\u201d \u6216\u201c\u6b20\u91c7\u6837\u201d\u72b6\u6001\u4e0b\u8868\u73b0\u7c7b\u4f3c\uff08\u6a21\u578b\u5728\u6709\u9650\u6570\u636e\u4e0a\u8bad\u7ec3\u591a\u4e2a epoch\uff0c\u76f4\u5230\u8fc7\u5ea6\u62df\u5408\uff1a\u4e0a\u56fe\u4e2d\u7684 \u201cReal World\u201d\uff09\uff0c\u5728\u201c\u6b20\u53c2\u6570\u5316\u201d \u6216\u8005 \u201c\u5728\u7ebf\u201d \u72b6\u6001\u4e0b\u4e5f\u662f\u5982\u6b64\uff08\u6a21\u578b\u8bad\u7ec3\u5355\u4e2a epoch\uff0c\u6bcf\u4e2a\u6837\u672c\u53ea\u770b\u4e00\u6b21\uff1a\u4e0a\u56fe\u4e2d\u7684 \u201cIdeal World\u201d\uff09\u3002\u56fe\u6e90\uff1a<a href=\"https:\/\/arxiv.org\/abs\/2010.08127\">https:\/\/arxiv.org\/abs\/2010.08127<\/a><\/p>\n<p><strong>\u603b\u7ed3<\/strong><\/p>\n<p>\u7edf\u8ba1\u5b66\u4e60\u5f53\u7136\u5728\u6df1\u5ea6\u5b66\u4e60\u4e2d\u53d1\u6325\u7740\u4f5c\u7528\u3002\u7136\u800c\uff0c\u5c3d\u7ba1\u4f7f\u7528\u4e86\u76f8\u4f3c\u7684\u672f\u8bed\u548c\u4ee3\u7801\uff0c\u4f46\u5c06\u6df1\u5ea6\u5b66\u4e60\u89c6\u4e3a\u7b80\u5355\u5730\u62df\u5408\u4e00\u4e2a\u6bd4\u7ecf\u5178\u6a21\u578b\u5177\u6709\u66f4\u591a\u53c2\u6570\u7684\u6a21\u578b\uff0c\u4f1a\u5ffd\u7565\u5f88\u591a\u5bf9\u5176\u6210\u529f\u81f3\u5173\u91cd\u8981\u7684\u4e1c\u897f\u3002\u6559\u5b66\u751f\u6570\u5b66\u7684\u6bd4\u55bb\u4e5f\u4e0d\u662f\u5b8c\u7f8e\u7684\u3002<\/p>\n<p>\u4e0e\u751f\u7269\u8fdb\u5316\u4e00\u6837\uff0c\u5c3d\u7ba1\u6df1\u5ea6\u5b66\u4e60\u5305\u542b\u8bb8\u591a\u590d\u7528\u7684\u89c4\u5219\uff08\u5982\u7ecf\u9a8c\u635f\u5931\u7684\u68af\u5ea6\u4e0b\u964d\uff09\uff0c\u4f46\u5b83\u4f1a\u4ea7\u751f\u9ad8\u5ea6\u590d\u6742\u7684\u7ed3\u679c\u3002\u4f3c\u4e4e\u5728\u4e0d\u540c\u7684\u65f6\u95f4\uff0c\u7f51\u7edc\u7684\u4e0d\u540c\u7ec4\u4ef6\u4f1a\u5b66\u4e60\u4e0d\u540c\u7684\u4e1c\u897f\uff0c\u5305\u62ec\u8868\u793a\u5b66\u4e60\u3001\u9884\u6d4b\u62df\u5408\u3001\u9690\u5f0f\u6b63\u5219\u5316\u548c\u7eaf\u566a\u58f0\u7b49\u3002\u7814\u7a76\u4eba\u5458\u4ecd\u5728\u5bfb\u627e\u5408\u9002\u7684\u89c6\u89d2\u63d0\u51fa\u6709\u5173\u6df1\u5ea6\u5b66\u4e60\u7684\u95ee\u9898\uff0c\u66f4\u4e0d\u7528\u8bf4\u56de\u7b54\u8fd9\u4e9b\u95ee\u9898\u3002<\/p>\n<p>\u539f\u6587\u94fe\u63a5\uff1a<a href=\"https:\/\/windowsontheory.org\/2022\/06\/20\/the-uneasy-relationship-between-deep-learning-and-classical-statistics\/\">https:\/\/windowsontheory.org\/2022\/06\/20\/the-uneasy-relationship-between-deep-learning-and-classical-statistics\/<\/a><\/p>\n<p>\u7f16\u8f91\uff1a\u9ec4\u7ee7\u5f66<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u6df1\u5ea6\u5b66\u4e60\u548c\u7ecf\u5178\u7edf\u8ba1\u5b66\u662f\u4e00\u56de\u4e8b\uff1f \u6765\u6e90\uff1a\u673a\u5668\u4e4b\u5fc3 \u672c\u6587\u7ea67000\u5b57\uff0c\u5efa\u8bae\u9605\u8bfb10+\u5206\u949f \u672c\u6587\u4f1a\u89e3\u91ca\u4e3a\u4ec0\u4e48\u6df1\u5ea6\u5b66\u4e60\u7684\u57fa\u7840\u5176\u5b9e\u4e0d\u540c\u4e8e\u7edf\u8ba1\u5b66\uff0c\u751a   \u2026 &#8230;<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"ngg_post_thumbnail":0},"categories":[273],"tags":[],"_links":{"self":[{"href":"\/index.php?rest_route=\/wp\/v2\/posts\/9326"}],"collection":[{"href":"\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=9326"}],"version-history":[{"count":7,"href":"\/index.php?rest_route=\/wp\/v2\/posts\/9326\/revisions"}],"predecessor-version":[{"id":9334,"href":"\/index.php?rest_route=\/wp\/v2\/posts\/9326\/revisions\/9334"}],"wp:attachment":[{"href":"\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=9326"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=9326"},{"taxonomy":"post_tag","embeddable":true,"href":"\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=9326"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}