{"id":1673,"date":"2025-02-04T17:18:16","date_gmt":"2025-02-04T09:18:16","guid":{"rendered":"https:\/\/www.forillusion.com\/?p=1673"},"modified":"2025-02-14T11:39:01","modified_gmt":"2025-02-14T03:39:01","slug":"3-11-underfit-overfit","status":"publish","type":"post","link":"https:\/\/www.forillusion.com\/index.php\/3-11-underfit-overfit\/","title":{"rendered":"3.11 \u6a21\u578b\u9009\u62e9\u3001\u6b20\u62df\u5408\u548c\u8fc7\u62df\u5408"},"content":{"rendered":"\n<p><div class=\"has-toc have-toc\"><\/div><\/p>\n\n\n\n<h2 class=\"wp-block-heading\">\u8bad\u7ec3\u8bef\u5dee\u548c\u6cdb\u5316\u8bef\u5dee<\/h2>\n\n\n\n<p>\u8bad\u7ec3\u8bef\u5dee\uff08training error\uff09\u6307\u6a21\u578b\u5728\u8bad\u7ec3\u6570\u636e\u96c6\u4e0a\u8868\u73b0\u51fa\u7684\u8bef\u5dee\uff0c\u6cdb\u5316\u8bef\u5dee\uff08generalization error\uff09\u6307\u6a21\u578b\u5728\u4efb\u610f\u4e00\u4e2a\u6d4b\u8bd5\u6570\u636e\u6837\u672c\u4e0a\u8868\u73b0\u51fa\u7684\u8bef\u5dee\u7684\u671f\u671b\uff0c\u5e76\u5e38\u5e38\u901a\u8fc7\u6d4b\u8bd5\u6570\u636e\u96c6\u4e0a\u7684\u8bef\u5dee\u6765\u8fd1\u4f3c\u3002\u8ba1\u7b97\u8bad\u7ec3\u8bef\u5dee\u548c\u6cdb\u5316\u8bef\u5dee\u53ef\u4ee5\u4f7f\u7528\u635f\u5931\u51fd\u6570\uff0c\u4f8b\u5982\u7ebf\u6027\u56de\u5f52\u7528\u5230\u7684\u5e73\u65b9\u635f\u5931\u51fd\u6570\u548csoftmax\u56de\u5f52\u7528\u5230\u7684\u4ea4\u53c9\u71b5\u635f\u5931\u51fd\u6570\u3002<\/p>\n\n\n\n<p>\u5728\u673a\u5668\u5b66\u4e60\u91cc\uff0c\u901a\u5e38\u5047\u8bbe\u8bad\u7ec3\u6570\u636e\u96c6\u548c\u6d4b\u8bd5\u6570\u636e\u96c6\u91cc\u7684\u6bcf\u4e00\u4e2a\u6837\u672c\u90fd\u662f\u4ece\u540c\u4e00\u4e2a\u6982\u7387\u5206\u5e03\u4e2d\u76f8\u4e92\u72ec\u7acb\u5730\u751f\u6210\u7684\u3002\u7531\u8bad\u7ec3\u6570\u636e\u96c6\u5b66\u5230\u7684\u6a21\u578b\u53c2\u6570\u4f1a\u4f7f\u6a21\u578b\u5728\u8bad\u7ec3\u6570\u636e\u96c6\u4e0a\u7684\u8868\u73b0\u4f18\u4e8e\u6216\u7b49\u4e8e\u5728\u6d4b\u8bd5\u6570\u636e\u96c6\u4e0a\u7684\u8868\u73b0\u3002\u7531\u4e8e\u65e0\u6cd5\u4ece\u8bad\u7ec3\u8bef\u5dee\u4f30\u8ba1\u6cdb\u5316\u8bef\u5dee\uff0c\u4e00\u5473\u5730\u964d\u4f4e\u8bad\u7ec3\u8bef\u5dee\u5e76\u4e0d\u610f\u5473\u7740\u6cdb\u5316\u8bef\u5dee\u4e00\u5b9a\u4f1a\u964d\u4f4e\u3002<\/p>\n\n\n\n<p>\u673a\u5668\u5b66\u4e60\u6a21\u578b\u5e94\u5173\u6ce8\u964d\u4f4e\u6cdb\u5316\u8bef\u5dee\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">\u6a21\u578b\u9009\u62e9<\/h2>\n\n\n\n<p>\u5728\u673a\u5668\u5b66\u4e60\u4e2d\uff0c\u901a\u5e38\u9700\u8981\u8bc4\u4f30\u82e5\u5e72\u5019\u9009\u6a21\u578b\u7684\u8868\u73b0\u5e76\u4ece\u4e2d\u9009\u62e9\u6a21\u578b\u3002\u8fd9\u4e00\u8fc7\u7a0b\u79f0\u4e3a\u6a21\u578b\u9009\u62e9\uff08model selection\uff09\u3002\u53ef\u4f9b\u9009\u62e9\u7684\u5019\u9009\u6a21\u578b\u53ef\u4ee5\u662f\u6709\u7740\u4e0d\u540c\u8d85\u53c2\u6570\u7684\u540c\u7c7b\u6a21\u578b\u3002\u4ee5\u591a\u5c42\u611f\u77e5\u673a\u4e3a\u4f8b\uff0c\u53ef\u4ee5\u9009\u62e9\u9690\u85cf\u5c42\u7684\u4e2a\u6570\uff0c\u4ee5\u53ca\u6bcf\u4e2a\u9690\u85cf\u5c42\u4e2d\u9690\u85cf\u5355\u5143\u4e2a\u6570\u548c\u6fc0\u6d3b\u51fd\u6570\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">\u9a8c\u8bc1\u6570\u636e\u96c6<\/h3>\n\n\n\n<p>\u4ece\u4e25\u683c\u610f\u4e49\u4e0a\u8bb2\uff0c\u6d4b\u8bd5\u96c6\u53ea\u80fd\u5728\u6240\u6709\u8d85\u53c2\u6570\u548c\u6a21\u578b\u53c2\u6570\u9009\u5b9a\u540e\u4f7f\u7528\u4e00\u6b21\u3002\u4e0d\u53ef\u4ee5\u4f7f\u7528\u6d4b\u8bd5\u6570\u636e\u9009\u62e9\u6a21\u578b\uff0c\u5982\u8c03\u53c2\u3002\u7531\u4e8e\u65e0\u6cd5\u4ece\u8bad\u7ec3\u8bef\u5dee\u4f30\u8ba1\u6cdb\u5316\u8bef\u5dee\uff0c\u56e0\u6b64\u4e5f\u4e0d\u5e94\u53ea\u4f9d\u8d56\u8bad\u7ec3\u6570\u636e\u9009\u62e9\u6a21\u578b\u3002\u9274\u4e8e\u6b64\uff0c\u53ef\u4ee5\u9884\u7559\u4e00\u90e8\u5206\u5728\u8bad\u7ec3\u6570\u636e\u96c6\u548c\u6d4b\u8bd5\u6570\u636e\u96c6\u4ee5\u5916\u7684\u6570\u636e\u6765\u8fdb\u884c\u6a21\u578b\u9009\u62e9\u3002\u8fd9\u90e8\u5206\u6570\u636e\u88ab\u79f0\u4e3a\u9a8c\u8bc1\u6570\u636e\u96c6\uff0c\u7b80\u79f0\u9a8c\u8bc1\u96c6\uff08validation set\uff09\u3002\u4f8b\u5982\uff0c\u53ef\u4ee5\u4ece\u7ed9\u5b9a\u7684\u8bad\u7ec3\u96c6\u4e2d\u968f\u673a\u9009\u53d6\u4e00\u5c0f\u90e8\u5206\u4f5c\u4e3a\u9a8c\u8bc1\u96c6\uff0c\u800c\u5c06\u5269\u4f59\u90e8\u5206\u4f5c\u4e3a\u771f\u6b63\u7684\u8bad\u7ec3\u96c6\u3002<\/p>\n\n\n\n<p>\u7136\u800c\u5728\u5b9e\u9645\u5e94\u7528\u4e2d\uff0c\u7531\u4e8e\u6570\u636e\u4e0d\u5bb9\u6613\u83b7\u53d6\uff0c\u6d4b\u8bd5\u6570\u636e\u6781\u5c11\u53ea\u4f7f\u7528\u4e00\u6b21\u5c31\u4e22\u5f03\u3002\u56e0\u6b64\uff0c\u5b9e\u8df5\u4e2d\u9a8c\u8bc1\u6570\u636e\u96c6\u548c\u6d4b\u8bd5\u6570\u636e\u96c6\u7684\u754c\u9650\u53ef\u80fd\u6bd4\u8f83\u6a21\u7cca\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">$K$\u6298\u4ea4\u53c9\u9a8c\u8bc1<\/h3>\n\n\n\n<p>\u7531\u4e8e\u9a8c\u8bc1\u6570\u636e\u96c6\u4e0d\u53c2\u4e0e\u6a21\u578b\u8bad\u7ec3\uff0c\u5f53\u8bad\u7ec3\u6570\u636e\u4e0d\u591f\u7528\u65f6\uff0c\u9884\u7559\u5927\u91cf\u7684\u9a8c\u8bc1\u6570\u636e\u663e\u5f97\u592a\u5962\u4f88\u3002\u4e00\u79cd\u6539\u5584\u7684\u65b9\u6cd5\u662f$K$\u6298\u4ea4\u53c9\u9a8c\u8bc1\uff08$K$-fold cross-validation\uff09\u3002\u5728$K$\u6298\u4ea4\u53c9\u9a8c\u8bc1\u4e2d\uff0c\u628a\u539f\u59cb\u8bad\u7ec3\u6570\u636e\u96c6\u5206\u5272\u6210$K$\u4e2a\u4e0d\u91cd\u5408\u7684\u5b50\u6570\u636e\u96c6\uff0c\u7136\u540e\u505a$K$\u6b21\u6a21\u578b\u8bad\u7ec3\u548c\u9a8c\u8bc1\u3002\u6bcf\u4e00\u6b21\uff0c\u4f7f\u7528\u4e00\u4e2a\u5b50\u6570\u636e\u96c6\u9a8c\u8bc1\u6a21\u578b\uff0c\u5e76\u4f7f\u7528\u5176\u4ed6$K-1$\u4e2a\u5b50\u6570\u636e\u96c6\u6765\u8bad\u7ec3\u6a21\u578b\u3002\u5728\u8fd9$K$\u6b21\u8bad\u7ec3\u548c\u9a8c\u8bc1\u4e2d\uff0c\u6bcf\u6b21\u7528\u6765\u9a8c\u8bc1\u6a21\u578b\u7684\u5b50\u6570\u636e\u96c6\u90fd\u4e0d\u540c\u3002\u6700\u540e\uff0c\u5bf9\u8fd9$K$\u6b21\u8bad\u7ec3\u8bef\u5dee\u548c\u9a8c\u8bc1\u8bef\u5dee\u5206\u522b\u6c42\u5e73\u5747\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">\u6b20\u62df\u5408\u548c\u8fc7\u62df\u5408<\/h2>\n\n\n\n<p>\u6b20\u62df\u5408\uff08underfitting\uff09\uff1a\u6a21\u578b\u65e0\u6cd5\u5f97\u5230\u8f83\u4f4e\u7684\u8bad\u7ec3\u8bef\u5dee\uff1b\u8fc7\u62df\u5408\uff08overfitting\uff09\uff1a\u6a21\u578b\u7684\u8bad\u7ec3\u8bef\u5dee\u8fdc\u5c0f\u4e8e\u5b83\u5728\u6d4b\u8bd5\u6570\u636e\u96c6\u4e0a\u7684\u8bef\u5dee\u3002\u6a21\u578b\u590d\u6742\u5ea6\u548c\u8bad\u7ec3\u6570\u636e\u96c6\u5927\u5c0f\u662f\u5f71\u54cd\u8fd9\u4e24\u79cd\u62df\u5408\u7684\u91cd\u8981\u56e0\u7d20\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">\u6a21\u578b\u590d\u6742\u5ea6<\/h3>\n\n\n\n<p>\u4ee5\u591a\u9879\u5f0f\u51fd\u6570\u62df\u5408\u4e3a\u4f8b\u3002\u7ed9\u5b9a\u4e00\u4e2a\u7531\u6807\u91cf\u6570\u636e\u7279\u5f81$x$\u548c\u5bf9\u5e94\u7684\u6807\u91cf\u6807\u7b7e$y$\u7ec4\u6210\u7684\u8bad\u7ec3\u6570\u636e\u96c6\uff0c\u591a\u9879\u5f0f\u51fd\u6570\u62df\u5408\u7684\u76ee\u6807\u662f\u627e\u4e00\u4e2a$K$\u9636\u591a\u9879\u5f0f\u51fd\u6570<\/p>\n\n\n\n<p>$$<br>\\hat{y} = b + \\sum_{k=1}^K x^k w_k<br>$$<\/p>\n\n\n\n<p>\u6765\u8fd1\u4f3c $y$\u3002\u5728\u4e0a\u5f0f\u4e2d\uff0c$w_k$\u662f\u6a21\u578b\u7684\u6743\u91cd\u53c2\u6570\uff0c$b$\u662f\u504f\u5dee\u53c2\u6570\u3002\u4e0e\u7ebf\u6027\u56de\u5f52\u76f8\u540c\uff0c\u591a\u9879\u5f0f\u51fd\u6570\u62df\u5408\u4e5f\u4f7f\u7528\u5e73\u65b9\u635f\u5931\u51fd\u6570\u3002\u7279\u522b\u5730\uff0c\u4e00\u9636\u591a\u9879\u5f0f\u51fd\u6570\u62df\u5408\u53c8\u53eb\u7ebf\u6027\u51fd\u6570\u62df\u5408\u3002<\/p>\n\n\n\n<p>\u56e0\u4e3a\u9ad8\u9636\u591a\u9879\u5f0f\u51fd\u6570\u6a21\u578b\u53c2\u6570\u66f4\u591a\uff0c\u6a21\u578b\u51fd\u6570\u7684\u9009\u62e9\u7a7a\u95f4\u66f4\u5927\uff0c\u6240\u4ee5\u9ad8\u9636\u591a\u9879\u5f0f\u51fd\u6570\u6bd4\u4f4e\u9636\u591a\u9879\u5f0f\u51fd\u6570\u7684\u590d\u6742\u5ea6\u66f4\u9ad8\u3002\u56e0\u6b64\uff0c\u9ad8\u9636\u591a\u9879\u5f0f\u51fd\u6570\u6bd4\u4f4e\u9636\u591a\u9879\u5f0f\u51fd\u6570\u66f4\u5bb9\u6613\u5728\u76f8\u540c\u7684\u8bad\u7ec3\u6570\u636e\u96c6\u4e0a\u5f97\u5230\u66f4\u4f4e\u7684\u8bad\u7ec3\u8bef\u5dee\u3002\u7ed9\u5b9a\u8bad\u7ec3\u6570\u636e\u96c6\uff0c\u6a21\u578b\u590d\u6742\u5ea6\u548c\u8bef\u5dee\u4e4b\u95f4\u7684\u5173\u7cfb\u901a\u5e38\u5982\u4e0b\u56fe\u6240\u793a\u3002\u7ed9\u5b9a\u8bad\u7ec3\u6570\u636e\u96c6\uff0c\u5982\u679c\u6a21\u578b\u7684\u590d\u6742\u5ea6\u8fc7\u4f4e\uff0c\u5f88\u5bb9\u6613\u51fa\u73b0\u6b20\u62df\u5408\uff1b\u5982\u679c\u6a21\u578b\u590d\u6742\u5ea6\u8fc7\u9ad8\uff0c\u5f88\u5bb9\u6613\u51fa\u73b0\u8fc7\u62df\u5408\u3002\u5e94\u5bf9\u6b20\u62df\u5408\u548c\u8fc7\u62df\u5408\u7684\u4e00\u4e2a\u529e\u6cd5\u662f\u9488\u5bf9\u6570\u636e\u96c6\u9009\u62e9\u5408\u9002\u590d\u6742\u5ea6\u7684\u6a21\u578b\u3002<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\"   class=\"lazyload\" data-src=\"https:\/\/cos.forillusion.top\/wp-content\/uploads\/2025\/02\/3.11_capacity_vs_error.png\" src=\"https:\/\/cdn.forillusion.com\/moezx\/img\/svg\/loader\/trans.ajax-spinner-preloader.svg\" onerror=\"imgError(this)\"  alt=\"\"\/><\/figure >\n<noscript><img decoding=\"async\" src=\"https:\/\/cos.forillusion.top\/wp-content\/uploads\/2025\/02\/3.11_capacity_vs_error.png\" alt=\"\"\/><\/figure><\/noscript>\n\n\n\n<h3 class=\"wp-block-heading\">\u8bad\u7ec3\u6570\u636e\u96c6\u5927\u5c0f<\/h3>\n\n\n\n<p>\u5f71\u54cd\u6b20\u62df\u5408\u548c\u8fc7\u62df\u5408\u7684\u53e6\u4e00\u4e2a\u91cd\u8981\u56e0\u7d20\u662f\u8bad\u7ec3\u6570\u636e\u96c6\u7684\u5927\u5c0f\u3002\u5982\u679c\u8bad\u7ec3\u6570\u636e\u96c6\u4e2d\u6837\u672c\u6570\u8fc7\u5c11\uff0c\u7279\u522b\u662f\u6bd4\u6a21\u578b\u53c2\u6570\u6570\u91cf\uff08\u6309\u5143\u7d20\u8ba1\uff09\u66f4\u5c11\u65f6\uff0c\u8fc7\u62df\u5408\u66f4\u5bb9\u6613\u53d1\u751f\u3002\u6b64\u5916\uff0c\u6cdb\u5316\u8bef\u5dee\u4e0d\u4f1a\u968f\u8bad\u7ec3\u6570\u636e\u96c6\u91cc\u6837\u672c\u6570\u91cf\u589e\u52a0\u800c\u589e\u5927\u3002\u901a\u5e38\u5e0c\u671b\u8bad\u7ec3\u6570\u636e\u96c6\u5927\u4e00\u4e9b\uff0c\u7279\u522b\u662f\u5728\u6a21\u578b\u590d\u6742\u5ea6\u8f83\u9ad8\u65f6\uff0c\u4f8b\u5982\u5c42\u6570\u8f83\u591a\u7684\u6df1\u5ea6\u5b66\u4e60\u6a21\u578b\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">\u591a\u9879\u5f0f\u51fd\u6570\u62df\u5408\u5b9e\u9a8c<\/h2>\n\n\n\n<p>\u4ee5\u591a\u9879\u5f0f\u51fd\u6570\u62df\u5408\u4e3a\u4f8b\u6765\u5b9e\u9a8c\u3002\u5bfc\u5165\u9700\u8981\u7684\u5e93\u3002<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>import torch\nimport numpy as np\nimport matplotlib.pyplot as plt\nfrom IPython import display<\/code><\/pre>\n\n\n\n<h3 class=\"wp-block-heading\">\u751f\u6210\u6570\u636e\u96c6<\/h3>\n\n\n\n<p>\u751f\u6210\u4e00\u4e2a\u4eba\u5de5\u6570\u636e\u96c6\u3002\u5728\u8bad\u7ec3\u6570\u636e\u96c6\u548c\u6d4b\u8bd5\u6570\u636e\u96c6\u4e2d\uff0c\u7ed9\u5b9a\u6837\u672c\u7279\u5f81$x$\uff0c\u4f7f\u7528\u5982\u4e0b\u7684\u4e09\u9636\u591a\u9879\u5f0f\u51fd\u6570\u6765\u751f\u6210\u8be5\u6837\u672c\u7684\u6807\u7b7e\uff1a<\/p>\n\n\n\n<p>$$<br>y = 1.2x - 3.4x^2 + 5.6x^3 + 5 + \\epsilon,<br>$$<\/p>\n\n\n\n<p>\u5176\u4e2d\u566a\u58f0\u9879$\\epsilon$\u670d\u4ece\u5747\u503c\u4e3a0\u3001\u6807\u51c6\u5dee\u4e3a0.01\u7684\u6b63\u6001\u5206\u5e03\u3002\u8bad\u7ec3\u6570\u636e\u96c6\u548c\u6d4b\u8bd5\u6570\u636e\u96c6\u7684\u6837\u672c\u6570\u90fd\u8bbe\u4e3a100\u3002<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>n_train, n_test, true_w, true_b = 100, 100, &#91;1.2, -3.4, 5.6], 5\nfeatures = torch.randn((n_train + n_test, 1)) # \u751f\u6210\u968f\u673a\u6570\u636ex\u503c\uff0c\u5f62\u72b6\u4e3a\uff08200\uff0c1\uff09\npoly_features = torch.cat((features, torch.pow(features, 2), torch.pow(features, 3)), 1)  # \u8ba1\u7b97x\u76841\u30012\u30013\u6b21\u65b9\u3002torch.cat\u662f\u5c06\u4e09\u4e2a\u5f20\u91cf\u62fc\u63a5\u5728\u4e00\u8d77\uff0c\u5f62\u72b6\u4e3a\uff08200\uff0c3\uff09\nlabels = (true_w&#91;0] * poly_features&#91;:, 0] + true_w&#91;1] * poly_features&#91;:, 1]\n          + true_w&#91;2] * poly_features&#91;:, 2] + true_b) # \u8ba1\u7b97y\u503c\nlabels += torch.tensor(np.random.normal(0, 0.01, size=labels.size()), dtype=torch.float) # \u52a0\u4e0a\u566a\u58f0<\/code><\/pre>\n\n\n\n<h3 class=\"wp-block-heading\">\u5b9a\u4e49\u3001\u8bad\u7ec3\u548c\u6d4b\u8bd5\u6a21\u578b<\/h3>\n\n\n\n<p>\u5b9a\u4e49\u7ed8\u56fe\u51fd\u6570\uff1a<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>def set_figsize(figsize=(3.5, 2.5)):\n    display.set_matplotlib_formats('png')\n    plt.rcParams&#91;'figure.figsize'] = figsize\n\ndef semilogy(x_vals, y_vals, x_label, y_label, x2_vals=None, y2_vals=None,\n             legend=None, figsize=(3.5, 2.5)):\n    set_figsize(figsize)\n    plt.xlabel(x_label)\n    plt.ylabel(y_label)\n    plt.semilogy(x_vals, y_vals) # y\u8f74\u4f7f\u7528\u5bf9\u6570\u5c3a\u5ea6\n    if x2_vals and y2_vals:\n        plt.semilogy(x2_vals, y2_vals, linestyle=':')\n        plt.legend(legend)\n    plt.show()<\/code><\/pre>\n\n\n\n<p>\u591a\u9879\u5f0f\u51fd\u6570\u62df\u5408\u4e5f\u4f7f\u7528\u5e73\u65b9\u635f\u5931\u51fd\u6570\u3002\u5c06\u5c1d\u8bd5\u4f7f\u7528\u4e0d\u540c\u590d\u6742\u5ea6\u7684\u6a21\u578b\u6765\u62df\u5408\u751f\u6210\u7684\u6570\u636e\u96c6\u3002\u5b9a\u4e49\u6a21\u578b\uff1a<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>num_epochs, loss = 100, torch.nn.MSELoss() # \u8bbe\u7f6e\u8fed\u4ee3\u5468\u671f\u6570\u548c\u635f\u5931\u51fd\u6570\n\ndef fit_and_plot(train_features, test_features, train_labels, test_labels):\n    net = torch.nn.Linear(train_features.shape&#91;-1], 1) # \u7ebf\u6027\u56de\u5f52\u6a21\u578b\n    # \u901a\u8fc7Linear\u6587\u6863\u53ef\u77e5\uff0cpytorch\u5df2\u7ecf\u5c06\u53c2\u6570\u521d\u59cb\u5316\u4e86\uff0c\u6240\u4ee5\u8fd9\u91cc\u5c31\u4e0d\u624b\u52a8\u521d\u59cb\u5316\u4e86\n\n    batch_size = min(10, train_labels.shape&#91;0]) # \u8bbe\u7f6e\u6279\u91cf\u5927\u5c0f\n    dataset = torch.utils.data.TensorDataset(train_features, train_labels) # \u8bbe\u7f6e\u6570\u636e\u96c6\n    train_iter = torch.utils.data.DataLoader(dataset, batch_size, shuffle=True) # \u8bbe\u7f6e\u6570\u636e\u8fed\u4ee3\u5668\n\n    train_labels = train_labels.view(-1, 1) # \u8f6c\u6362\u6807\u7b7e\u5f62\u72b6\n    test_labels = test_labels.view(-1, 1) # \u8f6c\u6362\u6807\u7b7e\u5f62\u72b6\n\n    optimizer = torch.optim.SGD(net.parameters(), lr=0.01) # \u8bbe\u7f6e\u4f18\u5316\u7b97\u6cd5\n    train_ls, test_ls = &#91;], &#91;] # \u8bad\u7ec3\u548c\u6d4b\u8bd5\u635f\u5931\n    for _ in range(num_epochs): # \u8bad\u7ec3\u6a21\u578b\uff0c\u8fed\u4ee3\u5468\u671f\n        for X, y in train_iter: # X\u662f\u7279\u5f81\uff0cy\u662f\u6807\u7b7e\n            l = loss(net(X), y.view(-1, 1)) # \u8ba1\u7b97\u635f\u5931\uff0cy.view(-1, 1)\u5c06y\u7684\u5f62\u72b6\u8f6c\u6362\u6210\u548c\u9884\u6d4b\u503cy\u7684\u5f62\u72b6\u4e00\u6837\n            optimizer.zero_grad() # \u68af\u5ea6\u6e05\u96f6\n            l.backward() # \u8ba1\u7b97\u68af\u5ea6\n            optimizer.step() # \u8fed\u4ee3\u6a21\u578b\u53c2\u6570\n        train_ls.append(loss(net(train_features), train_labels).item()) # \u8ba1\u7b97\u8bad\u7ec3\u635f\u5931\n        test_ls.append(loss(net(test_features), test_labels).item()) # \u8ba1\u7b97\u6d4b\u8bd5\u635f\u5931\n    print('\u6700\u7ec8\u5468\u671f\uff1a\u8bad\u7ec3\u635f\u5931 ', train_ls&#91;-1], '\u6d4b\u8bd5\u635f\u5931 ', test_ls&#91;-1])\n    print('weight:', net.weight.data,\n          '\\nbias:', net.bias.data)\n    semilogy(range(1, num_epochs + 1), train_ls, 'epochs', 'loss', range(1, num_epochs + 1), test_ls, &#91;'train', 'test'])<\/code><\/pre>\n\n\n\n<h3 class=\"wp-block-heading\">\u4e09\u9636\u591a\u9879\u5f0f\u51fd\u6570\u62df\u5408\uff08\u6b63\u5e38\uff09<\/h3>\n\n\n\n<p>\u5148\u4f7f\u7528\u4e0e\u6570\u636e\u751f\u6210\u51fd\u6570\u540c\u9636\u7684\u4e09\u9636\u591a\u9879\u5f0f\u51fd\u6570\u62df\u5408\u3002\u5b9e\u9a8c\u8868\u660e\uff0c\u8fd9\u4e2a\u6a21\u578b\u7684\u8bad\u7ec3\u8bef\u5dee\u548c\u5728\u6d4b\u8bd5\u6570\u636e\u96c6\u7684\u8bef\u5dee\u90fd\u8f83\u4f4e\u3002\u8bad\u7ec3\u51fa\u7684\u6a21\u578b\u53c2\u6570\u4e5f\u63a5\u8fd1\u771f\u5b9e\u503c\u3002<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>fit_and_plot(poly_features&#91;:n_train, :], poly_features&#91;n_train:, :], labels&#91;:n_train], labels&#91;n_train:]) # \u4e09\u9636\u591a\u9879\u5f0f\u51fd\u6570\uff0c\u4f20\u51653\u4e2a\u7279\u5f81\uff0c\u5373x\u76841\u30012\u30013\u6b21\u65b9\uff0c\u4ee3\u8868\u4e09\u9636\u591a\u9879\u5f0f\u51fd\u6570<\/code><\/pre>\n\n\n\n<p>\u8f93\u51fa\uff1a<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>\u6700\u7ec8\u5468\u671f\uff1a\u8bad\u7ec3\u635f\u5931  0.00012335127394180745 \u6d4b\u8bd5\u635f\u5931  9.706116543384269e-05\nweight: tensor(&#91;&#91; 1.2056, -3.4008,  5.5980]])\nbias: tensor(&#91;5.0013])<\/code><\/pre>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\"   class=\"lazyload\" data-src=\"https:\/\/cos.forillusion.top\/wp-content\/uploads\/2025\/02\/3.11_output1.png\" src=\"https:\/\/cdn.forillusion.com\/moezx\/img\/svg\/loader\/trans.ajax-spinner-preloader.svg\" onerror=\"imgError(this)\"  alt=\"\"\/><\/figure >\n<noscript><img decoding=\"async\" src=\"https:\/\/cos.forillusion.top\/wp-content\/uploads\/2025\/02\/3.11_output1.png\" alt=\"\"\/><\/figure><\/noscript>\n\n\n\n<h3 class=\"wp-block-heading\">\u7ebf\u6027\u51fd\u6570\u62df\u5408\uff08\u6b20\u62df\u5408\uff09<\/h3>\n\n\n\n<p>\u8be5\u6a21\u578b\u7684\u8bad\u7ec3\u8bef\u5dee\u5728\u8fed\u4ee3\u65e9\u671f\u4e0b\u964d\u540e\u4fbf\u5f88\u96be\u7ee7\u7eed\u964d\u4f4e\u3002\u5728\u5b8c\u6210\u6700\u540e\u4e00\u6b21\u8fed\u4ee3\u5468\u671f\u540e\uff0c\u8bad\u7ec3\u8bef\u5dee\u4f9d\u65e7\u5f88\u9ad8\u3002\u7ebf\u6027\u6a21\u578b\u5728\u975e\u7ebf\u6027\u6a21\u578b\uff08\u5982\u4e09\u9636\u591a\u9879\u5f0f\u51fd\u6570\uff09\u751f\u6210\u7684\u6570\u636e\u96c6\u4e0a\u5bb9\u6613\u6b20\u62df\u5408\u3002<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>fit_and_plot(features&#91;:n_train, :], features&#91;n_train:, :], labels&#91;:n_train], labels&#91;n_train:]) # \u7ebf\u6027\u51fd\u6570\uff0c\u4f20\u51651\u4e2a\u7279\u5f81\uff0c\u5373x\uff0c\u4ee3\u8868\u7ebf\u6027\u51fd\u6570<\/code><\/pre>\n\n\n\n<p>\u8f93\u51fa\uff1a<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>\u6700\u7ec8\u5468\u671f\uff1a\u8bad\u7ec3\u635f\u5931  183.3161163330078 \u6d4b\u8bd5\u635f\u5931  314.3257751464844\nweight: tensor(&#91;&#91;19.3992]])\nbias: tensor(&#91;4.1038])<\/code><\/pre>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\"   class=\"lazyload\" data-src=\"https:\/\/cos.forillusion.top\/wp-content\/uploads\/2025\/02\/3.11_output2.png\" src=\"https:\/\/cdn.forillusion.com\/moezx\/img\/svg\/loader\/trans.ajax-spinner-preloader.svg\" onerror=\"imgError(this)\"  alt=\"\"\/><\/figure >\n<noscript><img decoding=\"async\" src=\"https:\/\/cos.forillusion.top\/wp-content\/uploads\/2025\/02\/3.11_output2.png\" alt=\"\"\/><\/figure><\/noscript>\n\n\n\n<h3 class=\"wp-block-heading\">\u8bad\u7ec3\u6837\u672c\u4e0d\u8db3\uff08\u8fc7\u62df\u5408\uff09<\/h3>\n\n\n\n<p>\u4e8b\u5b9e\u4e0a\uff0c\u5373\u4fbf\u4f7f\u7528\u4e0e\u6570\u636e\u751f\u6210\u6a21\u578b\u540c\u9636\u7684\u4e09\u9636\u591a\u9879\u5f0f\u51fd\u6570\u6a21\u578b\uff0c\u5982\u679c\u8bad\u7ec3\u6837\u672c\u4e0d\u8db3\uff0c\u8be5\u6a21\u578b\u4f9d\u7136\u5bb9\u6613\u8fc7\u62df\u5408\u3002\u53ea\u4f7f\u7528\u4e24\u4e2a\u6837\u672c\u6765\u8bad\u7ec3\u6a21\u578b\u3002\u663e\u7136\uff0c\u8bad\u7ec3\u6837\u672c\u8fc7\u5c11\u4e86\uff0c\u751a\u81f3\u5c11\u4e8e\u6a21\u578b\u53c2\u6570\u7684\u6570\u91cf\u3002\u8fd9\u4f7f\u6a21\u578b\u663e\u5f97\u8fc7\u4e8e\u590d\u6742\uff0c\u4ee5\u81f3\u4e8e\u5bb9\u6613\u88ab\u8bad\u7ec3\u6570\u636e\u4e2d\u7684\u566a\u58f0\u5f71\u54cd\u3002\u5728\u8fed\u4ee3\u8fc7\u7a0b\u4e2d\uff0c\u5c3d\u7ba1\u8bad\u7ec3\u8bef\u5dee\u8f83\u4f4e\uff0c\u4f46\u662f\u6d4b\u8bd5\u6570\u636e\u96c6\u4e0a\u7684\u8bef\u5dee\u5374\u5f88\u9ad8\u3002\u8fd9\u662f\u5178\u578b\u7684\u8fc7\u62df\u5408\u73b0\u8c61\u3002<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>fit_and_plot(poly_features&#91;0:2, :], poly_features&#91;n_train:, :], labels&#91;0:2], labels&#91;n_train:]) # \u8fc7\u62df\u5408<\/code><\/pre>\n\n\n\n<p>\u8f93\u51fa\uff1a<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>\u6700\u7ec8\u5468\u671f\uff1a\u8bad\u7ec3\u635f\u5931  0.26395294070243835 \u6d4b\u8bd5\u635f\u5931  37.11482238769531\nweight: tensor(&#91;&#91;3.8149, 0.2624, 2.9220]])\nbias: tensor(&#91;1.2183])<\/code><\/pre>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\"   class=\"lazyload\" data-src=\"https:\/\/cos.forillusion.top\/wp-content\/uploads\/2025\/02\/3.11_output3.png\" src=\"https:\/\/cdn.forillusion.com\/moezx\/img\/svg\/loader\/trans.ajax-spinner-preloader.svg\" onerror=\"imgError(this)\"  alt=\"\"\/><\/figure >\n<noscript><img decoding=\"async\" src=\"https:\/\/cos.forillusion.top\/wp-content\/uploads\/2025\/02\/3.11_output3.png\" alt=\"\"\/><\/figure><\/noscript>\n","protected":false},"excerpt":{"rendered":"<p>\u8bad\u7ec3\u8bef\u5dee\u548c\u6cdb\u5316\u8bef\u5dee \u8bad\u7ec3\u8bef\u5dee\uff08training error\uff09\u6307\u6a21\u578b\u5728\u8bad\u7ec3\u6570\u636e\u96c6\u4e0a\u8868\u73b0\u51fa\u7684\u8bef\u5dee\uff0c\u6cdb\u5316\u8bef\u5dee\uff08generalizatio &#8230;<\/p>","protected":false},"author":1,"featured_media":1675,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[46,3],"tags":[45,44,12,22],"class_list":["post-1673","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-46","category-3","tag-45","tag-44","tag-12","tag-22"],"_links":{"self":[{"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/posts\/1673","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/comments?post=1673"}],"version-history":[{"count":1,"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/posts\/1673\/revisions"}],"predecessor-version":[{"id":1711,"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/posts\/1673\/revisions\/1711"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/media\/1675"}],"wp:attachment":[{"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/media?parent=1673"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/categories?post=1673"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/tags?post=1673"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}