{"id":555,"date":"2023-07-25T00:15:00","date_gmt":"2023-07-24T16:15:00","guid":{"rendered":"https:\/\/blog.cirno.fun\/?p=555"},"modified":"2025-03-19T00:32:11","modified_gmt":"2025-03-18T16:32:11","slug":"chap6-%e5%8f%8d%e5%90%91%e4%bc%a0%e6%92%ad","status":"publish","type":"post","link":"https:\/\/blog.sssn.tech\/?p=555","title":{"rendered":"[Chap6]-\u53cd\u5411\u4f20\u64ad"},"content":{"rendered":"<h2>\u4ee3\u4ef7\u51fd\u6570<\/h2>\n<p>\u9996\u5148\u505a\u51fa\u4e00\u4e9b\u7ea6\u5b9a\uff0c\u4ee5\u65b9\u4fbf\u540e\u9762\u8ba8\u8bba<\/p>\n<p>\u6211\u4eec\u5047\u8bbe\u795e\u7ecf\u7f51\u7edc\u7684\u8bad\u7ec3\u6837\u672c\u6709m\u4e2a\uff0c\u5206\u522b\u662f<code>(x^{(i)}, y^{(i)})<\/code>L\u8868\u793a\u795e\u7ecf\u7f51\u7edc\u7684\u5c42\u6570 <code>S_l<\/code>\u8868\u793a\u7b2cl\u5c42\u7684\u795e\u7ecf\u5143\u4e2a\u6570\uff0c\u81ea\u7136\u7684\uff0c<code>S_L<\/code>\u8868\u793a\u8f93\u51fa\u5c42\u7684\u795e\u7ecf\u5143\u4e2a\u6570<\/p>\n<p>\u795e\u7ecf\u7f51\u7edc\u7684\u5206\u7c7b\u6709\u4e24\u79cd\u60c5\u51b5\uff0c\u4e8c\u5143\u5206\u7c7b\u548c\u591a\u5143\u5206\u7c7b\uff0c\u6ce8\u610f\u4e0b\u56fe\u5728\u591a\u5206\u7c7b\u4e2d\u4f7f\u7528\u7684\u7684One-Hot\u7f16\u7801<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/blog.sssn.tech\/wp-content\/uploads\/2023\/07\/image-1690210262160.png\" alt=\"file\" \/><\/p>\n<p>\u73b0\u5728\u590d\u4e60\u4e00\u4e0b\u903b\u8f91\u56de\u5f52\u7684\u4ee3\u4ef7\u51fd\u6570<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/blog.sssn.tech\/wp-content\/uploads\/2023\/07\/image-1690210290931.png\" alt=\"file\" \/><\/p>\n<p>\u5728\u903b\u8f91\u56de\u5f52\u4e2d\uff0cx\u548cy\u90fd\u662f\u6807\u91cf\uff0c\u4f46\u662f\u5728\u795e\u7ecf\u7f51\u7edc\u4e2d\uff0cx\u662f\u5411\u91cf\uff0c\u7ef4\u5ea6\u662f\u7279\u5f81\u6570\u91cf\uff0c<code>h_\\theta(x)<\/code>\u662fK\u7ef4\u5411\u91cf\uff0cK\u662f\u5206\u7c7b\u6570\u76ee\uff0c\u56e0\u6b64<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/blog.sssn.tech\/wp-content\/uploads\/2023\/07\/image-1690210515048.png\" alt=\"file\" \/><\/p>\n<p>\u540e\u8005\u662f\u6211\u4eec\u505a\u7684\u7ea6\u5b9a<\/p>\n<p>\u6240\u4ee5\u795e\u7ecf\u7f51\u7edc\u7684\u4ee3\u4ef7\u51fd\u6570\u662f<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/blog.sssn.tech\/wp-content\/uploads\/2023\/07\/image-1690210567399.png\" alt=\"file\" \/>\u6ce8\u610f\u56fe\u4e2d\u6c42\u548c\u7684\u7b2c\u4e09\u5c42\u5e94\u8be5\u6539\u6210<code>s_(l+1)<\/code>\uff0c\u539f\u56e0\u4e0b\u9762\u4f1a\u89e3\u91ca<\/p>\n<p>\u5148\u8bf4\u524d\u534a\u90e8\u5206\u7684\u53d8\u5316\uff0c\u6bd4\u539f\u5148\u591a\u4e86\u4ecek=1\u5230k\u7684\u6c42\u548c\uff0c\u4ee3\u8868\u5bf9\u4e8e\u6bcf\u4e00\u4e2a\u7ec4\u6570\u636e\uff08x,y\uff09\uff08m\u662f\u5728\u679a\u4e3e\u6240\u6709\u6d4b\u8bd5\u6570\u636e\uff09\uff0c\u90fd\u9700\u8981\u5bf9\u6240\u6709\u8f93\u51fa\u8282\u70b9\u4ea7\u751f\u7684\u8bef\u5dee\u6c42\u548c\uff0c\u6ce8\u610fk\u662f\u5206\u7c7b\u6570\u76ee\uff0c\u81ea\u7136\u4e5f\u662f\u8f93\u51fa\u5c42\u7684\u8282\u70b9\u6570\u76ee<\/p>\n<p>\u518d\u8bf4\u540e\u9762\u6b63\u5219\u5316\u7684\u90e8\u5206\uff0c\u8fd9\u91cc\u5b9e\u9645\u4e0a\u8fd8\u662f\u5bf9\u6bcf\u4e00\u5c42\u9664\u4e86<code>\\theta_0<\/code>\u4ee5\u5916\u7684\u53c2\u6570\u6b63\u5219\u5316<\/p>\n<p>\u9996\u5148l\u679a\u4e3e\u5c42\u6570\uff0c\u6839\u636e\u53c2\u6570\u77e9\u9635\u7684\u5b9a\u4e49\uff0c<code>\\theta^{(l)}<\/code>\u8868\u793a\u4ecel\u5c42\u5230l+1\u5c42\u7684\u53c2\u6570\u77e9\u9635\uff0c\u81ea\u7136\u8f93\u51fa\u5c42\u5411\u540e\u7684\u53c2\u6570\u77e9\u9635\uff0cl\u7684\u8303\u56f4\u662f1\u5230L-1<\/p>\n<p>\u7136\u540eij\u679a\u4e3e\u53c2\u6570\u77e9\u9635\u7684\u5143\u7d20\uff0c\u770b\u4e00\u4e0b\u4e0b\u9762\u8fd9\u5f20\u56fe\uff0c\u4ee3\u8868\u7b2cl\u5c42\u7684\u7f51\u7edc\u60c5\u51b5<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/blog.sssn.tech\/wp-content\/uploads\/2023\/07\/image-1690213297541.png\" alt=\"file\" \/><\/p>\n<p>\u9996\u5148\u6ce8\u610f\uff0c\u53c2\u6570\u77e9\u9635\u7684\u7f16\u7801\u65b9\u5f0f\u662f\u4ece\u7ec8\u70b9\u5230\u8d77\u70b9\uff0c\u6bd4\u5982m1\u5230n2\u7684\u8fb9\u662ftheta[2][1]\uff0c\u8fd9\u662f\u4e3a\u4e86\u65b9\u4fbf\u5411\u91cf\u5316\uff0c\u770b\u5982\u4e0b\u56fe<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/blog.sssn.tech\/wp-content\/uploads\/2023\/07\/image-1690214991110.png\" alt=\"file\" \/><\/p>\n<p>\u7ea2\u8272\u90e8\u5206\u662f\u5ffd\u7565\u504f\u7f6e\u540e\u7684\u7ed3\u679c\uff0c\u5b9e\u9645\u4e0a\u4e0d\u5ffd\u7565\u4e5f\u4e0d\u5f71\u54cd\u7ed3\u679c<\/p>\n<p>\u6839\u636e\u60ef\u4f8b\uff0c\u504f\u6267\u5355\u5143(m0)\u7684\u51fa\u8fb9\u8fb9\u6743\uff0c\u5373\u84dd\u8272\u90e8\u5206\u4e0d\u53c2\u4e0e\u6b63\u5219\u5316\uff0c\u6240\u4ee5theta[0][i]\u4e0d\u53c2\u4e0e\u6b63\u5219\u5316\u3002<\/p>\n<p>\u800c\u524d\u4e00\u7ea7\u8282\u70b9\u6ca1\u6709\u5230\u4e0b\u4e00\u7ea7\u504f\u6267\u5355\u5143\u8282\u70b9\u7684\u51fa\u8fb9\uff08\u504f\u7f6e\u5355\u5143\u603b\u662f\u4e3a1\u7684\uff0c\u4ed6\u4e0d\u4f1a\u66f4\u65b0\uff0c\u81ea\u7136\u4e0d\u7528\u63a5\u53d7\u53c2\u6570\uff09\uff0c\u6240\u4ee5theta[i][0]\u4e0d\u53c2\u4e0e\u6b63\u5219\u5316<\/p>\n<p>\u6839\u636etheta\u77e9\u9635\u7684\u5f62\u72b6\u5b9a\u4e49\uff0c\u81ea\u7136\u6709i\u4ece1\u5230s<em>l\uff0c\u5373m\u5c42\u8282\u70b9\u6570\u76ee,\u800cj\u662f1\u5230s<\/em>(l+1)\uff0c\u5373n\u5c42\u8282\u70b9\u6570\u76ee<\/p>\n<h2>\u53cd\u5411\u4f20\u64ad<\/h2>\n<p>\u5570\u55e6\u8fd9\u4e48\u4e45\uff0c\u7ec8\u4e8e\u53ef\u4ee5\u8fdb\u5165\u6b63\u6587\u4e86\uff0c\u53cd\u5411\u4f20\u64ad\u662f\u795e\u7ecf\u7f51\u7edc\u7528\u6765\u66f4\u65b0\u53c2\u6570\u77e9\u9635\uff0c\u4ee5\u6b64\u6700\u5c0f\u5316\u4ee3\u4ef7\u51fd\u6570\u7684\u7b97\u6cd5\uff0c\u4ed6\u7684\u8fdb\u7a0b\u51e0\u4e4e\u662f\u6b63\u5411\u4f20\u64ad\u7684\u9006\u5e8f<\/p>\n<p>\u8981\u6700\u5c0f\u5316\u4ee3\u4ef7\u51fd\u6570\uff0c\u6211\u4eec\u9700\u8981\u77e5\u9053\u4ee3\u4ef7\u51fd\u6570\u5bf9\u6bcf\u4e2a\u53c2\u6570\u7684\u504f\u5bfc\u6570\uff0c\u4ee5\u6b64\u6765\u5b9e\u884c\u68af\u5ea6\u4e0b\u964d\u6216\u5176\u4ed6\u9ad8\u7ea7\u7684\u4f18\u5316\u6cd5\uff0c\u5373<\/p>\n<pre><code class=\"language-katex\">\\frac{\\partial}{\\partial \\Theta ^{(l)}_{ij}} J(\\Theta)<\/code><\/pre>\n<p>\u73b0\u5728\u5047\u8bbe\u6211\u4eec\u7684\u8bad\u7ec3\u96c6\u53ea\u6709\u4e00\u7ec4\u6570\u636e(x, y)\uff0c\u5728\u56fe\u4e2d\u7684\u795e\u7ecf\u7f51\u7edc\u4e2d\uff0c\u524d\u5411\u4f20\u64ad\u7684\u6267\u884c\u662f\u8fd9\u6837\u7684<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/blog.sssn.tech\/wp-content\/uploads\/2023\/07\/image-1690215732490.png\" alt=\"file\" \/><\/p>\n<p>\u6ce8\u610f\u4e0a\u56fe\u4e2d\u7528\u7684\u662f\u5411\u91cf\u5316\u7684\u8868\u8fbe<\/p>\n<p>\u6211\u4eec\u7528<code>\\delta^{(i)}<\/code>\u6765\u8868\u793a\u6bcf\u4e00\u5c42\u7684\u8bef\u5dee\uff0c\u663e\u7136\u5728\u8f93\u51fa\u5c42\u4e2d,\u5373\uff0cl=L\u65f6<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/blog.sssn.tech\/wp-content\/uploads\/2023\/07\/image-1690215979430.png\" alt=\"file\" \/><\/p>\n<p>\u6211\u4eec\u7528\u8fd9\u4e00\u5c42\u7684\u8bef\u5dee\u53cd\u5411\u8f6c\u64ad\u81f3\u524d\u4e00\u5c42\uff0c\u6765\u63a8\u7b97\u524d\u4e00\u5c42\u7684\u8bef\u5dee<\/p>\n<pre><code class=\"language-katex\">\\delta ^{(i)}=(\\Theta^{(i)})^T\\delta ^{(i+1)}*g&#039;(z^{(i)}) , i=1,2,...,L-1<\/code><\/pre>\n<p>\u8fd9\u91cc\u7684\u661f\u53f7\u662f\u70b9\u4e58\u7684\u610f\u601d<\/p>\n<p>\u5728\u4e0a\u5f0f\u4e2d<\/p>\n<pre><code class=\"language-katex\">g&#039;(z^{(i)}) =a^{(i)}(1-a^{(i)})<\/code><\/pre>\n<p>\u8fd9\u662f\u56e0\u4e3ag\u662fsigmoid\u51fd\u6570<\/p>\n<pre><code class=\"language-katex\">g(x) = \\frac{1}{1+e^{-x}}\n\\\\\ng&#039;(x) = g(x)(1-g(x))<\/code><\/pre>\n<p>sigmoid\u51fd\u6570\u7684\u6c42\u5bfc\u975e\u5e38\u5e38\u7528\uff0c\u9700\u8981\u7279\u522b\u8bb0\u5fc6\u4e00\u4e0b\uff0c\u5426\u5219\u5c31\u4f1a\u548c\u6211\u4e00\u6837\uff0c\u522b\u4eba\u4e00\u6b65\u8fc7\u7684\u6b65\u9aa4\u95ee\u534a\u5929\uff08orz\uff09<\/p>\n<p>\u5728\u6211\u4eec\u66f4\u65b0\u5b8c\u6210\u6240\u6709\u7684delta\u8868\u8fbe\u5f0f\u540e\uff0c\u4fbf\u53ef\u4ee5\u8ba1\u7b97\u4ee3\u4ef7\u51fd\u6570\u7684\u504f\u5bfc\u6570\u4e86\uff0c\u4ee5\u4e0b\u662f\u4e0d\u52a0\u6b63\u5219\u5316\u5904\u7406\u7684\u60c5\u51b5<\/p>\n<pre><code class=\"language-katex\">\\frac{\\partial}{\\partial \\Theta ^{(l)}_{ij}} J(\\Theta)=a_j^{(l)}\\delta_i^{l+1}<\/code><\/pre>\n<p>\u8fd9\u91cc\u63a8\u5bfc\u6bd4\u8f83\u590d\u6742\uff0c\u770b\u4e86\u4e2a\u534a\u61c2\u4e0d\u61c2\uff0c\u62bd\u65f6\u95f4\u4e13\u95e8\u5199\u4e00\u7bc7\u535a\u5ba2\uff0c\u73b0\u5728\u6682\u65f6\u5148\u5f80\u540e\u8fdb\u884c\uff0c\u6bd5\u7adf\u6b65\u6b65\u8ffd\u6c42\u4e25\u8c28\u7684\u8bc1\u660e\u662f\u5f88\u5f71\u54cd\u5b66\u4e60\u6548\u7387\u7684\uff08\u9003\uff09<\/p>\n<p>\u753b\u4e2a\u56fe\u76f4\u89c2\u7684\u770b\u4e00\u4e0b\uff0c\u53ef\u4ee5\u770b\u5230\u53cd\u5411\u4f20\u64ad\u5728\u9006\u5411\u7684\u4f20\u64ad\u8bef\u5dee<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/blog.sssn.tech\/wp-content\/uploads\/2023\/07\/image-1690217061585.png\" alt=\"file\" \/><\/p>\n<p>\u73b0\u5728\u63a8\u5e7f\u5230\u4e00\u822c\u60c5\u51b5\uff0c\u6211\u4eec\u8003\u8651\u6b63\u5219\u5316\u5904\u7406\uff0c\u5e76\u4e14\u6211\u4eec\u7684\u8bad\u7ec3\u96c6\u662f\u4e00\u4e2a\u77e9\u9635\uff0c\u6b64\u65f6\u7684\u8bef\u5dee\u5355\u5143\u4e5f\u662f\u4e00\u4e2a\u77e9\u9635\uff0c\u6211\u4eec\u7528<code>\\Delta_{ij}^{(l)}<\/code>\u8868\u793a\u7b2cl\u5c42\u7b2ci\u4e2a\u795e\u7ecf\u5143\u53d7\u5230\u7b2cj\u4e2a\u53c2\u6570\u5f71\u54cd\u7684\u8bef\u5dee,\u5373\u7b2cl\u5c42ij\u8fd9\u6761\u8fb9\u7684\u8bef\u5dee \u7b97\u6cd5\u8868\u793a\u5982\u4e0b<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/blog.sssn.tech\/wp-content\/uploads\/2023\/07\/image-1690217342060.png\" alt=\"file\" \/><\/p>\n<p>\u5373\u9996\u5148\u7528\u6b63\u5411\u4f20\u64ad\u65b9\u6cd5\u8ba1\u7b97\u51fa\u6bcf\u4e00\u5c42\u7684\u6fc0\u6d3b\u5355\u5143\uff0c\u5229\u7528\u8bad\u7ec3\u96c6\u7684\u7ed3\u679c\u4e0e\u795e\u7ecf\u7f51\u7edc\u9884\u6d4b\u7684\u7ed3\u679c\u6c42\u51fa\u6700\u540e\u4e00\u5c42\u7684\u8bef\u5dee\uff0c\u7136\u540e\u5229\u7528\u8be5\u8bef\u5dee\u8fd0\u7528\u53cd\u5411\u4f20\u64ad\u6cd5\u8ba1\u7b97\u51fa\u76f4\u81f3\u7b2c\u4e8c\u5c42\u7684\u6240\u6709\u8bef\u5dee<\/p>\n<p>\u7136\u540e\u7528\u4e0b\u9762\u65b9\u6cd5\u8ba1\u7b97D\uff0c\u6309\u7167\u662f\u5426\u6709\u6b63\u5219\u5316\u6765\u8ba8\u8bba\uff0c\u5f53j=0\u65f6\uff0c\u6309\u7167\u60ef\u4f8b\u4e0d\u6b63\u5219\u5316\uff08\u84dd\u8272\u7ebf\uff09<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/blog.sssn.tech\/wp-content\/uploads\/2023\/07\/image-1690217451728.png\" alt=\"file\" \/><\/p>\n<p>\u53ef\u4ee5\u8bc1\u660e<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/blog.sssn.tech\/wp-content\/uploads\/2023\/07\/image-1690278234642.png\" alt=\"file\" \/><\/p>\n<p>\u8bc1\u660e\u5f88\u590d\u6742\uff0c\u8fd9\u91cc\u4e0d\u5199\u4e86\uff0c\u6709\u65f6\u95f4\u4e13\u95e8\u7814\u7a76\u4e00\u4e0b<\/p>\n<p>\u8fd9\u8282\u96be\u5ea6\u7565\u5927\uff0c\u501f\u7528\u5434\u6069\u8fbe\u8001\u5e08\u7684\u8bdd\u8bf4\uff0c\u5927\u90e8\u5206\u4eba\u5728\u7b2c\u4e00\u6b21\u63a5\u89e6\u53cd\u5411\u4f20\u64ad\u7b97\u6cd5\u65f6\u90fd\u4f1a\u611f\u5230\u56f0\u60d1\uff0c\u5b83\u4e0d\u50cf\u7ebf\u6027\u56de\u5f52\u6216\u8005\u903b\u8f91\u56de\u5f52\u90a3\u6837&quot;mathematical clear&quot;\uff0c\u8fd9\u662f\u5b8c\u5168\u6b63\u5e38\u7684\uff0c\u5373\u4f7f\u662f\u5f88\u591a\u7814\u7a76\u4eba\u5458\uff0c\u5728\u5e94\u7528\u53cd\u5411\u4f20\u64ad\u7b97\u6cd5\u8fbe\u6210\u5f88\u591a\u76ee\u7684\u540e\uff0c\u4e5f\u4e0d\u5b8c\u5168\u6e05\u695a\u4ed6\u7684\u5b8c\u6574\u539f\u7406\uff08\u6743\u5f53\u8c26\u865a\uff09<\/p>\n<p>\u4e0b\u4e00\u8282\u5c06\u4f1a\u4ecb\u7ecd\u5bf9\u53cd\u5411\u4f20\u64ad\u7b97\u6cd5\u7684\u611f\u6027\u7406\u89e3\u548c\u4e00\u4e9b\u5728\u5b9e\u65bd\u7b97\u6cd5\u8fc7\u7a0b\u4e2d\u7684\u6ce8\u610f\u4e8b\u9879<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u4ee3\u4ef7\u51fd\u6570 \u9996\u5148\u505a\u51fa\u4e00\u4e9b\u7ea6\u5b9a\uff0c\u4ee5\u65b9\u4fbf\u540e\u9762\u8ba8\u8bba \u6211\u4eec\u5047\u8bbe\u795e\u7ecf\u7f51\u7edc\u7684\u8bad\u7ec3\u6837\u672c\u6709m\u4e2a\uff0c\u5206 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