{"id":60410,"date":"2026-01-15T14:57:29","date_gmt":"2026-01-15T06:57:29","guid":{"rendered":"https:\/\/www.wsisp.com\/helps\/60410.html"},"modified":"2026-01-15T14:57:29","modified_gmt":"2026-01-15T06:57:29","slug":"%e9%80%bb%e8%be%91%e5%9b%9e%e5%bd%92%e7%9a%84%e5%af%b9%e6%95%b0%e6%8d%9f%e5%a4%b1","status":"publish","type":"post","link":"https:\/\/www.wsisp.com\/helps\/60410.html","title":{"rendered":"\u903b\u8f91\u56de\u5f52\u7684\u5bf9\u6570\u635f\u5931"},"content":{"rendered":"<h3>\u4ec0\u4e48\u662f\u5bf9\u6570&#xff1f;\u4e00\u53e5\u8bdd\u8bb2\u900f<\/h3>\n<p><img decoding=\"async\" alt=\"$\\\\log_b x$\" class=\"mathcode\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2026\/01\/20260115065727-69688fd785519.png\" \/>\u00a0\u7684\u610f\u601d\u5c31\u662f&#xff1a;\u8981\u628a b \u4e58\u81ea\u5df1\u591a\u5c11\u6b21&#xff0c;\u624d\u80fd\u5f97\u5230 x\u3002<\/p>\n<p>\u66f4\u6b63\u5f0f\u4e00\u70b9&#xff1a;<\/p>\n<p><img decoding=\"async\" alt=\"$\\\\log_b x &#061; y \\\\quad \\\\Longleftrightarrow \\\\quad b^y &#061; x$\" class=\"mathcode\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2026\/01\/20260115065727-69688fd78c4cc.png\" \/><\/p>\n<p>\u4e5f\u5c31\u662f\u8bf4&#xff1a;\u5bf9\u6570\u662f\u5728\u201c\u53cd\u95ee\u6307\u6570\u201d\u3002<\/p>\n<ul>\n<li>\n<p>\u6307\u6570&#xff1a;\u7ed9\u4f60 y&#xff0c;\u7b97 <img decoding=\"async\" alt=\"$b^y$\" class=\"mathcode\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2026\/01\/20260115065727-69688fd7937fe.png\" \/>\u00a0\u5f97\u5230 x<\/p>\n<\/li>\n<li>\n<p>\u5bf9\u6570&#xff1a;\u7ed9\u4f60 x&#xff0c;\u53cd\u8fc7\u6765\u95ee\u201c\u6307\u6570 y\u00a0\u662f\u591a\u5c11\u201d<\/p>\n<\/li>\n<\/ul>\n<h3>\u4f60\u53ef\u4ee5\u628a\u201c\u5bf9\u6570\u201d\u5f53\u6210\u4e00\u79cd\u5e38\u7528\u7684\u201c\u538b\u7f29\u5c3a\u5ea6\u201d<\/h3>\n<p>\u73b0\u5b9e\u91cc\u5f88\u591a\u91cf\u8de8\u5ea6\u5de8\u5927&#xff08;\u6bd4\u5982\u58f0\u97f3\u5206\u8d1d\u3001\u5730\u9707\u9707\u7ea7\u3001\u6570\u636e\u5927\u5c0f&#xff09;&#xff0c;\u7528\u5bf9\u6570\u53ef\u4ee5\u628a\u201c\u4e58\u6cd5\u589e\u957f\u201d\u53d8\u6210\u201c\u52a0\u6cd5\u589e\u957f\u201d&#xff0c;\u66f4\u597d\u770b\u3001\u66f4\u597d\u6bd4\u8f83\u3002<\/p>\n<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"315\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2026\/01\/20260115065727-69688fd799168.png\" width=\"478\" \/><\/p>\n<p>\u8fd9\u5f20\u56fe\u8bb2\u7684\u5c31\u662f\u903b\u8f91\u56de\u5f52&#xff08;Logistic Regression&#xff09;\u6700\u6838\u5fc3\u7684\u635f\u5931\u51fd\u6570&#xff1a;\u5bf9\u6570\u635f\u5931&#xff08;Log Loss \/ Logarithmic Loss&#xff09;&#xff0c;\u4e5f\u53eb \u4ea4\u53c9\u71b5\u635f\u5931&#xff08;Cross Entropy Loss&#xff09;\u3002\u6211\u7528\u5f88\u901a\u4fd7\u7684\u65b9\u5f0f\u628a\u5b83\u8bb2\u900f\u3002<\/p>\n<hr \/>\n<h3>1&#xff09;\u903b\u8f91\u56de\u5f52\u5230\u5e95\u5728\u8f93\u51fa\u4ec0\u4e48&#xff1f;<\/h3>\n<p>\u903b\u8f91\u56de\u5f52\u4e0d\u662f\u76f4\u63a5\u8f93\u51fa\u201c0 \u6216 1\u201d&#xff0c;\u800c\u662f\u8f93\u51fa\u4e00\u4e2a\u6982\u7387&#xff1a;<\/p>\n<p><img decoding=\"async\" alt=\"$a &#061; p(y&#061;1\\\\mid x)$\" class=\"mathcode\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2026\/01\/20260115065727-69688fd7aaa76.png\" \/><\/p>\n<ul>\n<li>\n<p>a \u8d8a\u63a5\u8fd1 1&#xff1a;\u6a21\u578b\u8d8a\u786e\u4fe1\u6837\u672c\u662f 1<\/p>\n<\/li>\n<li>\n<p>a \u8d8a\u63a5\u8fd1 0&#xff1a;\u6a21\u578b\u8d8a\u786e\u4fe1\u6837\u672c\u662f 0<\/p>\n<\/li>\n<\/ul>\n<p>\u8fd9\u4e2a a \u901a\u5e38\u6765\u81ea sigmoid&#xff1a;<\/p>\n<p><img decoding=\"async\" alt=\"$a&#061;\\\\sigma(z)&#061;\\\\frac{1}{1&#043;e^{-z}},\\\\quad z&#061;w^\\\\top x&#043;b$\" class=\"mathcode\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2026\/01\/20260115065727-69688fd7b0846.png\" \/><\/p>\n<hr \/>\n<h3>2&#xff09;\u5bf9\u6570\u635f\u5931\u957f\u4ec0\u4e48\u6837&#xff1f;&#xff08;\u56fe\u91cc\u90a3\u4e24\u6761\u66f2\u7ebf&#xff09;<\/h3>\n<p>\u56fe\u4e0a\u6709\u4e24\u6761\u66f2\u7ebf&#xff1a;<\/p>\n<h4>\u2705 \u5f53\u771f\u5b9e\u6807\u7b7e y&#061;1\u00a0\u65f6&#xff08;\u84dd\u7ebf&#xff09;<\/h4>\n<p>\u4f60\u5e0c\u671b\u6a21\u578b\u7ed9\u51fa a\u00a0\u8d8a\u5927\u8d8a\u597d&#xff08;\u8d8a\u63a5\u8fd1 1 \u8d8a\u5bf9&#xff09;<\/p>\n<p>\u635f\u5931\u5b9a\u4e49\u4e3a&#xff1a;<\/p>\n<p><img decoding=\"async\" alt=\"$L&#061;-\\\\log(a)$\" class=\"mathcode\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2026\/01\/20260115065727-69688fd7b7b73.png\" \/><\/p>\n<ul>\n<li>\n<p>\u5982\u679c a&#061;1&#xff1a;\u2212log\u2061(1)&#061;0&#xff0c;\u5b8c\u7f8e&#xff0c;\u635f\u5931 0<\/p>\n<\/li>\n<li>\n<p>\u5982\u679c a&#061;0.1&#xff1a;\u2212log\u2061(0.1)\u22482.30&#xff0c;\u9519\u5f97\u5f88\u79bb\u8c31&#xff0c;\u635f\u5931\u5927<\/p>\n<\/li>\n<li>\n<p>\u5982\u679c a\u21920&#xff1a;\u2212log\u2061(a)\u2192&#043;\u221e&#xff0c;\u201c\u81ea\u4fe1\u5730\u9519\u201d\u4f1a\u88ab\u6781\u5ea6\u60e9\u7f5a<\/p>\n<\/li>\n<\/ul>\n<p>\u8fd9\u5c31\u662f\u84dd\u7ebf\u5de6\u9ad8\u53f3\u4f4e\u7684\u539f\u56e0\u3002<\/p>\n<hr \/>\n<h4>\u2705 \u5f53\u771f\u5b9e\u6807\u7b7e y&#061;0\u00a0\u65f6&#xff08;\u7ea2\u7ebf&#xff09;<\/h4>\n<p>\u4f60\u5e0c\u671b\u6a21\u578b\u7ed9\u51fa a \u8d8a\u5c0f\u8d8a\u597d&#xff08;\u8d8a\u63a5\u8fd1 0 \u8d8a\u5bf9&#xff09;<\/p>\n<p>\u6b64\u65f6\u635f\u5931\u5b9a\u4e49\u4e3a&#xff1a;<\/p>\n<p><img decoding=\"async\" alt=\"$L&#061;-\\\\log(1-a)$\" class=\"mathcode\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2026\/01\/20260115065727-69688fd7bd51b.png\" \/><\/p>\n<p>\u56e0\u4e3a\u00a0<img decoding=\"async\" alt=\"$1-a &#061; p(y&#061;0\\\\mid x)$\" class=\"mathcode\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2026\/01\/20260115065727-69688fd7c253e.png\" \/><\/p>\n<ul>\n<li>\n<p>\u5982\u679c a&#061;0&#xff1a;\u2212log\u2061(1)&#061;0&#xff0c;\u5b8c\u7f8e<\/p>\n<\/li>\n<li>\n<p>\u5982\u679c a&#061;0.9&#xff1a;\u2212log\u2061(0.1)\u22482.30&#xff0c;\u9519\u5f97\u5f88\u79bb\u8c31<\/p>\n<\/li>\n<li>\n<p>\u5982\u679c a\u21921&#xff1a;\u2212log\u2061(1\u2212a)\u2192&#043;\u221e&#xff0c;\u81ea\u4fe1\u9519\u540c\u6837\u4e25\u91cd<\/p>\n<\/li>\n<\/ul>\n<p>\u8fd9\u5c31\u662f\u7ea2\u7ebf\u53f3\u9ad8\u5de6\u4f4e\u7684\u539f\u56e0\u3002<\/p>\n<hr \/>\n<h3>3&#xff09;\u628a\u4e24\u79cd\u60c5\u51b5\u5408\u8d77\u6765&#xff0c;\u5c31\u662f\u4e00\u4e2a\u7edf\u4e00\u516c\u5f0f<\/h3>\n<p>\u4f60\u770b\u5230\u56fe\u4e0a\u90a3\u4e2a\u5206\u6bb5\u5f62\u5f0f&#xff1a;<\/p>\n<p><img decoding=\"async\" alt=\"$\\\\begin{array}{c} L(a,y)&#061; \\\\begin{cases} -\\\\log(a), &amp; y&#061;1\\\\\\\\ -\\\\log(1-a), &amp; y&#061;0 \\\\end{cases} \\\\end{array}$\" class=\"mathcode\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2026\/01\/20260115065727-69688fd7c8709.png\" \/><\/p>\n<p>\u53ef\u4ee5\u5408\u5e76\u6210\u4e00\u4e2a\u66f4\u5e38\u7528\u7684\u5f62\u5f0f&#xff1a;<\/p>\n<p><img decoding=\"async\" alt=\"$\\\\boxed{ L(a,y)&#061; -\\\\left[y\\\\log(a)&#043;(1-y)\\\\log(1-a)\\\\right] }$\" class=\"mathcode\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2026\/01\/20260115065727-69688fd7d04fc.png\" \/><\/p>\n<p>\u8fd9\u5c31\u662f\u903b\u8f91\u56de\u5f52\u7684\u5bf9\u6570\u635f\u5931 \/ \u4ea4\u53c9\u71b5\u635f\u5931\u3002<\/p>\n<hr \/>\n<h3>4&#xff09;\u4e3a\u4ec0\u4e48\u8981\u7528 \u201clog\u201d&#xff1f;\u76f4\u89c9\u975e\u5e38\u91cd\u8981&#xff01;<\/h3>\n<p>\u4f60\u53ef\u4ee5\u628a\u5b83\u7406\u89e3\u6210\u4e00\u53e5\u8bdd&#xff1a;<\/p>\n<p>\u6a21\u578b\u8d8a\u201c\u81ea\u4fe1\u5730\u7b54\u9519\u201d&#xff0c;\u5c31\u7f5a\u5f97\u8d8a\u72e0\u3002<\/p>\n<h4>\u4e3e\u4e2a\u7279\u522b\u76f4\u89c2\u7684\u4f8b\u5b50<\/h4>\n<p>\u771f\u5b9e\u662f y&#061;1&#xff1a;<\/p>\n<table>\n<tr>\u9884\u6d4b\u6982\u7387 a\u635f\u5931 \u2212log\u2061(a)\u00a0\u611f\u89c9<\/tr>\n<tbody>\n<tr>\n<td>0.9<\/td>\n<td>0.105<\/td>\n<td>\u201c\u57fa\u672c\u7b54\u5bf9\u201d<\/td>\n<\/tr>\n<tr>\n<td>0.6<\/td>\n<td>0.511<\/td>\n<td>\u201c\u6709\u70b9\u72b9\u8c6b\u201d<\/td>\n<\/tr>\n<tr>\n<td>0.1<\/td>\n<td>2.303<\/td>\n<td>\u201c\u975e\u5e38\u81ea\u4fe1\u4f46\u9519\u4e86\u201d<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\u4f60\u4f1a\u53d1\u73b0&#xff1a;<\/p>\n<ul>\n<li>\n<p>\u4ece 0.9 \u2192 0.6&#xff0c;\u635f\u5931\u589e\u52a0\u4e00\u70b9\u70b9&#xff08;\u8fd8\u80fd\u63a5\u53d7&#xff09;<\/p>\n<\/li>\n<li>\n<p>\u4ece 0.6 \u2192 0.1&#xff0c;\u635f\u5931\u66b4\u6da8&#xff08;\u9519\u5f97\u592a\u79bb\u8c31&#xff09;<\/p>\n<\/li>\n<\/ul>\n<p>\u8fd9\u5c31\u662f log \u7684\u201c\u60e9\u7f5a\u52a0\u901f\u201d\u6548\u679c\u3002<\/p>\n<hr \/>\n<h3>5&#xff09;\u5b83\u5176\u5b9e\u7b49\u4ef7\u4e8e\u201c\u6700\u5927\u4f3c\u7136\u201d\u7684\u63a8\u5bfc\u7ed3\u679c<\/h3>\n<p>\u5982\u679c\u4f60\u628a\u6a21\u578b\u8f93\u51fa\u770b\u6210\u6982\u7387&#xff1a;<\/p>\n<ul>\n<li>\n<p>\u5f53 y&#061;1&#xff0c;\u6a21\u578b\u7ed9\u51fa\u6982\u7387 a<\/p>\n<\/li>\n<li>\n<p>\u5f53 y&#061;0&#xff0c;\u6a21\u578b\u7ed9\u51fa\u6982\u7387 1\u2212a<\/p>\n<\/li>\n<\/ul>\n<p>\u90a3\u4e48\u4e00\u4e2a\u6837\u672c\u7684\u201c\u6a21\u578b\u8ba4\u4e3a\u5b83\u53d1\u751f\u7684\u6982\u7387\u201d\u53ef\u4ee5\u5199\u6210&#xff1a;<\/p>\n<p><img decoding=\"async\" alt=\"$P(y\\\\mid x)&#061;a^y(1-a)^{1-y}$\" class=\"mathcode\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2026\/01\/20260115065727-69688fd7d9a72.png\" \/><\/p>\n<p>\u8bad\u7ec3\u65f6&#xff0c;\u6211\u4eec\u5e0c\u671b\u201c\u8fd9\u4e9b\u771f\u5b9e\u6807\u7b7e\u51fa\u73b0\u7684\u6982\u7387\u5c3d\u91cf\u5927\u201d<br \/>\n\u2192 \u6700\u5927\u5316\u6982\u7387&#xff08;\u4f3c\u7136&#xff09;<br \/>\n\u2192 \u7b49\u4ef7\u4e8e \u6700\u5c0f\u5316\u8d1f\u5bf9\u6570\u4f3c\u7136<\/p>\n<p><img decoding=\"async\" alt=\"$-\\\\log P(y\\\\mid x) &#061; -\\\\left[y\\\\log(a)&#043;(1-y)\\\\log(1-a)\\\\right]$\" class=\"mathcode\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2026\/01\/20260115065727-69688fd7e0f38.png\" \/><\/p>\n<p>\u6240\u4ee5\u5bf9\u6570\u635f\u5931\u4e0d\u662f\u62cd\u8111\u888b\u5b9a\u7684&#xff0c;\u5b83\u662f\u975e\u5e38\u81ea\u7136\u3001\u5408\u7406\u7684\u7edf\u8ba1\u7ed3\u679c\u3002<\/p>\n<hr \/>\n<h3>6&#xff09;\u4e00\u53e5\u8bdd\u603b\u7ed3\u56fe\u7684\u542b\u4e49&#xff08;\u4f60\u770b\u61c2\u5c31\u8d62\u4e86&#xff09;<\/h3>\n<ul>\n<li>\n<p>\u84dd\u7ebf \u2212log\u2061(a)&#xff1a;\u771f\u5b9e\u662f 1 \u65f6&#xff0c;a \u8d8a\u5c0f\u8d8a\u60e8<\/p>\n<\/li>\n<li>\n<p>\u7ea2\u7ebf \u2212log\u2061(1\u2212a)&#xff1a;\u771f\u5b9e\u662f 0 \u65f6&#xff0c;a \u8d8a\u5927\u8d8a\u60e8<\/p>\n<\/li>\n<li>\n<p>\u5728 0.5 \u9644\u8fd1\u4e24\u6761\u7ebf\u4ea4\u53c9&#xff1a;\u8868\u793a\u201c\u4f60\u4e0d\u786e\u5b9a\u201d\u65f6\u635f\u5931\u90fd\u5dee\u4e0d\u591a<\/p>\n<\/li>\n<\/ul>\n<p>\u6211\u4eec\u7ee7\u7eed&#xff0c;\u7528\u4e00\u4e2a\u8d85\u5177\u4f53\u3001\u80fd\u624b\u7b97\u7684\u5c0f\u4f8b\u5b50&#xff0c;\u628a\u903b\u8f91\u56de\u5f52\u7684 log loss \u4ece\u5934\u7b97\u5230\u5c3e&#xff08;\u8fd8\u987a\u4fbf\u770b\u61c2\u201cmini-batch \u5e73\u5747\u635f\u5931\u201d&#xff09;\u3002<\/p>\n<hr \/>\n<h2>1&#xff09;\u5148\u5b9a\u4e00\u4e2a mini-batch&#xff1a;3 \u4e2a\u6837\u672c<\/h2>\n<p>\u6211\u4eec\u5047\u8bbe\u6a21\u578b\u5df2\u7ecf\u7b97\u51fa\u4e86\u6bcf\u4e2a\u6837\u672c\u201c\u662f\u6b63\u7c7b y&#061;1\u201d\u7684\u6982\u7387 a\u3002<\/p>\n<table border=\"1\" cellpadding=\"1\" cellspacing=\"1\" style=\"width:500px\">\n<tbody>\n<tr>\n<td>\u6837\u672c<\/td>\n<td>\u771f\u5b9e\u6807\u7b7e y<\/td>\n<td>\u6a21\u578b\u9884\u6d4b\u6982\u7387 a&#061;p(y&#061;1\u2223x)<\/td>\n<\/tr>\n<tr>\n<td>1<\/td>\n<td>1<\/td>\n<td>0.9<\/td>\n<\/tr>\n<tr>\n<td>2<\/td>\n<td>1<\/td>\n<td>0.6<\/td>\n<\/tr>\n<tr>\n<td>3<\/td>\n<td>0<\/td>\n<td>0.2<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<hr \/>\n<h2>2&#xff09;\u6309\u516c\u5f0f\u9010\u4e2a\u7b97 log loss<\/h2>\n<p>\u7edf\u4e00\u516c\u5f0f&#xff1a;<\/p>\n<p><img decoding=\"async\" alt=\"$L(a,y)&#061; -\\\\left[y\\\\log(a)&#043;(1-y)\\\\log(1-a)\\\\right]$\" class=\"mathcode\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2026\/01\/20260115065727-69688fd7ec8d1.png\" \/><\/p>\n<p>\u9ed8\u8ba4 log \u662f\u81ea\u7136\u5bf9\u6570 ln\u2061\u3002<\/p>\n<hr \/>\n<h3>\u2705 \u6837\u672c1&#xff1a;y&#061;1,\u00a0a&#061;0.9<\/h3>\n<p><img decoding=\"async\" alt=\"$L_1 &#061; -\\\\log(0.9) \\\\approx 0.105$\" class=\"mathcode\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2026\/01\/20260115065728-69688fd80259f.png\" \/><\/p>\n<p>\u89e3\u91ca&#xff1a;\u6a21\u578b\u9884\u6d4b 90% \u662f\u6b63\u7c7b&#xff0c;\u800c\u4e14\u771f\u7684\u662f\u6b63\u7c7b \u2192 \u635f\u5931\u5f88\u5c0f<\/p>\n<hr \/>\n<h3>\u2705 \u6837\u672c2&#xff1a;y&#061;1,\u00a0a&#061;0.6<\/h3>\n<p><img decoding=\"async\" alt=\"$L_2 &#061; -\\\\log(0.6) \\\\approx 0.511$\" class=\"mathcode\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2026\/01\/20260115065728-69688fd8091a5.png\" \/><\/p>\n<p>\u89e3\u91ca&#xff1a;\u6a21\u578b\u4e5f\u731c\u5bf9\u4e86&#xff0c;\u4f46\u6ca1\u90a3\u4e48\u81ea\u4fe1 \u2192 \u635f\u5931\u6bd4\u6837\u672c1\u5927<\/p>\n<hr \/>\n<h3>\u2705 \u6837\u672c3&#xff1a;y&#061;0,\u00a0a&#061;0.2<\/h3>\n<p>\u6ce8\u610f&#xff1a;\u771f\u5b9e\u662f 0&#xff0c;\u5c31\u8981\u7528 \u2212log\u2061(1\u2212a)<\/p>\n<p><img decoding=\"async\" alt=\"$L_3 &#061; -\\\\log(1-0.2) &#061; -\\\\log(0.8) \\\\approx 0.223$\" class=\"mathcode\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2026\/01\/20260115065728-69688fd80f882.png\" \/><\/p>\n<p>\u89e3\u91ca&#xff1a;\u6a21\u578b\u7ed9\u6b63\u7c7b\u6982\u7387 0.2&#xff0c;\u610f\u601d\u662f\u201c\u6211\u89c9\u5f97\u4f60 80% \u662f\u8d1f\u7c7b\u201d<br \/>\n\u800c\u771f\u5b9e\u786e\u5b9e\u662f\u8d1f\u7c7b \u2192 \u635f\u5931\u4e5f\u5c0f<\/p>\n<hr \/>\n<h2>3&#xff09;mini-batch \u7684\u603b\u635f\u5931 &#061; \u5e73\u5747\u635f\u5931<\/h2>\n<p>\u903b\u8f91\u56de\u5f52\u8bad\u7ec3\u65f6\u6700\u5e38\u7528\u7684 batch loss&#xff1a;<\/p>\n<p><img decoding=\"async\" alt=\"$J &#061; \\\\frac{1}{m}\\\\sum_{i&#061;1}^{m}L_i$\" class=\"mathcode\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2026\/01\/20260115065728-69688fd8174e7.png\" \/><\/p>\n<p>\u8fd9\u91cc m&#061;3<\/p>\n<p><img decoding=\"async\" alt=\"$J&#061;\\\\frac{0.105&#043;0.511&#043;0.223}{3} &#061;\\\\frac{0.839}{3} \\\\approx 0.280$\" class=\"mathcode\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2026\/01\/20260115065728-69688fd81bee0.png\" \/><\/p>\n<p>\u2705 \u8fd9\u5c31\u662f\u8fd9\u4e00\u4e2a mini-batch \u7684\u4ea4\u53c9\u71b5\u635f\u5931\u3002<\/p>\n<hr \/>\n<h2>4&#xff09;\u611f\u53d7\u4e00\u4e0b\u201c\u81ea\u4fe1\u9519\u4e86\u4f1a\u88ab\u7f5a\u7206\u201d\u7684\u5a01\u529b<\/h2>\n<p>\u6211\u4eec\u628a\u6837\u672c3\u6539\u6210\u4e00\u4e2a\u201c\u5f88\u79bb\u8c31\u7684\u9884\u6d4b\u201d&#xff1a;<\/p>\n<p>\u771f\u5b9e\u4ecd\u7136 y&#061;0&#xff0c;\u4f46\u6a21\u578b\u5374\u7ed9 a&#061;0.99&#xff08;\u6781\u5ea6\u76f8\u4fe1\u5b83\u662f\u6b63\u7c7b&#xff09;<\/p>\n<p>\u90a3\u4e48&#xff1a;<\/p>\n<p><img decoding=\"async\" alt=\"$L&#061;-\\\\log(1-0.99) &#061; -\\\\log(0.01)\\\\approx 4.605$\" class=\"mathcode\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2026\/01\/20260115065728-69688fd824dc9.png\" \/><\/p>\n<p>\u4f60\u770b&#xff1a;<\/p>\n<ul>\n<li>\n<p>\u6b63\u786e\u4e14\u5408\u7406&#xff1a;a&#061;0.2 \u2192 \u635f\u5931 0.223<\/p>\n<\/li>\n<li>\n<p>\u81ea\u4fe1\u5730\u9519&#xff1a;a&#061;0.99 \u2192 \u635f\u5931 4.605&#xff08;\u76f4\u63a5\u7206\u70b8&#xff09;<\/p>\n<\/li>\n<\/ul>\n<p>&#x1f4cc; \u8fd9\u5c31\u662f log loss \u7684\u7075\u9b42&#xff1a;<\/p>\n<p>\u4f60\u53ef\u4ee5\u4e0d\u786e\u5b9a&#xff0c;\u4f46\u4f60\u4e0d\u80fd\u201c\u81ea\u4fe1\u5730\u9519\u201d\u3002<\/p>\n<hr \/>\n<h2>5&#xff09;\u4f60\u56fe\u91cc\u7684\u4e24\u6761\u66f2\u7ebf&#xff0c;\u73b0\u5728\u5c31\u5b8c\u5168\u901a\u4e86<\/h2>\n<ul>\n<li>\n<p>\u5f53 y&#061;1&#xff1a;\u635f\u5931 \u2212log\u2061(a)<\/p>\n<ul>\n<li>\n<p>a\u21921&#xff1a;\u635f\u5931 \u21920<\/p>\n<\/li>\n<li>\n<p>a\u21920&#xff1a;\u635f\u5931 \u2192&#043;\u221e<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li>\n<p>\u5f53 y&#061;0&#xff1a;\u635f\u5931 \u2212log\u2061(1\u2212a)<\/p>\n<ul>\n<li>\n<p>a\u21920&#xff1a;\u635f\u5931 \u21920<\/p>\n<\/li>\n<li>\n<p>a\u21921&#xff1a;\u635f\u5931 \u2192&#043;\u221e<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p>\u6240\u4ee5\u4f60\u770b\u5230\u7684\u5c31\u662f&#xff1a;<\/p>\n<ul>\n<li>\n<p>\u84dd\u7ebf&#xff08;y&#061;1&#xff09;\u5de6\u8fb9\u9ad8\u3001\u53f3\u8fb9\u4f4e<\/p>\n<\/li>\n<li>\n<p>\u7ea2\u7ebf&#xff08;y&#061;0&#xff09;\u5de6\u8fb9\u4f4e\u3001\u53f3\u8fb9\u9ad8<\/p>\n<\/li>\n<\/ul>\n<hr \/>\n<h2>6&#xff09;\u518d\u9001\u4f60\u4e00\u4e2a\u201c\u8d85\u7ea7\u8bb0\u5fc6\u53e3\u8bc0\u201d<\/h2>\n<p>\u4f60\u53ea\u8981\u8bb0\u4f4f\u4e00\u53e5\u8bdd\u5c31\u591f\u4e86&#xff1a;<\/p>\n<p>\u2705 \u771f\u5b9e\u662f\u4ec0\u4e48&#xff0c;\u5c31\u770b\u6a21\u578b\u7ed9\u5b83\u7684\u6982\u7387\u6709\u591a\u5927&#xff1b;\u6982\u7387\u8d8a\u5c0f&#xff0c;\u60e9\u7f5a\u8d8a\u5927&#xff0c;\u800c\u4e14\u662f log \u7ea7\u7206\u70b8\u3002<\/p>\n<ul>\n<li>\n<p>\u771f\u5b9e\u662f 1 \u2192 \u770b a<\/p>\n<\/li>\n<li>\n<p>\u771f\u5b9e\u662f 0 \u2192 \u770b 1\u2212a<\/p>\n<\/li>\n<\/ul>\n<p>\u6211\u4eec\u4e0b\u4e00\u6b65\u7ee7\u7eed\u8bb2\u66f4\u5173\u952e\u7684\u4e00\u70b9&#xff1a;<br \/>\n\u4e3a\u4ec0\u4e48\u903b\u8f91\u56de\u5f52\u8bad\u7ec3\u65f6&#xff0c;\u7ecf\u5e38\u4f1a\u51fa\u73b0\u4e00\u4e2a\u7279\u522b\u6f02\u4eae\u7684\u7ed3\u679c&#xff1a;<\/p>\n<p><img decoding=\"async\" alt=\"$\\\\frac{\\\\partial L}{\\\\partial z} &#061; a - y$\" class=\"mathcode\" src=\"2026-01-15z5wuw424wsa.png\" \/><\/p>\n<p>\u4e5f\u5c31\u662f\u201c\u8bef\u5dee &#061; \u9884\u6d4b &#8211; \u771f\u5b9e\u201d&#xff0c;\u68af\u5ea6\u5f62\u5f0f\u6781\u5176\u7b80\u6d01&#xff08;\u53cd\u5411\u4f20\u64ad\u91cc\u975e\u5e38\u723d&#xff09;\u3002<\/p>\n<p>\u6211\u4eec\u7ee7\u7eed\u628a\u6700\u5173\u952e\u3001\u6700\u201c\u723d\u201d\u7684\u4e00\u6b65\u8bb2\u900f&#xff1a;<\/p>\n<p>\u4e3a\u4ec0\u4e48\u903b\u8f91\u56de\u5f52 &#043; \u5bf9\u6570\u635f\u5931\u7684\u68af\u5ea6&#xff0c;\u4f1a\u53d8\u6210\u8d85\u7ea7\u7b80\u6d01\u7684&#xff1a;<br \/>\n<img decoding=\"async\" alt=\"$\\\\displaystyle \\\\frac{\\\\partial L}{\\\\partial z}&#061;a-y$\" class=\"mathcode\" src=\"2026-01-15imylsmbfpqc.png\" \/><\/p>\n<p>\u8fd9\u4e00\u6b65\u4e00\u65e6\u61c2\u4e86&#xff0c;\u4f60\u5c31\u80fd\u771f\u6b63\u660e\u767d&#xff1a;<br \/>\n\u903b\u8f91\u56de\u5f52\u4e3a\u4ec0\u4e48\u8fd9\u4e48\u597d\u8bad\u7ec3\u3001\u4e3a\u4ec0\u4e48\u68af\u5ea6\u4e0b\u964d\u8fd9\u4e48\u987a\u3002<\/p>\n<hr \/>\n<h2>0&#xff09;\u5148\u628a\u53d8\u91cf\u5173\u7cfb\u753b\u51fa\u6765&#xff08;\u94fe\u5f0f\u6cd5\u5219\u901a\u9053&#xff09;<\/h2>\n<p>\u903b\u8f91\u56de\u5f52\u662f&#xff1a;<\/p>\n<p><img decoding=\"async\" alt=\"$z &#061; w^\\\\top x &#043; b$\" class=\"mathcode\" src=\"2026-01-15fouln5jypls.png\" \/><\/p>\n<p><img decoding=\"async\" alt=\"$a &#061; \\\\sigma(z)&#061;\\\\frac{1}{1&#043;e^{-z}}$\" class=\"mathcode\" src=\"2026-01-15shgpl2x4wws.png\" \/><\/p>\n<p><img decoding=\"async\" alt=\"$L(a,y)&#061; -\\\\left[y\\\\log(a)&#043;(1-y)\\\\log(1-a)\\\\right]$\" class=\"mathcode\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2026\/01\/20260115065727-69688fd7ec8d1.png\" \/><\/p>\n<p>\u6240\u4ee5\u8ba1\u7b97\u68af\u5ea6\u65f6\u5c31\u662f\u4e00\u6761\u94fe&#xff1a;<\/p>\n<p><img decoding=\"async\" alt=\"$z \\\\rightarrow a \\\\rightarrow L$\" class=\"mathcode\" src=\"2026-01-15usqui3bquyh.png\" \/><\/p>\n<p>\u6211\u4eec\u60f3\u8981\u7684\u662f&#xff1a;<\/p>\n<p><img decoding=\"async\" alt=\"$\\\\frac{\\\\partial L}{\\\\partial z} &#061; \\\\frac{\\\\partial L}{\\\\partial a}\\\\cdot\\\\frac{\\\\partial a}{\\\\partial z}$\" class=\"mathcode\" src=\"2026-01-151vdwmebvm0q.png\" \/><\/p>\n<hr \/>\n<h2>1&#xff09;\u5148\u6c42\u00a0<img decoding=\"async\" alt=\"$\\\\frac{\\\\partial L}{\\\\partial a}$\" class=\"mathcode\" src=\"2026-01-15wwdbhlslyyp.png\" \/><\/h2>\n<p><img decoding=\"async\" alt=\"$L&#061; -\\\\left[y\\\\log(a)&#043;(1-y)\\\\log(1-a)\\\\right]$\" class=\"mathcode\" src=\"2026-01-15mbaofmbwhws.png\" \/><\/p>\n<p>\u5bf9 a\u00a0\u6c42\u5bfc&#xff1a;<\/p>\n<ul>\n<li>\n<p><img decoding=\"async\" alt=\"\\\\frac{d}{da}\\\\log(a)&#061;\\\\frac{1}{a}\" class=\"mathcode\" src=\"2026-01-154gxysx4wp5b.png\" \/><\/p>\n<\/li>\n<li>\n<p><img decoding=\"async\" alt=\"\\\\frac{d}{da}\\\\log(1-a)&#061;\\\\frac{-1}{1-a}\" class=\"mathcode\" src=\"2026-01-15ppzaizgoue2.png\" \/><\/p>\n<\/li>\n<\/ul>\n<p>\u6240\u4ee5&#xff1a;<\/p>\n<p><img decoding=\"async\" alt=\"$\\\\frac{\\\\partial L}{\\\\partial a} &#061; -\\\\left(y\\\\cdot \\\\frac{1}{a}&#043;(1-y)\\\\cdot\\\\frac{-1}{1-a}\\\\right)$\" class=\"mathcode\" src=\"2026-01-15rorqo01u4mk.png\" \/><\/p>\n<p>\u6574\u7406\u4e00\u4e0b&#xff1a;<\/p>\n<p><img decoding=\"async\" alt=\"$\\\\frac{\\\\partial L}{\\\\partial a} &#061; -\\\\frac{y}{a}&#043;\\\\frac{1-y}{1-a}$\" class=\"mathcode\" src=\"2026-01-15rheawqj41tk.png\" \/><\/p>\n<hr \/>\n<h2>2&#xff09;\u518d\u6c42 <img decoding=\"async\" alt=\"$\\\\frac{\\\\partial a}{\\\\partial z}$\" class=\"mathcode\" src=\"2026-01-15prlldnnneyq.png\" \/>\u200b&#xff08;sigmoid \u7684\u5bfc\u6570&#xff09;<\/h2>\n<h4>a&#061;\u03c3(z)<\/h4>\n<p>\u4e00\u4e2a\u975e\u5e38\u7ecf\u5178\u7684\u7ed3\u8bba&#xff1a;<\/p>\n<p><img decoding=\"async\" alt=\"$\\\\frac{\\\\partial a}{\\\\partial z}&#061;a(1-a)$\" class=\"mathcode\" src=\"2026-01-15e0x1ivgypp3.png\" \/><\/p>\n<hr \/>\n<h2>3&#xff09;\u94fe\u5f0f\u6cd5\u5219\u76f8\u4e58&#xff1a;<img decoding=\"async\" alt=\"$\\\\frac{\\\\partial L}{\\\\partial z}$\" class=\"mathcode\" src=\"2026-01-15hpc1od5d53q.png\" \/><\/h2>\n<p><img decoding=\"async\" alt=\"$\\\\frac{\\\\partial L}{\\\\partial z} &#061; \\\\left(-\\\\frac{y}{a}&#043;\\\\frac{1-y}{1-a}\\\\right)\\\\cdot a(1-a)$\" class=\"mathcode\" src=\"2026-01-15o3whhex3mfc.png\" \/><\/p>\n<p>\u628a a(1\u2212a) \u5206\u522b\u4e58\u8fdb\u53bb&#xff1a;<\/p>\n<p><img decoding=\"async\" alt=\"$\\\\frac{\\\\partial L}{\\\\partial z} &#061; -\\\\frac{y}{a}\\\\cdot a(1-a)&#043;\\\\frac{1-y}{1-a}\\\\cdot a(1-a)$\" class=\"mathcode\" src=\"2026-01-153lmvetnfks2.png\" \/><\/p>\n<p>\u7ea6\u6389&#xff1a;<\/p>\n<p><img decoding=\"async\" alt=\"$&#061; -y(1-a) &#043; (1-y)a$\" class=\"mathcode\" src=\"2026-01-15a4hmk5bpihr.png\" \/><\/p>\n<p>\u5c55\u5f00&#xff1a;<\/p>\n<p><img decoding=\"async\" alt=\"$&#061; -y &#043; ya &#043; a - ya$\" class=\"mathcode\" src=\"2026-01-15rhcz5geptiu.png\" \/><\/p>\n<p>\u4e2d\u95f4 ya\u00a0\u62b5\u6d88\u4e86&#xff0c;\u5269\u4e0b&#xff1a;<\/p>\n<p><img decoding=\"async\" alt=\"$\\\\boxed{\\\\frac{\\\\partial L}{\\\\partial z}&#061;a-y}$\" class=\"mathcode\" src=\"2026-01-15wx2javfn3bb.png\" \/><\/p>\n<p>\u2705 \u8fd9\u5c31\u662f\u795e\u5947\u7684\u7b80\u6d01\u7ed3\u679c&#xff01;<\/p>\n<hr \/>\n<h2>4&#xff09;\u8fd9\u4e2a\u7ed3\u679c\u4e3a\u4ec0\u4e48\u201c\u7279\u522b\u6709\u76f4\u89c9\u201d&#xff1f;<\/h2>\n<p><img decoding=\"async\" alt=\"$\\\\frac{\\\\partial L}{\\\\partial z}&#061;a-y$\" class=\"mathcode\" src=\"2026-01-15ahzscjbg32g.png\" \/><\/p>\n<p>\u5b83\u8bf4\u7684\u5c31\u662f\u4e00\u53e5\u8bdd&#xff1a;<\/p>\n<p>\u68af\u5ea6 &#061; \u9884\u6d4b\u6982\u7387 \u2212 \u771f\u5b9e\u6807\u7b7e<\/p>\n<ul>\n<li>\n<p>\u5982\u679c\u771f\u5b9e y&#061;1&#xff0c;\u4f46\u4f60\u9884\u6d4b a&#061;0.2<br \/>\n<img decoding=\"async\" alt=\"$\\\\Rightarrow a-y &#061; 0.2-1&#061;-0.8$\" class=\"mathcode\" src=\"2026-01-15jki5vhzrf0t.png\" \/>&#xff08;\u8d1f\u6570\u5f88\u5927&#xff09;<br \/>\n&#x1f449; \u8bf4\u660e\u4f60\u5e94\u8be5\u628a z\u00a0\u5f80\u4e0a\u63a8&#xff08;\u8ba9 a\u00a0\u53d8\u5927&#xff09;<\/p>\n<\/li>\n<li>\n<p>\u5982\u679c\u771f\u5b9e y&#061;0&#xff0c;\u4f46\u4f60\u9884\u6d4b a&#061;0.9<br \/>\n<img decoding=\"async\" alt=\"$\\\\Rightarrow a-y &#061; 0.9-0&#061;0.9$\" class=\"mathcode\" src=\"2026-01-15siq2vjykra4.png\" \/>&#xff08;\u6b63\u6570\u5f88\u5927&#xff09;<br \/>\n&#x1f449; \u8bf4\u660e\u4f60\u5e94\u8be5\u628a z\u00a0\u5f80\u4e0b\u538b&#xff08;\u8ba9 a\u00a0\u53d8\u5c0f&#xff09;<\/p>\n<\/li>\n<\/ul>\n<p>\u6240\u4ee5\u5b83\u7279\u522b\u50cf\u4f60\u719f\u6089\u7684\u201c\u8bef\u5dee\u201d&#xff1a;<\/p>\n<p><img decoding=\"async\" alt=\"$\\\\text{error} &#061; \\\\text{prediction} - \\\\text{target}$\" class=\"mathcode\" src=\"2026-01-15cwxqa5sci5t.png\" \/><\/p>\n<hr \/>\n<h2>5&#xff09;\u518d\u8fdb\u4e00\u6b65&#xff1a;\u6c42\u5230 w\u00a0\u548c b \u7684\u68af\u5ea6&#xff08;\u8bad\u7ec3\u8981\u7528&#xff09;<\/h2>\n<p>\u56e0\u4e3a&#xff1a;<\/p>\n<p><img decoding=\"async\" alt=\"$z&#061;w^\\\\top x&#043;b$\" class=\"mathcode\" src=\"2026-01-15chhmeqe304c.png\" \/><\/p>\n<p>\u6240\u4ee5&#xff1a;<\/p>\n<p><img decoding=\"async\" alt=\"$\\\\frac{\\\\partial z}{\\\\partial w}&#061;x,\\\\quad \\\\frac{\\\\partial z}{\\\\partial b}&#061;1$\" class=\"mathcode\" src=\"2026-01-15drx1eqzlp5t.png\" \/><\/p>\n<p>\u94fe\u5f0f\u6cd5\u5219&#xff1a;<\/p>\n<p><img decoding=\"async\" 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