{"id":76171,"date":"2026-02-21T21:34:34","date_gmt":"2026-02-21T13:34:34","guid":{"rendered":"https:\/\/www.wsisp.com\/helps\/76171.html"},"modified":"2026-02-21T21:34:34","modified_gmt":"2026-02-21T13:34:34","slug":"%e4%bb%8e%e9%a2%84%e6%b5%8b%e5%88%b0%e5%8e%8b%e7%bc%a9%ef%bc%9a%e7%90%86%e8%a7%a3-softmax-%e5%9b%9e%e5%bd%92%e4%b8%8e%e4%ba%a4%e5%8f%89%e7%86%b5%e8%83%8c%e5%90%8e%e7%9a%84%e7%9b%b4%e8%a7%89","status":"publish","type":"post","link":"https:\/\/www.wsisp.com\/helps\/76171.html","title":{"rendered":"\u4ece\u9884\u6d4b\u5230\u538b\u7f29\uff1a\u7406\u89e3 Softmax \u56de\u5f52\u4e0e\u4ea4\u53c9\u71b5\u80cc\u540e\u7684\u76f4\u89c9"},"content":{"rendered":"<p>\u201c\u5206\u7c7b\u4e0d\u662f\u95ee\u2018\u591a\u5c11\u2019&#xff0c;\u800c\u662f\u95ee\u2018\u54ea\u4e00\u4e2a\u2019\u3002\u201d<\/p>\n<p>\u5728\u673a\u5668\u5b66\u4e60\u7684\u4e16\u754c\u91cc&#xff0c;\u7ebf\u6027\u56de\u5f52\u6559\u4f1a\u6211\u4eec\u5982\u4f55\u9884\u6d4b\u8fde\u7eed\u503c\u2014\u2014\u623f\u4ef7\u3001\u6e29\u5ea6\u3001\u80dc\u573a\u6570\u3002\u4f46\u73b0\u5b9e\u4e2d\u7684\u8bb8\u591a\u95ee\u9898\u672c\u8d28\u4e0a\u662f\u79bb\u6563\u9009\u62e9&#xff1a;\u8fd9\u5c01\u90ae\u4ef6\u662f\u5783\u573e\u90ae\u4ef6\u5417&#xff1f;\u8fd9\u5f20\u56fe\u662f\u732b\u8fd8\u662f\u72d7&#xff1f;\u7528\u6237\u4f1a\u70b9\u51fb\u8fd9\u4e2a\u5e7f\u544a\u5417&#xff1f;<\/p>\n<p>\u8fd9\u5c31\u662f\u5206\u7c7b\u4efb\u52a1\u7684\u821e\u53f0&#xff0c;\u800c Softmax \u56de\u5f52&#xff0c;\u6b63\u662f\u5904\u7406\u591a\u7c7b\u522b\u5206\u7c7b\u95ee\u9898\u6700\u57fa\u7840\u3001\u4e5f\u6700\u4f18\u96c5\u7684\u5de5\u5177\u4e4b\u4e00\u3002\u7136\u800c&#xff0c;Softmax \u4e0d\u53ea\u662f\u4e00\u4e2a\u6570\u5b66\u6280\u5de7&#xff1b;\u5b83\u7684\u80cc\u540e&#xff0c;\u6df1\u690d\u4e8e\u4fe1\u606f\u8bba\u4e0e\u6982\u7387\u5efa\u6a21\u7684\u575a\u5b9e\u571f\u58e4\u3002\u672c\u6587\u5c06\u5e26\u4f60\u4ece\u76f4\u89c9\u51fa\u53d1&#xff0c;\u4e00\u6b65\u6b65\u63ed\u5f00 Softmax \u4e0e\u4ea4\u53c9\u71b5\u635f\u5931\u7684\u795e\u79d8\u9762\u7eb1\u3002<\/p>\n<hr \/>\n<h3>\u4e00\u3001\u4e3a\u4ec0\u4e48\u4e0d\u80fd\u76f4\u63a5\u7528\u7ebf\u6027\u8f93\u51fa\u505a\u5206\u7c7b&#xff1f;<\/h3>\n<p>\u5047\u8bbe\u6211\u4eec\u6709\u4e00\u4e2a\u56fe\u50cf\u5206\u7c7b\u4efb\u52a1&#xff0c;\u7c7b\u522b\u4e3a {\u732b, \u9e21, \u72d7}\u3002\u6211\u4eec\u53ef\u4ee5\u6784\u5efa\u4e00\u4e2a\u7ebf\u6027\u6a21\u578b&#xff0c;\u5bf9\u6bcf\u4e2a\u7c7b\u522b\u8f93\u51fa\u4e00\u4e2a\u201c\u5f97\u5206\u201d&#xff08;logit&#xff09;&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">o1&#061;w1\u22a4x&#043;b1(\u732b)o2&#061;w2\u22a4x&#043;b2(\u9e21)o3&#061;w3\u22a4x&#043;b3(\u72d7)<br \/>\n\\\\begin{aligned}<br \/>\no_1 &amp;&#061; \\\\mathbf{w}_1^\\\\top \\\\mathbf{x} &#043; b_1 \\\\quad (\\\\text{\u732b}) \\\\\\\\<br \/>\no_2 &amp;&#061; \\\\mathbf{w}_2^\\\\top \\\\mathbf{x} &#043; b_2 \\\\quad (\\\\text{\u9e21}) \\\\\\\\<br \/>\no_3 &amp;&#061; \\\\mathbf{w}_3^\\\\top \\\\mathbf{x} &#043; b_3 \\\\quad (\\\\text{\u72d7})<br \/>\n\\\\end{aligned}<br \/>\n<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 4.6773em;vertical-align: -2.0887em\"><\/span><span class=\"mord\"><span class=\"mtable\"><span class=\"col-align-r\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 2.5887em\"><span class=\"\" style=\"top: -4.6896em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathnormal\">o<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3011em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.1304em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathnormal\">o<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3011em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -1.5713em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathnormal\">o<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3011em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">3<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 2.0887em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"col-align-l\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 2.5887em\"><span class=\"\" style=\"top: -4.6896em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\"><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mord\"><span class=\"mord mathbf\" style=\"margin-right: 0.016em\">w<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8991em\"><span class=\"\" style=\"top: -2.453em;margin-left: -0.016em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><span class=\"\" style=\"top: -3.113em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">\u22a4<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.247em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mord mathbf\">x<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">b<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3011em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 1em\"><\/span><span class=\"mopen\">(<\/span><span class=\"mord text\"><span class=\"mord cjk_fallback\">\u732b<\/span><\/span><span class=\"mclose\">)<\/span><\/span><\/span><span class=\"\" style=\"top: -3.1304em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\"><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mord\"><span class=\"mord mathbf\" style=\"margin-right: 0.016em\">w<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8991em\"><span class=\"\" style=\"top: -2.453em;margin-left: -0.016em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><span class=\"\" style=\"top: -3.113em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">\u22a4<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.247em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mord mathbf\">x<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">b<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3011em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 1em\"><\/span><span class=\"mopen\">(<\/span><span class=\"mord text\"><span class=\"mord cjk_fallback\">\u9e21<\/span><\/span><span class=\"mclose\">)<\/span><\/span><\/span><span class=\"\" style=\"top: -1.5713em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\"><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mord\"><span class=\"mord mathbf\" style=\"margin-right: 0.016em\">w<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8991em\"><span class=\"\" style=\"top: -2.453em;margin-left: -0.016em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">3<\/span><\/span><\/span><span class=\"\" style=\"top: -3.113em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">\u22a4<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.247em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mord mathbf\">x<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">b<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3011em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">3<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 1em\"><\/span><span class=\"mopen\">(<\/span><span class=\"mord text\"><span class=\"mord cjk_fallback\">\u72d7<\/span><\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 2.0887em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u4f46\u8fd9\u4e9b <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">ojo_j<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.7167em;vertical-align: -0.2861em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">o<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3117em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0572em\">j<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2861em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u662f\u672a\u89c4\u8303\u5316\u7684\u5b9e\u6570\u2014\u2014\u53ef\u80fd\u4e3a\u8d1f&#xff0c;\u603b\u548c\u4e5f\u4e0d\u4e3a 1\u3002\u5b83\u4eec\u4e0d\u80fd\u76f4\u63a5\u89e3\u91ca\u4e3a\u6982\u7387&#xff0c;\u56e0\u4e3a\u8fdd\u53cd\u4e86\u6982\u7387\u7684\u57fa\u672c\u516c\u7406\u3002<\/p>\n<p>\u6211\u4eec\u9700\u8981\u4e00\u4e2a\u51fd\u6570&#xff0c;\u80fd\u628a\u4efb\u610f\u5b9e\u6570\u5411\u91cf <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">o\\\\mathbf{o}<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.4444em\"><\/span><span class=\"mord mathbf\">o<\/span><\/span><\/span><\/span><\/span> \u8f6c\u6362\u4e3a\u4e00\u4e2a\u5408\u6cd5\u7684\u6982\u7387\u5206\u5e03 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">y^\\\\hat{\\\\mathbf{y}}<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.9023em;vertical-align: -0.1944em\"><\/span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.7079em\"><span class=\"\" style=\"top: -3em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord mathbf\" style=\"margin-right: 0.016em\">y<\/span><\/span><span class=\"\" style=\"top: -3.0134em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"accent-body\" style=\"left: -0.1944em\"><span class=\"mord\">^<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.1944em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>&#xff0c;\u5373&#xff1a;<\/p>\n<ul>\n<li>\u6bcf\u4e2a <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">y^j\u22650\\\\hat{y}_j \\\\geq 0<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.9805em;vertical-align: -0.2861em\"><\/span><span class=\"mord\"><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.6944em\"><span class=\"\" style=\"top: -3em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">y<\/span><\/span><span class=\"\" style=\"top: -3em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"accent-body\" style=\"left: -0.1944em\"><span class=\"mord\">^<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.1944em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3117em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0359em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0572em\">j<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2861em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2265<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6444em\"><\/span><span class=\"mord\">0<\/span><\/span><\/span><\/span><\/span><\/li>\n<li><span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">\u2211jy^j&#061;1\\\\sum_j \\\\hat{y}_j &#061; 1<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1.1858em;vertical-align: -0.4358em\"><\/span><span class=\"mop\"><span class=\"mop op-symbol small-op\" style=\"position: relative;top: 0em\">\u2211<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.162em\"><span class=\"\" style=\"top: -2.4003em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0572em\">j<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.4358em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord\"><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.6944em\"><span class=\"\" style=\"top: -3em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">y<\/span><\/span><span class=\"\" style=\"top: -3em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"accent-body\" style=\"left: -0.1944em\"><span class=\"mord\">^<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.1944em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3117em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0359em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0572em\">j<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2861em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6444em\"><\/span><span class=\"mord\">1<\/span><\/span><\/span><\/span><\/span><\/li>\n<\/ul>\n<p>Softmax \u51fd\u6570 \u6b63\u662f\u4e3a\u6b64\u800c\u751f&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">y^j&#061;exp\u2061(oj)\u2211kexp\u2061(ok)<br \/>\n\\\\hat{y}_j &#061; \\\\frac{\\\\exp(o_j)}{\\\\sum_k \\\\exp(o_k)}<br \/>\n<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.9805em;vertical-align: -0.2861em\"><\/span><span class=\"mord\"><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.6944em\"><span class=\"\" style=\"top: -3em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">y<\/span><\/span><span class=\"\" style=\"top: -3em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"accent-body\" style=\"left: -0.1944em\"><span class=\"mord\">^<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.1944em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3117em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0359em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0572em\">j<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2861em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 2.4127em;vertical-align: -0.9857em\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.427em\"><span class=\"\" style=\"top: -2.314em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mop\"><span class=\"mop op-symbol small-op\" style=\"position: relative;top: 0em\">\u2211<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.1864em\"><span class=\"\" style=\"top: -2.4003em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0315em\">k<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2997em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mop\">exp<\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\">o<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3361em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0315em\">k<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em\"><\/span><\/span><span class=\"\" style=\"top: -3.677em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mop\">exp<\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\">o<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3117em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0572em\">j<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2861em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.9857em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u5b83\u901a\u8fc7\u6307\u6570\u5316\u786e\u4fdd\u975e\u8d1f&#xff0c;\u518d\u901a\u8fc7\u5f52\u4e00\u5316\u786e\u4fdd\u548c\u4e3a 1\u3002\u66f4\u91cd\u8981\u7684\u662f&#xff0c;\u5b83\u4fdd\u6301\u53ef\u5bfc\u6027&#xff0c;\u4f7f\u5f97\u68af\u5ea6\u4e0b\u964d\u6210\u4e3a\u53ef\u80fd\u3002<\/p>\n<p>&#x1f4a1; \u6709\u8da3\u4e8b\u5b9e&#xff1a;Softmax \u7684\u601d\u60f3\u53ef\u8ffd\u6eaf\u81f3\u793e\u4f1a\u79d1\u5b66\u5bb6\u9093\u80af\u00b7\u5362\u65af&#xff08;Duncan Luce&#xff09;1959 \u5e74\u63d0\u51fa\u7684\u9009\u62e9\u516c\u7406\u2014\u2014\u4eba\u4eec\u5728\u591a\u4e2a\u9009\u9879\u4e2d\u505a\u9009\u62e9\u7684\u6982\u7387&#xff0c;\u6b63\u6bd4\u4e8e\u5404\u9009\u9879\u201c\u5438\u5f15\u529b\u201d\u7684\u6307\u6570\u3002\u8fd9\u4e0e softmax \u5b8c\u7f8e\u5951\u5408&#xff01;<\/p>\n<hr \/>\n<h3>\u4e8c\u3001\u5982\u4f55\u8bad\u7ec3\u6a21\u578b&#xff1f;\u6700\u5927\u4f3c\u7136\u4f30\u8ba1\u767b\u573a<\/h3>\n<p>\u6709\u4e86\u6982\u7387\u8f93\u51fa <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">y^&#061;P(y\u2223x)\\\\hat{\\\\mathbf{y}} &#061; P(y \\\\mid \\\\mathbf{x})<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.9023em;vertical-align: -0.1944em\"><\/span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.7079em\"><span class=\"\" style=\"top: -3em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord mathbf\" style=\"margin-right: 0.016em\">y<\/span><\/span><span class=\"\" style=\"top: -3.0134em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"accent-body\" style=\"left: -0.1944em\"><span class=\"mord\">^<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.1944em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">y<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2223<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathbf\">x<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span>&#xff0c;\u6211\u4eec\u81ea\u7136\u5e0c\u671b&#xff1a;\u6a21\u578b\u5bf9\u771f\u5b9e\u6807\u7b7e\u7684\u9884\u6d4b\u6982\u7387\u8d8a\u9ad8\u8d8a\u597d\u3002<\/p>\n<p>\u5047\u8bbe\u6211\u4eec\u6709 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">nn<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.4306em\"><\/span><span class=\"mord mathnormal\">n<\/span><\/span><\/span><\/span><\/span> \u4e2a\u72ec\u7acb\u6837\u672c&#xff0c;\u771f\u5b9e\u6807\u7b7e\u7528\u72ec\u70ed\u7f16\u7801\u8868\u793a&#xff08;\u5982\u201c\u9e21\u201d \u2192 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">[0,1,0][0,1,0]<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mopen\">[<\/span><span class=\"mord\">0<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord\">1<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord\">0<\/span><span class=\"mclose\">]<\/span><\/span><\/span><\/span><\/span>&#xff09;\u3002\u90a3\u4e48&#xff0c;\u6574\u4e2a\u6570\u636e\u96c6\u88ab\u89c2\u6d4b\u5230\u7684\u8054\u5408\u6982\u7387&#xff08;\u5373\u4f3c\u7136&#xff09;\u4e3a&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">P(Y\u2223X)&#061;\u220fi&#061;1nP(y(i)\u2223x(i))<br \/>\nP(\\\\mathbf{Y} \\\\mid \\\\mathbf{X}) &#061; \\\\prod_{i&#061;1}^n P(\\\\mathbf{y}^{(i)} \\\\mid \\\\mathbf{x}^{(i)})<br \/>\n<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathbf\" style=\"margin-right: 0.0288em\">Y<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2223<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathbf\">X<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 2.9291em;vertical-align: -1.2777em\"><\/span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.6514em\"><span class=\"\" style=\"top: -1.8723em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">i<\/span><span class=\"mrel mtight\">&#061;<\/span><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.05em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"\"><span class=\"mop op-symbol large-op\">\u220f<\/span><\/span><\/span><span class=\"\" style=\"top: -4.3em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">n<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.2777em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathbf\" style=\"margin-right: 0.016em\">y<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.938em\"><span class=\"\" style=\"top: -3.113em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mopen mtight\">(<\/span><span class=\"mord mathnormal mtight\">i<\/span><span class=\"mclose mtight\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2223<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1.188em;vertical-align: -0.25em\"><\/span><span class=\"mord\"><span class=\"mord mathbf\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.938em\"><span class=\"\" style=\"top: -3.113em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mopen mtight\">(<\/span><span class=\"mord mathnormal mtight\">i<\/span><span class=\"mclose mtight\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u4e3a\u4ec0\u4e48\u8981\u76f8\u4e58&#xff1f;<br \/>\n\u56e0\u4e3a\u6837\u672c\u72ec\u7acb&#xff01;\u72ec\u7acb\u4e8b\u4ef6\u540c\u65f6\u53d1\u751f\u7684\u6982\u7387\u7b49\u4e8e\u5404\u81ea\u6982\u7387\u7684\u4e58\u79ef\u3002\u8fd9\u662f\u7edf\u8ba1\u5efa\u6a21\u7684\u57fa\u77f3\u3002<\/p>\n<p>\u4e3a\u4e86\u4fbf\u4e8e\u4f18\u5316&#xff0c;\u6211\u4eec\u53d6\u8d1f\u5bf9\u6570&#xff0c;\u5f97\u5230\u8d1f\u5bf9\u6570\u4f3c\u7136&#xff08;Negative Log-Likelihood&#xff09;&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">\u2212log\u2061P(Y\u2223X)&#061;\u2211i&#061;1n\u2212log\u2061P(y(i)\u2223x(i))<br \/>\n-\\\\log P(\\\\mathbf{Y} \\\\mid \\\\mathbf{X}) &#061; \\\\sum_{i&#061;1}^n -\\\\log P(\\\\mathbf{y}^{(i)} \\\\mid \\\\mathbf{x}^{(i)})<br \/>\n<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord\">\u2212<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mop\">lo<span style=\"margin-right: 0.0139em\">g<\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathbf\" style=\"margin-right: 0.0288em\">Y<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2223<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathbf\">X<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 2.9291em;vertical-align: -1.2777em\"><\/span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.6514em\"><span class=\"\" style=\"top: -1.8723em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">i<\/span><span class=\"mrel mtight\">&#061;<\/span><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.05em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"\"><span class=\"mop op-symbol large-op\">\u2211<\/span><\/span><\/span><span class=\"\" style=\"top: -4.3em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">n<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.2777em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord\">\u2212<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mop\">lo<span style=\"margin-right: 0.0139em\">g<\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathbf\" style=\"margin-right: 0.016em\">y<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.938em\"><span class=\"\" style=\"top: -3.113em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mopen mtight\">(<\/span><span class=\"mord mathnormal mtight\">i<\/span><span class=\"mclose mtight\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2223<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1.188em;vertical-align: -0.25em\"><\/span><span class=\"mord\"><span class=\"mord mathbf\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.938em\"><span class=\"\" style=\"top: -3.113em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mopen mtight\">(<\/span><span class=\"mord mathnormal mtight\">i<\/span><span class=\"mclose mtight\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u7531\u4e8e <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">y(i)\\\\mathbf{y}^{(i)}<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1.0824em;vertical-align: -0.1944em\"><\/span><span class=\"mord\"><span class=\"mord mathbf\" style=\"margin-right: 0.016em\">y<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.888em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mopen mtight\">(<\/span><span class=\"mord mathnormal mtight\">i<\/span><span class=\"mclose mtight\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u662f one-hot \u5411\u91cf&#xff0c;\u4e0a\u5f0f\u7b80\u5316\u4e3a&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">l(y,y^)&#061;\u2212\u2211jyjlog\u2061y^j&#061;\u2212log\u2061y^true\u00a0class<br \/>\nl(\\\\mathbf{y}, \\\\hat{\\\\mathbf{y}}) &#061; -\\\\sum_j y_j \\\\log \\\\hat{y}_j &#061; -\\\\log \\\\hat{y}_{\\\\text{true class}}<br \/>\n<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0197em\">l<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathbf\" style=\"margin-right: 0.016em\">y<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.7079em\"><span class=\"\" style=\"top: -3em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord mathbf\" style=\"margin-right: 0.016em\">y<\/span><\/span><span class=\"\" style=\"top: -3.0134em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"accent-body\" style=\"left: -0.1944em\"><span class=\"mord\">^<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.1944em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 2.4638em;vertical-align: -1.4138em\"><\/span><span class=\"mord\">\u2212<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.05em\"><span class=\"\" style=\"top: -1.8723em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0572em\">j<\/span><\/span><\/span><span class=\"\" style=\"top: -3.05em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"\"><span class=\"mop op-symbol large-op\">\u2211<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.4138em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">y<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3117em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0359em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0572em\">j<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2861em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mop\">lo<span style=\"margin-right: 0.0139em\">g<\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord\"><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.6944em\"><span class=\"\" style=\"top: -3em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">y<\/span><\/span><span class=\"\" style=\"top: -3em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"accent-body\" style=\"left: -0.1944em\"><span class=\"mord\">^<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.1944em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3117em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0359em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0572em\">j<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2861em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.8889em;vertical-align: -0.1944em\"><\/span><span class=\"mord\">\u2212<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mop\">lo<span style=\"margin-right: 0.0139em\">g<\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord\"><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.6944em\"><span class=\"\" style=\"top: -3em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">y<\/span><\/span><span class=\"\" style=\"top: -3em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"accent-body\" style=\"left: -0.1944em\"><span class=\"mord\">^<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.1944em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3361em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0359em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord text mtight\"><span class=\"mord mtight\">true\u00a0class<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u8fd9\u5c31\u662f\u8457\u540d\u7684 \u4ea4\u53c9\u71b5\u635f\u5931&#xff08;Cross-Entropy Loss&#xff09;\u3002<\/p>\n<hr \/>\n<h3>\u4e09\u3001\u4ea4\u53c9\u71b5 &#061; \u9884\u671f\u201c\u60ca\u5f02\u201d&#xff1f;\u4fe1\u606f\u8bba\u89c6\u89d2<\/h3>\n<p>\u8fd9\u91cc&#xff0c;\u4fe1\u606f\u8bba\u4e3a\u6211\u4eec\u63d0\u4f9b\u4e86\u6df1\u523b\u7684\u6d1e\u89c1\u3002<\/p>\n<p>\u514b\u52b3\u5fb7\u00b7\u9999\u519c\u63d0\u51fa&#xff1a;\u4e00\u4e2a\u4e8b\u4ef6\u7684\u4fe1\u606f\u91cf&#xff08;\u60ca\u5f02\u7a0b\u5ea6&#xff09;\u4e0e\u5176\u53d1\u751f\u6982\u7387\u6210\u53cd\u6bd4&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">I(x)&#061;\u2212log\u2061P(x)<br \/>\nI(x) &#061; -\\\\log P(x)<br \/>\n<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0785em\">I<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord\">\u2212<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mop\">lo<span style=\"margin-right: 0.0139em\">g<\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<ul>\n<li>\u592a\u9633\u5347\u8d77&#xff1f;\u6982\u7387 \u22481 \u2192 \u4fe1\u606f\u91cf \u22480&#xff08;\u4e0d\u610f\u5916&#xff09;<\/li>\n<li>\u5f69\u7968\u4e2d\u5956&#xff1f;\u6982\u7387 \u22480 \u2192 \u4fe1\u606f\u91cf\u5f88\u5927&#xff08;\u5f88\u610f\u5916&#xff09;<\/li>\n<\/ul>\n<p>\u90a3\u4e48&#xff0c;\u5e73\u5747\u6bcf\u6b21\u89c2\u5bdf\u5e26\u6765\u7684\u201c\u60ca\u5f02\u201d\u662f\u591a\u5c11&#xff1f;\u8fd9\u5c31\u662f\u71b5&#xff08;Entropy&#xff09;&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">H(P)&#061;Ex\u223cP[\u2212log\u2061P(x)]&#061;\u2212\u2211jP(j)log\u2061P(j)<br \/>\nH(P) &#061; \\\\mathbb{E}_{x \\\\sim P}[-\\\\log P(x)] &#061; -\\\\sum_j P(j) \\\\log P(j)<br \/>\n<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0813em\">H<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord\"><span class=\"mord mathbb\">E<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3283em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">x<\/span><span class=\"mrel mtight\">\u223c<\/span><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.1389em\">P<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mopen\">[<\/span><span class=\"mord\">\u2212<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mop\">lo<span style=\"margin-right: 0.0139em\">g<\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mclose\">)]<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 2.4638em;vertical-align: -1.4138em\"><\/span><span class=\"mord\">\u2212<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.05em\"><span class=\"\" style=\"top: -1.8723em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0572em\">j<\/span><\/span><\/span><span class=\"\" style=\"top: -3.05em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"\"><span class=\"mop op-symbol large-op\">\u2211<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.4138em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0572em\">j<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mop\">lo<span style=\"margin-right: 0.0139em\">g<\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0572em\">j<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u71b5\u662f\u5f53\u4f60\u7684\u6982\u7387\u6a21\u578b\u5b8c\u5168\u5339\u914d\u771f\u5b9e\u4e16\u754c\u65f6\u7684\u5e73\u5747\u4fe1\u606f\u91cf&#xff0c;\u4e5f\u662f\u65e0\u635f\u538b\u7f29\u7684\u7406\u8bba\u6781\u9650\u3002<\/p>\n<p>\u73b0\u5728&#xff0c;\u5982\u679c\u6211\u4eec\u7528\u4e00\u4e2a\u9519\u8bef\u7684\u6a21\u578b <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">QQ<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8778em;vertical-align: -0.1944em\"><\/span><span class=\"mord mathnormal\">Q<\/span><\/span><\/span><\/span><\/span> \u53bb\u63cf\u8ff0\u771f\u5b9e\u5206\u5e03 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">PP<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><\/span><\/span><\/span><\/span>&#xff0c;\u5e73\u5747\u201c\u60ca\u5f02\u201d\u5c31\u53d8\u6210\u4e86&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">H(P,Q)&#061;Ex\u223cP[\u2212log\u2061Q(x)]&#061;\u2212\u2211jP(j)log\u2061Q(j)<br \/>\nH(P, Q) &#061; \\\\mathbb{E}_{x \\\\sim P}[-\\\\log Q(x)] &#061; -\\\\sum_j P(j) \\\\log Q(j)<br \/>\n<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0813em\">H<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\">Q<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord\"><span class=\"mord mathbb\">E<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3283em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">x<\/span><span class=\"mrel mtight\">\u223c<\/span><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.1389em\">P<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mopen\">[<\/span><span class=\"mord\">\u2212<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mop\">lo<span style=\"margin-right: 0.0139em\">g<\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\">Q<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mclose\">)]<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 2.4638em;vertical-align: -1.4138em\"><\/span><span class=\"mord\">\u2212<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.05em\"><span class=\"\" style=\"top: -1.8723em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0572em\">j<\/span><\/span><\/span><span class=\"\" style=\"top: -3.05em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"\"><span class=\"mop op-symbol large-op\">\u2211<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.4138em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0572em\">j<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mop\">lo<span style=\"margin-right: 0.0139em\">g<\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\">Q<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0572em\">j<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u8fd9\u6b63\u662f\u4ea4\u53c9\u71b5&#xff01;<\/p>\n<ul>\n<li>\u5f53 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">Q&#061;PQ &#061; P<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8778em;vertical-align: -0.1944em\"><\/span><span class=\"mord mathnormal\">Q<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><\/span><\/span><\/span><\/span> \u65f6&#xff0c;<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">H(P,Q)&#061;H(P)H(P, Q) &#061; H(P)<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0813em\">H<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\">Q<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0813em\">H<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span>&#xff08;\u6700\u5c0f\u53ef\u80fd\u503c&#xff09;<\/li>\n<li>\u5f53 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">Q\u2260PQ \\\\neq P<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8889em;vertical-align: -0.1944em\"><\/span><span class=\"mord mathnormal\">Q<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\"><span class=\"mrel\"><span class=\"mord vbox\"><span class=\"thinbox\"><span class=\"rlap\"><span class=\"strut\" style=\"height: 0.8889em;vertical-align: -0.1944em\"><\/span><span class=\"inner\"><span class=\"mord\"><span class=\"mrel\">\ue020<\/span><\/span><\/span><span class=\"fix\"><\/span><\/span><\/span><\/span><\/span><span class=\"mrel\">&#061;<\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><\/span><\/span><\/span><\/span> \u65f6&#xff0c;\u4ea4\u53c9\u71b5\u66f4\u5927<\/li>\n<\/ul>\n<p>\u56e0\u6b64&#xff0c;\u6700\u5c0f\u5316\u4ea4\u53c9\u71b5\u635f\u5931&#xff0c;\u672c\u8d28\u4e0a\u662f\u5728\u8ba9\u6a21\u578b\u7684\u9884\u6d4b\u5206\u5e03 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">QQ<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8778em;vertical-align: -0.1944em\"><\/span><span class=\"mord mathnormal\">Q<\/span><\/span><\/span><\/span><\/span> \u5c3d\u53ef\u80fd\u63a5\u8fd1\u771f\u5b9e\u5206\u5e03 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">PP<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><\/span><\/span><\/span><\/span>&#xff0c;\u4ece\u800c\u51cf\u5c11\u201c\u60ca\u5f02\u201d&#xff0c;\u63d0\u9ad8\u9884\u6d4b\u51c6\u786e\u6027\u3002<\/p>\n<p>\u2705 \u538b\u7f29\u4e0e\u9884\u6d4b\u7684\u7edf\u4e00&#xff1a;<br \/>\n\u5982\u679c\u4f60\u80fd\u51c6\u786e\u9884\u6d4b\u4e0b\u4e00\u4e2a\u7b26\u53f7&#xff08;\u5373 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">Q\u2248PQ \\\\approx P<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8778em;vertical-align: -0.1944em\"><\/span><span class=\"mord mathnormal\">Q<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2248<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><\/span><\/span><\/span><\/span>&#xff09;&#xff0c;\u4f60\u5c31\u80fd\u7528\u63a5\u8fd1 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">H(P)H(P)<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0813em\">H<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span> \u6bd4\u7279\/\u7b26\u53f7\u6765\u538b\u7f29\u6570\u636e\u3002<br \/>\n\u53cd\u4e4b&#xff0c;\u5206\u7c7b\u6a21\u578b\u8d8a\u51c6&#xff0c;\u5176\u4ea4\u53c9\u71b5\u8d8a\u4f4e&#xff0c;\u8bf4\u660e\u5b83\u5bf9\u4e16\u754c\u7684\u201c\u4e0d\u786e\u5b9a\u6027\u201d\u8d8a\u5c0f\u3002<\/p>\n<hr \/>\n<h3>\u56db\u3001Softmax \u56de\u5f52\u7684\u4f18\u96c5\u6027\u8d28<\/h3>\n<p>\u5c3d\u7ba1 Softmax \u662f\u975e\u7ebf\u6027\u7684&#xff0c;\u4f46\u6574\u4e2a\u6a21\u578b\u4ecd\u662f\u7ebf\u6027\u6a21\u578b\u2014\u2014\u56e0\u4e3a\u6700\u7ec8\u51b3\u7b56\u8fb9\u754c\u7531 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">Wx&#043;b\\\\mathbf{W}\\\\mathbf{x} &#043; \\\\mathbf{b}<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.7694em;vertical-align: -0.0833em\"><\/span><span class=\"mord mathbf\">Wx<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6944em\"><\/span><span class=\"mord mathbf\">b<\/span><\/span><\/span><\/span><\/span> \u51b3\u5b9a\u3002<\/p>\n<p>\u66f4\u5999\u7684\u662f&#xff0c;\u4ea4\u53c9\u71b5\u635f\u5931\u5bf9 logits \u7684\u68af\u5ea6\u6781\u5176\u7b80\u6d01&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">\u2202l\u2202oj&#061;y^j\u2212yj<br \/>\n\\\\frac{\\\\partial l}{\\\\partial o_j} &#061; \\\\hat{y}_j &#8211; y_j<br \/>\n<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 2.3435em;vertical-align: -0.9721em\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.3714em\"><span class=\"\" style=\"top: -2.314em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\" style=\"margin-right: 0.0556em\">\u2202<\/span><span class=\"mord\"><span class=\"mord mathnormal\">o<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3117em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0572em\">j<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2861em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em\"><\/span><\/span><span class=\"\" style=\"top: -3.677em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\" style=\"margin-right: 0.0556em\">\u2202<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0197em\">l<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.9721em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.9805em;vertical-align: -0.2861em\"><\/span><span class=\"mord\"><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.6944em\"><span class=\"\" style=\"top: -3em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">y<\/span><\/span><span class=\"\" style=\"top: -3em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"accent-body\" style=\"left: -0.1944em\"><span class=\"mord\">^<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.1944em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3117em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0359em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0572em\">j<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2861em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.7167em;vertical-align: -0.2861em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">y<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3117em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0359em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0572em\">j<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2861em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u5373&#xff1a;\u9884\u6d4b\u6982\u7387\u4e0e\u771f\u5b9e\u6807\u7b7e\u7684\u5dee\u3002\u8fd9\u4e0e\u7ebf\u6027\u56de\u5f52\u4e2d <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">(y^\u2212y)(\\\\hat{y} &#8211; y)<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mopen\">(<\/span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.6944em\"><span class=\"\" style=\"top: -3em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">y<\/span><\/span><span class=\"\" style=\"top: -3em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"accent-body\" style=\"left: -0.1944em\"><span class=\"mord\">^<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.1944em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">y<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span> \u7684\u5f62\u5f0f\u60ca\u4eba\u5730\u76f8\u4f3c&#xff0c;\u4f53\u73b0\u4e86\u6307\u6570\u65cf\u5206\u5e03\u6a21\u578b\u7684\u7edf\u4e00\u6027\u3002<\/p>\n<hr \/>\n<h3>\u603b\u7ed3&#xff1a;Softmax \u56de\u5f52\u7684\u4e09\u91cd\u8eab\u4efd<\/h3>\n<table>\n<tr>\u89c6\u89d2\u89e3\u91ca<\/tr>\n<tbody>\n<tr>\n<td>\u5de5\u7a0b\u89c6\u89d2<\/td>\n<td>\u5c06\u7ebf\u6027\u8f93\u51fa\u8f6c\u4e3a\u6982\u7387&#xff0c;\u652f\u6301\u591a\u5206\u7c7b<\/td>\n<\/tr>\n<tr>\n<td>\u7edf\u8ba1\u89c6\u89d2<\/td>\n<td>\u6700\u5927\u4f3c\u7136\u4f30\u8ba1\u4e0b\u7684\u6700\u4f18\u5206\u7c7b\u5668<\/td>\n<\/tr>\n<tr>\n<td>\u4fe1\u606f\u8bba\u89c6\u89d2<\/td>\n<td>\u6700\u5c0f\u5316\u6a21\u578b\u5bf9\u771f\u5b9e\u4e16\u754c\u7684\u201c\u9884\u671f\u60ca\u5f02\u201d&#xff08;\u4ea4\u53c9\u71b5&#xff09;<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Softmax \u56de\u5f52\u8fdc\u4e0d\u6b62\u662f\u4e00\u4e2a\u516c\u5f0f\u3002\u5b83\u662f\u9884\u6d4b\u3001\u538b\u7f29\u4e0e\u4fe1\u606f\u4e09\u5927\u6982\u5ff5\u4ea4\u6c47\u7684\u5178\u8303\u3002\u7406\u89e3\u5b83&#xff0c;\u5c31\u662f\u7406\u89e3\u73b0\u4ee3\u673a\u5668\u5b66\u4e60\u5982\u4f55\u7528\u6982\u7387\u8bed\u8a00\u63cf\u8ff0\u4e16\u754c\u3002<\/p>\n<hr \/>\n<p>\u5ef6\u4f38\u601d\u8003&#xff1a;<br \/>\n\u5982\u679c\u4f60\u5bf9\u201c\u71b5\u201d\u548c\u201c\u538b\u7f29\u201d\u7684\u5173\u7cfb\u611f\u5174\u8da3&#xff0c;\u4e0d\u59a8\u601d\u8003&#xff1a;\u4e3a\u4ec0\u4e48 ZIP \u6587\u4ef6\u5bf9\u6587\u672c\u538b\u7f29\u6548\u679c\u597d&#xff0c;\u4f46\u5bf9\u5df2\u7ecf\u538b\u7f29\u8fc7\u7684 JPG \u56fe\u50cf\u51e0\u4e4e\u65e0\u6548&#xff1f;\u7b54\u6848\u5c31\u5728\u6570\u636e\u7684\u53ef\u9884\u6d4b\u6027\u4e0e\u71b5\u4e4b\u4e2d\u3002<\/p>\n<hr \/>\n<p>\u53c2\u8003&#xff1a;\u672c\u6587\u5185\u5bb9\u4e3b\u8981\u57fa\u4e8e\u300a\u52a8\u624b\u5b66\u6df1\u5ea6\u5b66\u4e60\u300b&#xff08;Dive into Deep Learning&#xff09;\u7b2c 3.4 \u8282\u53ca\u9644\u5f55\u4fe1\u606f\u8bba\u7ae0\u8282&#xff0c;\u5e76\u878d\u5408\u4e86\u9999\u519c\u4fe1\u606f\u8bba\u7684\u6838\u5fc3\u601d\u60f3\u3002<\/p>\n<hr \/>\n<p>\u5e0c\u671b\u8fd9\u7bc7\u535a\u5ba2\u80fd\u5e2e\u52a9\u8bfb\u8005\u4e0d\u4ec5\u201c\u4f1a\u7528\u201d 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