{"id":1122,"date":"2018-08-11T23:08:58","date_gmt":"2018-08-11T15:08:58","guid":{"rendered":"http:\/\/www.whudj.cn\/?p=1122"},"modified":"2018-08-12T20:01:27","modified_gmt":"2018-08-12T12:01:27","slug":"%e9%ab%98%e6%96%af-%e7%89%9b%e9%a1%bf%e6%b3%95guass-newton-algorithm%e4%b8%8e%e8%8e%b1%e6%96%87%e8%b4%9d%e6%a0%bc-%e9%a9%ac%e5%a4%b8%e7%89%b9%e6%96%b9%e6%b3%95levenberg-marquardt-algorithm","status":"publish","type":"post","link":"http:\/\/www.whudj.cn\/?p=1122","title":{"rendered":"\u9ad8\u65af-\u725b\u987f\u6cd5(Guass-Newton Algorithm)\u4e0e\u83b1\u6587\u8d1d\u683c-\u9a6c\u5938\u7279\u65b9\u6cd5(Levenberg\u2013Marquardt algorithm)\u6c42\u89e3\u975e\u7ebf\u6027\u6700\u5c0f\u4e8c\u4e58\u95ee\u9898"},"content":{"rendered":"

\u4f17\u6240\u5468\u77e5\uff0c\u6700\u5c0f\u4e8c\u4e58\u6cd5\u901a\u8fc7\u6700\u5c0f\u5316\u8bef\u5dee\u5e73\u65b9\u548c\u83b7\u5f97\u6700\u4f73\u51fd\u6570\u3002\u6709\u65f6\u5019\u4f60\u53ef\u80fd\u4ea7\u751f\u7591\u95ee\uff0c\u4e3a\u4ec0\u4e48\u4e0d\u80fd\u901a\u8fc7\u5176\u4ed6\u65b9\u5f0f\u83b7\u5f97\u6700\u4f18\u51fd\u6570\uff0c\u6bd4\u5982\u8bf4\u6700\u5c0f\u5316\u8bef\u5dee\u7684\u7edd\u5bf9\u503c\u7684\u548c\uff1f\u672c\u6587\u4e2d\uff0c\u6211\u5c06\u4f1a\u4ece\u6982\u7387\u7684\u89d2\u5ea6\u89e3\u91ca\u6700\u5c0f\u4e8c\u4e58\u6cd5\u7684\u4f9d\u636e\uff08\u53c2\u8003\u81eaandrew ng \u300a\u673a\u5668\u5b66\u4e60\u8bfe\u7a0b\u300b \u7b2c\u4e09\u8bb2\uff09\u3002\u6700\u5c0f\u4e8c\u4e58\u95ee\u9898\u53ef\u4ee5\u5206\u4e3a\u7ebf\u6027\u6700\u5c0f\u4e8c\u4e58\u548c\u975e\u7ebf\u6027\u6700\u5c0f\u4e8c\u4e58\u4e24\u7c7b\uff0c\u672c\u6587\u7684\u76ee\u6807\u662f\u4ecb\u7ecd\u4e24\u79cd\u7ecf\u5178\u7684\u6700\u5c0f\u4e8c\u4e58\u95ee\u9898\u89e3\u6cd5\uff1a\u9ad8\u65af\u725b\u987f\u6cd5\u4e0e\u83b1\u6587\u8d1d\u683c-\u9a6c\u5938\u7279\u65b9\u6cd5\u3002\u5b9e\u9645\u4e0a\uff0c\u540e\u8005\u662f\u5bf9\u524d\u8005\u4ee5\u53ca\u68af\u5ea6\u4e0b\u964d\u6cd5\u7684\u7efc\u5408\u3002<\/p>\n

\u6700\u5c0f\u4e8c\u4e58\u6cd5\u7684\u6982\u7387\u89e3\u91ca(probabilistic interpretation)<\/strong><\/span><\/p>\n

\u4ee5\u7ebf\u6027\u56de\u5f52\u4e3a\u4f8b\uff0c\u5047\u8bbe\u6700\u4f73\u51fd\u6570\u4e3a \\(y=\\mathbf{\\theta}^T\\mathbf{x}\\) (\\(\\theta,x\\)\u4e3a\u5411\u91cf), \u5bf9\u4e8e\u6bcf\u5bf9\u89c2\u6d4b\u7ed3\u679c\\((x^{(i)},y^{(i)})\\)\uff0c\u90fd\u6709$$y^{(i)}=\\theta^Tx^{(i)}+\\epsilon^{(i)}$$\u5176\u4e2d \\(\\epsilon\\)\u4e3a\u8bef\u5dee\uff0c\u57fa\u4e8e\u4e00\u79cd\u5408\u7406\u7684\u5047\u8bbe\uff08\u4e2d\u5fc3\u6781\u9650\u5b9a\u7406\uff09\uff0c\u6211\u4eec\u53ef\u4ee5\u8ba4\u4e3a\u8bef\u5dee\u7684\u5206\u5e03\u670d\u4ece\u6b63\u6001\u5206\u5e03(\u53c8\u79f0\u9ad8\u65af\u5206\u5e03)\uff0c\u5373 \\(\\epsilon \\sim N(0,\\sigma^2)\\) \uff0c\u90a3\u4e48\uff0c\u6211\u4eec\u53ef\u4ee5\u8ba4\u4e3a\\(y^{(i)} \\sim N(\\theta^Tx^{(i)},\\sigma^2)\\),\u6839\u636e\u6b63\u6001\u5206\u5e03\u7684\u6982\u7387\u516c\u5f0f$$P(y^{(i)}|x^{(i)};\\theta)=\\frac{1}{\\sqrt{2\\pi}\\sigma}exp(-\\frac{(y^{(i)}-\\theta^Tx^{(i)})^2}{2\\sigma^2})$$\u5728\u7edf\u8ba1\u5b66\u4e2d\uff0c\u5c06\u6240\u6709\u7684\\(P(y|x)\\)\u7d2f\u4e58\u4f5c\u4e3a\\(\\theta\\)\u7684\u4f3c\u7136\u51fd\u6570,\u7528\u4ee5\u8861\u91cf\\(\\theta\\)\u7684\u6216\u7136\u6027(likelihood)$$L(\\theta)=P(y|x;\\theta)=\\prod _{i=0}^m \\frac{1}{\\sqrt{2\\pi}\\sigma}exp(-\\frac{(y^{(i)}-\\theta^Tx^{(i)})^2}{2\\sigma^2})$$\u6700\u4f73\u7684\u53c2\u6570\\(\\theta\\)\u5e94\u8be5\u662f\u4f7f\u5f97\u6240\u6709\u6570\u636e\u51fa\u73b0\u7684\u6982\u7387\u6700\u5927\u7684\u90a3\u4e2a\uff0c\u8fd9\u4e2a\u8fc7\u7a0b\u79f0\u4e4b\u4e3a\u6700\u5927\u4f3c\u7136\u4f30\u8ba1(maximum likelihood estimation)\u3002\u4e3a\u4e86\u6570\u5b66\u8ba1\u7b97\u4e0a\u7684\u4fbf\u5229\uff0c\u91c7\u7528\u5355\u8c03\u51fd\u6570:log\u51fd\u6570\\(l(\\theta)=log(L(\\theta))\\)\uff0c\u79f0\u4e3a\u5bf9\u6570\u4f3c\u7136\u51fd\u6570\u4ee3\u8868\\(L(\\theta)\\):$$l(\\theta)=log\\prod _{i=0}^m \\frac{1}{\\sqrt{2\\pi}\\sigma}exp(-\\frac{(y^{(i)}-\\theta^Tx^{(i)})^2}{2\\sigma^2}) \\\\ =\\sum_{i=0}^mlog\\frac{1}{\\sqrt{2\\pi}\\sigma}exp(-\\frac{(y^{(i)}-\\theta^Tx^{(i)})^2}{2\\sigma^2})\\\\=mlog\\frac{1}{\\sqrt{2\\pi}\\sigma} – \\frac{1}{2\\sigma^2}\\sum_{i=0}^m(y^{(i)}-\\theta^Tx^{(i)})^2$$\u9700\u8981\u6307\u51fa\u7684\u662f\uff0c\u5728Andrew ng\u7684\u8bb2\u89e3\u8ba4\u4e3a\\(\\sigma\\)\u4e0d\u5f71\u54cd\\(\\theta\\)\u7684\u51b3\u5b9a\uff0c\u56e0\u6b64\uff0c\u6700\u5927\u5316\\(l(\\theta)\\)\u7b49\u540c\u4e8e\u6700\u5c0f\u5316\\(\\sum_{i=0}^m(y^{(i)}-\\theta^Tx^{(i)})^2\\)\u3002\u6700\u5c0f\u4e8c\u4e58\u6cd5\u901a\u8fc7\u6700\u5c0f\u5316\u8bef\u5dee\u5e73\u65b9\u548c\u5f97\u5230\u6700\u4f73\u51fd\u6570\u7684\u65b9\u6cd5\u5b58\u5728\u6982\u7387\u8bba\u65b9\u9762\u7684\u57fa\u7840\u3002<\/p>\n

 <\/p>\n

\u9ad8\u65af-\u725b\u987f\u6cd5(Guass-Newton Algorithm)<\/strong><\/span><\/p>\n

\u9ad8\u65af-\u725b\u987f\u6cd5\u662f\u5728\u725b\u987f\u6cd5<\/a>\u57fa\u7840\u4e0a\u8fdb\u884c\u4fee\u6539\u5f97\u5230\u7684\uff0c\u7528\u6765(\u4ec5\u7528\u4e8e)\u89e3\u51b3\u975e\u7ebf\u6027\u6700\u5c0f\u4e8c\u4e58\u95ee\u9898\u3002\u9ad8\u65af-\u725b\u987f\u6cd5\u76f8\u8f83\u725b\u987f\u6cd5\u7684\u6700\u5927\u4f18\u70b9\u662f\u4e0d\u9700\u8981\u8ba1\u7b97\u4e8c\u9636\u5bfc\u6570\u77e9\u9635(Hessian\u77e9\u9635)\uff0c\u5f53\u7136\uff0c\u8fd9\u9879\u597d\u5904\u7684\u4ee3\u4ef7\u662f\u5176\u4ec5\u9002\u7528\u4e8e\u6700\u5c0f\u4e8c\u4e58\u95ee\u9898\u3002\u5982\u4e0b\u662f\u5176\u63a8\u5bfc\u8fc7\u7a0b\uff1a<\/p>\n

\u6700\u5c0f\u4e8c\u4e58\u65b9\u6cd5\u7684\u76ee\u6807\u662f\u4ee4\u6b8b\u5dee\u7684\u5e73\u65b9\u548c\u6700\u5c0f\uff1a$$f(\\theta) = \\frac{1}{2}\\sum_{i=0}^mr(\\mathbf{x_i})^2$$\u91c7\u7528\u725b\u987f\u6cd5\u6c42\u89e3\\(f(\\theta)\\)\u7684\u6700\u5c0f\u503c\uff0c\u9700\u8981\u8ba1\u7b97\u5176\u68af\u5ea6\u5411\u91cf\u4e0eHessian\u77e9\u9635\u3002$$\\nabla_\\theta f =\\frac{\\partial f}{\\partial \\theta}=\\sum r_i\\frac{\\partial r_i}{\\partial \\theta}=\\begin{bmatrix}
\nr(x_1)&r(x_2)&\\dots&r(r_m)\\end{bmatrix} \\begin{bmatrix}
\n\\nabla_\\theta r(x_1)^T \\\\
\n\\nabla_\\theta r(x_2)^T \\\\
\n\\vdots\\\\
\n\\nabla_\\theta r(x_m)^T \\\\
\n\\end{bmatrix}$$<\/p>\n

\u5176\u4e2d$$J_r(\\theta)=\\begin{bmatrix}\\frac{\\partial r_j}{\\partial\\theta_i}\\end{bmatrix}_{j=1,\\dots,m;i=1,\\dots,n}=\\begin{bmatrix}
\n\\nabla_\\theta r(x_1)^T \\\\
\n\\nabla_\\theta r(x_2)^T \\\\
\n\\vdots\\\\
\n\\nabla_\\theta r(x_m)^T \\\\\\
\n\\end{bmatrix}$$\u79f0\u4e3a\\(r\\)\u7684\u96c5\u5404\u6bd4(Jacobian)\u77e9\u9635\u3002\u56e0\u6b64\u4e0a\u5f0f\u53ef\u4ee5\u5199\u4f5c$$\\nabla_\\theta f = r^TJ_r=J_r^Tr$$\u5176\u4e2d\\(r=\\begin{bmatrix}r(x_1)&r(x_2)&\\dots&r_m\\end{bmatrix}^T\\)\u3002\u518d\u770bHessian\u77e9\u9635\u7684\u8ba1\u7b97\uff1a<\/p>\n

$$H = \\begin{bmatrix}\\frac{\\partial^2f}{\\partial \\theta^2} \\end{bmatrix}=\\sum\\begin{bmatrix}r_i\\frac{\\partial^2r_i}{\\partial\\theta^2}+(\\frac{\\partial r_i}{\\partial \\theta})(\\frac{\\partial r_i}{\\partial \\theta})^T \\end{bmatrix}$$\u89c2\u5bdf\u4e8c\u9636\u5bfc\u6570\u9879\\(r_i\\frac{\\partial^2r_i}{\\partial\\theta^2}\\)\uff0c\u56e0\u4e3a\u6b8b\u5dee \\(r_i\\approx 0\\),\u56e0\u6b64\u6211\u4eec\u53ef\u4ee5\u8ba4\u4e3a\u6b64\u9879\u63a5\u8fd1\u4e8e0\u800c\u820d\u53bb\u3002\u6240\u4ee5Hessian\u77e9\u9635\u53ef\u4ee5\u8fd1\u4f3c\u5199\u6210\uff1a$$H\\approx\\sum\\begin{bmatrix}(\\frac{\\partial r_i}{\\partial\\theta})(\\frac{\\partial r_i}{\\partial\\theta})^T\\end{bmatrix}=J_r^TJ_r$$<\/p>\n

\u8fd9\u91cc\u6211\u4eec\u53ef\u4ee5\u770b\u5230\u9ad8\u65af-\u725b\u987f\u6cd5\u76f8\u5bf9\u4e8e\u725b\u987f\u6cd5\u7684\u4e0d\u540c\u5c31\u662f\u5728\u4e8e\u91c7\u7528\u4e86\u8fd1\u4f3c\u7684Hessian\u77e9\u9635\u964d\u4f4e\u4e86\u8ba1\u7b97\u7684\u96be\u5ea6\uff0c\u4f46\u662f\u540c\u65f6\uff0c\u820d\u53bb\u9879\u4ec5\u9002\u7528\u4e8e\u6700\u5c0f\u4e8c\u4e58\u95ee\u9898\u4e2d\u6b8b\u5dee\u8f83\u5c0f\u7684\u60c5\u5f62\u3002<\/p>\n

\u5c06\u68af\u5ea6\u5411\u91cf\uff0cHessian\u77e9\u9635(\u8fd1\u4f3c)\u5e26\u5165\u725b\u987f\u6cd5\u6c42\u6839\u516c\u5f0f\uff0c\u5f97\u5230\u9ad8\u65af-\u725b\u987f\u6cd5\u7684\u8fed\u4ee3\u5f0f\uff1a$$\\theta_i = \\theta_{i-1}-{(J_r^TJ_r)}^{-1}J_r^Tr$$\u53ea\u9700\u8981\u8ba1\u7b97\u51fa\\(m\\times n \\)\u7684Jacobian\u77e9\u9635\u4fbf\u53ef\u4ee5\u8fdb\u884c\u9ad8\u65af-\u725b\u987f\u6cd5\u7684\u8fed\u4ee3\uff0c\u8ba1\u7b97\u5df2\u7ecf\u7b97\u662f\u975e\u5e38\u7b80\u4fbf\u7684\u4e86\u3002<\/p>\n

Levenberg-Marquart \u7b97\u6cd5<\/strong><\/span><\/p>\n

\u4e0e\u725b\u987f\u6cd5\u4e00\u6837\uff0c\u5f53\u521d\u59cb\u503c\u8ddd\u79bb\u6700\u5c0f\u503c\u8f83\u8fdc\u65f6\uff0c\u9ad8\u65af-\u725b\u987f\u6cd5\u7684\u5e76\u4e0d\u80fd\u4fdd\u8bc1\u6536\u655b\u3002\u5e76\u4e14\u5f53\\(J_r^TJ_r\\)\u8fd1\u4f3c\u5947\u5f02\u7684\u65f6\u5019\uff0c\u9ad8\u65af\u725b\u987f\u6cd5\u4e5f\u4e0d\u80fd\u6b63\u786e\u6536\u655b\u3002Levenberg-Marquart \u7b97\u6cd5\u662f\u5bf9\u4e0a\u8ff0\u7f3a\u70b9\u7684\u6539\u8fdb\u3002L-M\u65b9\u6cd5\u662f\u5bf9\u68af\u5ea6\u4e0b\u964d\u6cd5\u4e0e\u9ad8\u65af-\u725b\u987f\u6cd5\u8fdb\u884c\u7ebf\u6027\u7ec4\u5408\u4ee5\u5145\u5206\u5229\u7528\u4e24\u79cd\u7b97\u6cd5\u7684\u4f18\u52bf\u3002\u901a\u8fc7\u5728Hessian\u77e9\u9635\u4e2d\u52a0\u5165\u963b\u5c3c\u7cfb\u6570\\(\\lambda\\)\u6765\u63a7\u5236\u6bcf\u4e00\u6b65\u8fed\u4ee3\u7684\u6b65\u957f\u4ee5\u53ca\u65b9\u5411\uff1a$$(H+\\lambda I)\\epsilon = -J_r^Tr$$<\/p>\n