{"id":3014,"date":"2022-02-20T21:45:08","date_gmt":"2022-02-20T13:45:08","guid":{"rendered":"https:\/\/egonlin.com\/?p=3014"},"modified":"2022-02-20T21:45:08","modified_gmt":"2022-02-20T13:45:08","slug":"%e7%ac%ac%e4%b8%80%e8%8a%82%ef%bc%9a%e9%80%bb%e8%be%91%e5%9b%9e%e5%bd%92","status":"publish","type":"post","link":"https:\/\/egonlin.com\/?p=3014","title":{"rendered":"\u7b2c\u4e00\u8282\uff1a\u903b\u8f91\u56de\u5f52"},"content":{"rendered":"

\u903b\u8f91\u56de\u5f52<\/h1>\n

  \u867d\u7136\u903b\u8f91\u56de\u5f52\u7684\u540d\u5b57\u91cc\u6709\u201c\u56de\u5f52\u201d\u4e24\u4e2a\u5b57\uff0c\u4f46\u662f\u5b83\u5e76\u4e0d\u662f\u4e00\u4e2a\u56de\u5f52\u7b97\u6cd5\uff0c\u4e8b\u5b9e\u4e0a\u5b83\u662f\u4e00\u4e2a\u5206\u7c7b\u7b97\u6cd5\u3002<\/p>\n

\u903b\u8f91\u56de\u5f52\u5b66\u4e60\u76ee\u6807<\/h1>\n
    \n
  1. \u4e8c\u5143\u903b\u8f91\u56de\u5f52\u7684\u76ee\u6807\u51fd\u6570<\/li>\n
  2. \u6700\u5c0f\u5316\u4e8c\u5143\u903b\u8f91\u56de\u5f52\u76ee\u6807\u51fd\u6570<\/li>\n
  3. \u4e8c\u5143\u903b\u8f91\u56de\u5f52\u7684\u6b63\u5219\u5316<\/li>\n
  4. \u591a\u5143\u903b\u8f91\u56de\u5f52<\/li>\n
  5. \u903b\u8f91\u56de\u5f52\u7684\u6d41\u7a0b<\/li>\n
  6. \u903b\u8f91\u56de\u5f52\u7684\u4f18\u7f3a\u70b9<\/li>\n<\/ol>\n

      \u66fe\u7ecf\u5728\u611f\u77e5\u673a\u5f15\u5165\u65f6\u6211\u4eec\u8bb2\u8fc7\uff0c\u64cd\u573a\u4e0a\u7537\u751f\u548c\u5973\u751f\u7531\u4e8e\u53d7\u4f20\u7edf\u601d\u60f3\u7684\u5f71\u54cd\uff0c\u7537\u751f\u548c\u5973\u751f\u5206\u5f00\u7ad9\u7740\uff0c\u5e76\u4e14\u56e0\u4e3a\u7537\u751f\u548c\u5973\u751f\u6563\u4e71\u5728\u64cd\u573a\u4e0a\u5448\u7ebf\u6027\u53ef\u5206\u7684\u72b6\u6001\uff0c\u56e0\u6b64\u6211\u4eec\u603b\u53ef\u4ee5\u901a\u8fc7\u611f\u77e5\u673a\u7b97\u6cd5\u627e\u5230\u4e00\u6761\u76f4\u7ebf\u628a\u7537\u751f\u548c\u5973\u751f\u5206\u5f00\uff0c\u5e76\u4e14\u6700\u7ec8\u53ef\u4ee5\u5f97\u5230\u611f\u77e5\u673a\u6a21\u578b\u4e3a
    \n$$
    \nf(x)=sign((w^*)^Tx)
    \n$$
    \n  \u5982\u679c\u4f60\u7ec6\u5fc3\u70b9\u4f1a\u53d1\u73b0\uff0c\u7531\u4e8e\u611f\u77e5\u6a21\u578b\u4f7f\u7528\u7684\u662fsign\u51fd\u6570\uff0c\u5982\u679c\u5f53\u8ba1\u7b97\u4e00\u4e2a\u6837\u672c\u70b9$w^Tx=0.001$\u7684\u65f6\u5019\uff0c$sign(w^Tx)=1$\uff0c\u5373\u8be5\u51fd\u6570\u4f1a\u628a\u8be5\u6837\u672c\u5f52\u4e3a$1$\uff0c\u4f46\u662f\u4e3a\u4ec0\u4e48\u4ed6\u4e0d\u80fd\u662f$0$\u7c7b\u5462\uff1f\u5e76\u4e14\u7531\u4e8esign\u51fd\u6570\u5728$x=0$\u5904\u6709\u4e00\u4e2a\u9636\u8dc3\uff0c\u5373\u51fd\u6570\u4e0d\u8fde\u7eed\uff0c\u8be5\u51fd\u6570\u5728\u6570\u5b66\u4e0a\u4e5f\u662f\u4e0d\u65b9\u4fbf\u5904\u7406\u7684\u3002<\/p>\n

      \u7531\u6b64\u903b\u8f91\u51fd\u6570\u4f7f\u7528sigmoid\u51fd\u6570\u5bf9$w^Tx$\u505a\u5904\u7406\uff0c\u5e76\u4e14\u628asigmoid\u51fd\u6570\u5f97\u5230\u7684\u503c$\\hat{y}$\u5f53\u6210\u6982\u7387\u8fdb\u884c\u4e0b\u4e00\u6b65\u5904\u7406\uff0c\u8fd9\u4e5f\u6b63\u662f\u903b\u8f91\u56de\u5f52\u5bf9\u611f\u77e5\u673a\u7684\u6539\u8fdb\u3002<\/p>\n

      \u4e0a\u8ff0\u6574\u4e2a\u8fc7\u7a0b\u5176\u5b9e\u5c31\u662f\u903b\u8f91\u56de\u5f52\u4e00\u6b65\u4e00\u6b65\u88ab\u5047\u60f3\u51fa\u6765\u7684\u7684\u4e00\u4e2a\u8fc7\u7a0b\uff0c\u63a5\u4e0b\u6765\u5c06\u4ece\u7406\u8bba\u5c42\u9762\u62bd\u8c61\u7684\u8bb2\u89e3\u903b\u8f91\u56de\u5f52\u3002<\/p>\n

    # \u611f\u77e5\u673a\u5f15\u5165\u56fe\u4f8b\nimport numpy as np\nimport matplotlib.pyplot as plt\nfrom matplotlib.font_manager import FontProperties\n\n%matplotlib inline\nfont = FontProperties(fname='\/Library\/Fonts\/Heiti.ttc')\n\nnp.random.seed(1)\nx1 = np.random.random(20)+1.5\ny1 = np.random.random(20)+0.5\nx2 = np.random.random(20)+3\ny2 = np.random.random(20)+0.5\n\n# \u4e00\u884c\u4e8c\u5217\u7b2c\u4e00\u4e2a\nplt.subplot(121)\nplt.scatter(x1, y1, s=50, color='b', label='\u7537\u5b69(+1)')\nplt.scatter(x2, y2, s=50, color='r', label='\u5973\u5b69(-1)')\nplt.vlines(2.8, 0, 2, colors="r", linestyles="-", label='$wx+b=0$')\nplt.title('\u7ebf\u6027\u53ef\u5206', fontproperties=font, fontsize=20)\nplt.xlabel('x')\nplt.legend(prop=font)\n\n# \u4e00\u884c\u4e8c\u5217\u7b2c\u4e8c\u4e2a\nplt.subplot(122)\nplt.scatter(x1, y1, s=50, color='b', label='\u7537\u5b69(+1)')\nplt.scatter(x2, y2, s=50, color='r', label='\u5973\u5b69(-1)')\nplt.scatter(3.5, 1, s=50, color='b')\nplt.title('\u7ebf\u6027\u4e0d\u53ef\u5206', fontproperties=font, fontsize=20)\nplt.xlabel('x')\nplt.legend(prop=font)\nplt.show()<\/code><\/pre>\n
    \"\"<\/div><\/p>\n
    \u903b\u8f91\u56de\u5f52\u8be6\u89e3<\/h1>\n
    \u7ebf\u6027\u56de\u5f52\u4e0e\u903b\u8f91\u56de\u5f52<\/h2>\n
      \u7ebf\u6027\u56de\u5f52\u7684\u5047\u8bbe\u51fd\u6570\u4e3a
    \n$$
    \n\\hat{y} = \\omega^Tx
    \n$$
    \n\u6b64\u65f6\u7684$\\hat{y}$\u662f\u8fde\u7eed\u503c\uff0c\u6240\u4ee5\u5b83\u662f\u4e00\u4e2a\u56de\u5f52\u6a21\u578b\uff0c\u5982\u679c$\\hat{y}$\u662f\u79bb\u6563\u503c\u5462\uff1f<\/p>\n
      \u53ef\u80fd\u4f60\u5df2\u7ecf\u60f3\u5230\u4e86\uff0c\u5bf9\u5047\u8bbe\u51fd\u6570\u5f97\u5230\u7684\u8fde\u7eed\u503c\u518d\u505a\u4e00\u6b21\u8f6c\u6362\uff0c\u5373$g(\\hat{y})$\uff0c\u5e76\u4e14\u4ee4$g(\\hat{y})$\u51fd\u6570\u503c\u5728$\\hat{y}$\u5c5e\u4e8e\u67d0\u4e2a\u533a\u95f4\u65f6\u4e3a$c_1$\u7c7b\uff1b$\\hat{y}$\u5c5e\u4e8e\u53e6\u4e00\u4e2a\u533a\u95f4\u65f6\u4e3a$c_2$\u7c7b\uff0c\u8fd9\u6837\u5c31\u80fd\u5f97\u5230\u4e00\u4e2a\u4e8c\u5143\u5206\u7c7b\u6a21\u578b\u3002<\/p>\n
    \u4e8c\u5143\u903b\u8f91\u56de\u5f52\u7684\u5047\u8bbe\u51fd\u6570<\/h2>\n
      \u4e0a\u4e00\u8282\u8bb2\u5230\u5bf9\u7ebf\u6027\u56de\u5f52\u7684\u7ed3\u679c\u901a\u8fc7\u51fd\u6570$g$\u505a\u4e00\u6b21\u8f6c\u6362\u5373\u53ef\u5f97\u903b\u8f91\u56de\u5f52\u3002\u5728\u903b\u8f91\u56de\u5f52\u5f53\u4e2d\u51fd\u6570$g$\u901a\u5e38\u4f7f\u7528Sigmoid\u51fd\u6570\u66ff\u4ee3\uff0c\u5373\u51fd\u6570$g$\u4e3a
    \n$$
    \ng(z) = {\\frac{1}{1+e^{-z}}}
    \n$$<\/p>\n
    \u8ba9\u6b65\u6bd4<\/h3>\n
    \"\"<\/div><\/p>\n
      \u8ba9\u6b65\u6bd4\u53ef\u4ee5\u7406\u89e3\u6210\u6709\u5229\u4e8e\u67d0\u4e00\u7279\u5b9a\u4e8b\u4ef6\u7684\u6982\u7387\uff0c\u53ef\u4ee5\u5b9a\u4e49\u4e3a
    \n$$
    \n{\\frac{p}{1-p}}
    \n$$
    \n  \u5728\u5df2\u77e5\u4e8c\u5206\u7c7b\u95ee\u9898\u7684\u60c5\u51b5\u4e0b\u6bcf\u4e2a\u5206\u7c7b\u7684\u6982\u7387\u5206\u522b\u4e3a$\\hat{y_i}$\u548c$1-\\hat{y_i}$\uff0c\u53ef\u4ee5\u5b9a\u4e49logit\u51fd\u6570\uff0c\u5373\u8ba9\u6b65\u6bd4\u7684\u5bf9\u6570\u5f62\u5f0f\uff08log-odds\uff09\u4e3a
    \n$$
    \n\\begin{align}
    \n\\log{it}(\\hat{y_i}) & = \\log{\\frac{p(y=1|x,\\omega)}{p(y=0|x,\\omega)}} \\
    \n& = \\log{\\frac{\\hat{y_i}}{1-\\hat{y_i}}} \\
    \n& = \\log{\\frac{{\\frac{1}{1+e^{-\\omega^Tx}}}}{{\\frac{-\\omega^Tx}{1+e^{-\\omega^Tx}}}}} \\
    \n& = \\omega^Tx
    \n\\end{align}
    \n$$
    \n\u5176\u4e2d$\\log{it}(p)$\u51fd\u6570\u7b49\u4e8e\u4e8b\u4ef6\u53d1\u751f\u7684\u6982\u7387\u9664\u4ee5\u4e0d\u53d1\u751f\u7684\u6982\u7387\u53d6\u5bf9\u6570\uff0c\u5373\u8868\u793a\u7279\u5f81\u503c\u548c\u5bf9\u6570\u6982\u7387\u4e4b\u95f4\u7684\u7ebf\u6027\u5173\u7cfb\u3002<\/p>\n
      \u7136\u800c\u7279\u5f81\u503c\u548c\u5bf9\u6570\u6982\u7387\u4e4b\u95f4\u7684\u7ebf\u6027\u5173\u7cfb\u5e76\u4e0d\u91cd\u8981\uff0c\u91cd\u8981\u7684\u662f\u9884\u6d4b\u503c\u4e0e\u5b83\u53d1\u751f\u7684\u6982\u7387\u7684\u5173\u7cfb\uff0c\u5373\u8ba9\u6b65\u6bd4\u7684\u9006\u5f62\u5f0f\uff0c\u4e5f\u5c31\u662f\u4e0a\u8ff0\u8bf4\u5230\u7684Sigmoid\u51fd\u6570\u3002
    \n$$
    \nw^Tx = \\log{\\frac{p}{1-p}} \\
    \ne^{w^Tx} = {\\frac{p}{1-p}} \\
    \np = {\\frac{1}{1+e^{-w^Tx}}} \\
    \n$$<\/p>\n
    Sigmoid\u51fd\u6570\u56fe\u50cf<\/h3>\n
    # Sigmoid\u51fd\u6570\u56fe\u50cf\u56fe\u4f8b\nimport numpy as np\nimport matplotlib.pyplot as plt\n%matplotlib inline\n\ndef sigmoid(z):\n    return 1 \/ (1 + np.exp(-z))\n\nax = plt.subplot(111)\n\n# \u63cf\u7ed8\u5341\u5b57\u7ebf\nax.spines['right'].set_color('none')\nax.spines['top'].set_color('none')\nax.xaxis.set_ticks_position('bottom')\nax.spines['bottom'].set_position(('data', 0))\nax.yaxis.set_ticks_position('left')\nax.spines['left'].set_position(('data', 0))\n\nz = np.linspace(-10, 10, 256)\nhat_y = sigmoid(z)\nplt.plot(z, hat_y, c='r', label='Sigmoid')\n\n# \u63cf\u7ed8y=0.5\u548cy=1.0\u4e24\u6761\u76f4\u7ebf\nplt.yticks([0.0, 0.5, 1.0])\nax = plt.gca()\nax.yaxis.grid(True)\n\nplt.xlabel('z')\nplt.ylabel('$\\hat{y}$')\nplt.legend()\nplt.show()<\/code><\/pre>\n
    \"\"<\/div><\/p>\n
      \u4e0a\u56fe\u4e3aSigmoid\u51fd\u6570\u56fe\u50cf\uff0c\u53ef\u4ee5\u770b\u51fa\u5f53$z$\u8d8b\u4e8e\u6b63\u65e0\u7a77\u65f6\uff0c$g(z)$\u8d8b\u4e8e1\uff1b\u5f53$z$\u8d8b\u4e8e\u8d1f\u65e0\u7a77\u65f6\uff0c$g(z)$\u8d8b\u4e8e0\uff0c\u8fd9\u4e2a\u5c5e\u6027\u975e\u5e38\u9002\u5408\u4e0a\u8ff0\u6240\u8bf4\u7684\u5206\u7c7b\u6a21\u578b\u3002<\/p>\n
      \u56e0\u6b64\u53ef\u4ee5\u628a\u7ebf\u6027\u51fd\u6570\u7684\u5047\u8bbe\u51fd\u6570\u770b\u6210\u662fSigmoid\u51fd\u6570\u7684\u81ea\u53d8\u91cf\uff0c\u5373\u903b\u8f91\u56de\u5f52\u7684\u5047\u8bbe\u51fd\u6570\u4e3a
    \n$$
    \n\\hat{y} = {\\frac{1}{1+e^{-\\omega^Tx}}}
    \n$$
    \n  \u867d\u7136\u6539\u53d8\u4e86\u903b\u8f91\u56de\u5f52\u7684\u5047\u8bbe\u51fd\u6570\uff0c\u4f46\u662fSigmoid\u51fd\u6570\u7684\u8f93\u51fa\u503c\u662f$(0,1)$\u8303\u56f4\u5185\u7684\u8fde\u7eed\u503c\uff0c\u5e76\u4e0d\u662f\u4e8c\u5143\u5206\u7c7b\u6a21\u578b\u60f3\u8981\u7684\u4e8c\u5143\u79bb\u6563\u503c\uff0c\u56e0\u6b64\u9700\u8981\u5bf9\u903b\u8f91\u56de\u5f52\u7684\u5047\u8bbe\u51fd\u6570\u505a\u8fdb\u4e00\u6b65\u5904\u7406\uff0c\u5373
    \n$$
    \n\\begin{cases}
    \n\\hat{y}>0.5\u5373\\omega^Tx>0, \\quad y=1 \\
    \n\\hat{y}<0.5\u5373\\omega^Tx<0, \\quad y=0 \\
    \n\\end{cases}
    \n$$
    \n\u5982\u679c$\\hat{y}=0.5\u5373\\omega^Tx=0$\uff0c\u4e0d\u5728\u903b\u8f91\u56de\u5f52\u6a21\u578b\u7684\u8ba8\u8bba\u8303\u56f4\u5185\uff0c\u4e00\u822c\u800c\u8a00\u89c6\u5177\u4f53\u60c5\u51b5\u800c\u5b9a\u3002<\/p>\n
    \u4e8c\u5143\u903b\u8f91\u56de\u5f52\u7684\u76ee\u6807\u51fd\u6570<\/h2>\n
      \u5f97\u5230\u4e86\u903b\u8f91\u56de\u5f52\u7684\u5047\u8bbe\u51fd\u6570\uff0c\u5219\u9700\u8981\u901a\u8fc7\u6700\u5c0f\u5316\u76ee\u6807\u51fd\u6570\u5373\u6700\u5c0f\u5316\u8bef\u5dee\u627e\u5230\u6700\u5408\u9002\u7684\u53c2\u6570$\\omega$\u3002<\/p>\n
      \u7531\u4e8e\u7ebf\u6027\u56de\u5f52\u662f\u9884\u6d4b\u8fde\u7eed\u503c\u7684\u6a21\u578b\uff0c\u56e0\u6b64\u53ef\u4ee5\u4f7f\u7528\u5747\u65b9\u8bef\u5dee\u4ee3\u4ef7\u51fd\u6570\u3002\u4f46\u662f\u903b\u8f91\u56de\u5f52\u662f\u9884\u6d4b\u79bb\u6563\u503c\u7684\u6a21\u578b\uff0c\u56e0\u6b64\u53ef\u4ee5\u4f7f\u7528\u6781\u5927\u4f3c\u7136\u4f30\u8ba1\u63a8\u5bfc\u51fa\u903b\u8f91\u56de\u5f52\u7684\u76ee\u6807\u51fd\u6570\u3002<\/p>\n
      \u4e0a\u4e00\u8282\u5047\u8bbe\u903b\u8f91\u56de\u5f52\u7684\u8f93\u51fa\u4e3a\u7c7b\u522b$0$\u6216\u7c7b\u522b$1$\uff0c\u7528\u6982\u7387\u8868\u8fbe\u65b9\u5f0f\u4e3a
    \n$$
    \n\\begin{align}
    \n& p(y=1|x,\\omega)=\\pi(x) \\
    \n& p(y=0|x,\\omega)=1-\\pi(x)
    \n\\end{align}
    \n$$
    \n  \u7531\u4e8e$y$\u53ea\u53ef\u80fd\u662f$0$\u6216$1$\uff0c\u5219\u53ef\u4ee5\u628a\u4e0a\u8ff0\u4e24\u4e2a\u516c\u5f0f\u8054\u7acb\u53ef\u5f97$y$\u7684\u6982\u7387\u5206\u5e03\u51fd\u6570\u8868\u8fbe\u5f0f
    \n$$
    \np(y|x,\\omega) = (\\pi(x))^y(1-\\pi(x))^{(1-y)}
    \n$$
    \n  \u901a\u8fc7$y$\u7684\u6982\u7387\u5206\u5e03\u51fd\u6570\u8868\u8fbe\u5f0f\u5373\u53ef\u5f97\u4f3c\u7136\u51fd\u6570\u4e3a
    \n$$
    \nL(\\omega) = \\prod_{i=1}^m [\\pi(x_i)]^{y_i}[(1-\\pi{x_i})]^{(1-y_i)}
    \n$$
    \n\u5176\u4e2d$m$\u4e3a\u6837\u672c\u7684\u4e2a\u6570\u3002<\/p>\n
      \u901a\u8fc7\u4f3c\u7136\u51fd\u6570\u5f97\u5230\u5bf9\u6570\u4f3c\u7136\u51fd\u6570\u5373\u76ee\u6807\u51fd\u6570\uff08\u6ce8\uff1a\u8be5\u76ee\u6807\u51fd\u6570\u4e0e\u4ea4\u53c9\u71b5\u635f\u5931\u51fd\u6570\u7684\u5f62\u5f0f\u4e00\u81f4\uff0c\u4e8c\u5143\u903b\u8f91\u56de\u5f52\u53ef\u4ee5\u7406\u89e3\u4e3a\u4ea4\u53c9\u71b5\u635f\u5931\u51fd\u6570\u4e24\u4e2a\u7c7b\u53d8\u91cf\u7684\u7279\u6b8a\u5f62\u5f0f\uff0c\u4e3a
    \n$$
    \n\\begin{align}
    \nJ(\\omega) & = \\log{L(\\omega)} \\
    \n& = \\sum_{i=1}^m [y_i\\log\\pi(x_i)+(1-y_i)\\log(1-\\pi(x_i))]
    \n\\end{align}
    \n$$
    \n\u5bf9$J(\\omega)$\u6c42\u6781\u5927\u503c\uff0c\u5373\u53ef\u5f97\u5230$\\omega$\u7684\u4f30\u8ba1\u503c\u3002<\/p>\n
      \u4e00\u822c\u91c7\u7528\u68af\u5ea6\u4e0a\u5347\u6cd5\u6216\u62df\u725b\u987f\u6cd5\u6c42\u89e3$\\omega$\u7684\u4f30\u8ba1\u503c\u3002<\/p>\n
    \u4e0d\u540c\u6837\u672c\u5206\u7c7b\u7684\u4ee3\u4ef7<\/h3>\n
    [rml_readmore]\uff1a
    \n  \u903b\u8f91\u56de\u5f52\u7684\u76ee\u6807\u51fd\u6570\u4e3a
    \n$$
    \nJ(\\omega) = \\sum<\/em>{i=1}^m [y_i\\log\\pi(x_i)+(1-y_i)\\log(1-\\pi(x_i))]
    \n$$
    \n  \u5bf9\u4e8e\u4e8c\u5206\u7c7b\u95ee\u9898\uff0c\u53ef\u4ee5\u6c42\u51fa$y=1$\u548c$y=0$\u7684\u4ee3\u4ef7\u51fd\u6570
    \n$$
    \nJ(\\omega) =
    \n\\begin{cases}
    \n-\\log\\pi(x) \\quad if y=1 \\
    \n-\\log(1-\\pi(x)) \\quad if y=0 \\
    \n\\end{cases}
    \n$$<\/p>\n
    # \u4e0d\u540c\u6837\u672c\u5b9e\u4f8b\u5206\u7c7b\u7684\u4ee3\u4ef7\u56fe\u4f8b\ndef cost_1(z):\n    return -np.log(sigmoid(z))\n\ndef cost_0(z):\n    return -np.log(1-sigmoid(z))\n\nz = np.linspace(-10, 10, 256)\npi_x = sigmoid(z)\nc1 = [cost_1(i) for i in z]\nc0 = [cost_0(i) for i in z]\nplt.plot(pi_x, c1, c='r', linestyle='--',\n         label='$J(\\omega), \\quad if \\quad y=1$')\nplt.plot(pi_x, c0, c='g', label='$J(\\omega), \\quad if \\quad y=0$')\nplt.xlabel('$\\pi(x)$')\nplt.ylabel('$J(\\omega)$')\nplt.legend()\nplt.show()<\/code><\/pre>\n
    \"\"<\/div><\/p>\n
      \u4e0a\u56fe\u53ef\u4ee5\u770b\u51fa\u5982\u679c\u6b63\u786e\u5730\u9884\u6d4b\u6837\u672c\u5c5e\u4e8e\u7b2c$1$\u7c7b\uff0c\u4ee3\u4ef7\u4f1a\u63a5\u8fd10\uff08\u865a\u7ebf\uff09\uff1b\u5982\u679c\u6b63\u786e\u7684\u9884\u6d4b$y=0$\uff08\u5b9e\u7ebf\uff09\uff0c\u4ee3\u4ef7\u4e5f\u4f1a\u63a5\u8fd10\u3002\u5982\u679c\u9884\u6d4b\u9519\u8bef\uff0c\u4ee3\u4ef7\u5219\u4f1a\u8d8b\u8fd1\u4e8e\u65e0\u7a77\u5927\uff0c\u5373\u7528\u8d8a\u6765\u8d8a\u5927\u7684\u4ee3\u4ef7\u60e9\u7f5a\u9519\u8bef\u7684\u9884\u6d4b\u3002<\/p>\n
    \"\u5728\u8fd9\u91cc\u63d2\u5165\u56fe\u7247\u63cf\u8ff0\"<\/div><\/p>\n
    \u4e8c\u5143\u903b\u8f91\u56de\u5f52\u76ee\u6807\u51fd\u6570\u6700\u5927\u5316<\/h2>\n
    \u68af\u5ea6\u4e0a\u5347\u6cd5<\/h3>\n
      \u4e8c\u5143\u903b\u8f91\u56de\u5f52\u7684\u76ee\u6807\u51fd\u6570\u4e3a
    \n$$
    \nJ(\\omega) =  \\sum_{i=1}^m [y_i\\ln\\pi(x_i)+(1-y_i)\\ln(1-\\pi(x_i))]
    \n$$
    \n  \u5f97\u5230\u4e8c\u5143\u903b\u8f91\u56de\u5f52\u7684\u76ee\u6807\u51fd\u6570\uff0c\u6211\u4eec\u9700\u8981\u6700\u5927\u5316\u4f3c\u7136\u51fd\u6570\uff0c\u5373\u6700\u5927\u5316\u4e8c\u5143\u903b\u8f91\u56de\u5f52\u7684\u76ee\u6807\u51fd\u6570\u3002<\/p>\n
      \u76ee\u6807\u51fd\u6570\u5bf9$\\omega$\u7684\u504f\u5bfc\u4e3a
    \n$$
    \n\\begin{align}
    \n{\\frac{\\partial{J(\\omega)}}{\\partial{\\omegaj}}} & = \\sum<\/em>{i=1}^m({\\frac{y_i}{\\pi(x_i)}}-{\\frac{1-y_i}{1-\\pi(x_i)}}){\\frac{\\partial{\\pi(x_i)}}{\\partial{\\omegaj}}} \\
    \n& = \\sum<\/em>{i=1}^m({\\frac{y_i}{g(\\omega^Tx_i)}}-{\\frac{1-y_i}{1-g(\\omega^Tx_i)}}){\\frac{\\partial{g(\\omega^Tx_i)}}{\\partial{\\omegaj}}} \\
    \n& = \\sum<\/em>{i=1}^m({\\frac{y_i}{g(\\omega^Tx_i)}}-{\\frac{1-y_i}{1-g(\\omega^Tx_i)}})g(\\omega^Tx_i)(1-g(\\omega^Tx_i)){\\frac{\\partial{\\omega^Tx_i}}{\\partial{\\omegaj}}} \\
    \n& = \\sum<\/em>{i=1}^m (y_i(1-g(\\omega^Tx_i))-(1-y_i)g(\\omega^Tx_i)){x_i}j \\
    \n& = \\sum<\/em>{i=1}^m (y_i – g(\\omega^Tx_i)){x_i}_j
    \n\\end{align}
    \n$$
    \n\u5176\u4e2d$i$\u4e3a\u7b2c$i$\u4e2a\u6837\u672c\uff0c$j$\u4e3a\u7b2c$j$\u4e2a\u7279\u5f81\u3002<\/p>\n
      \u903b\u8f91\u56de\u5f52\u53c2\u6570\u7684\u5b66\u4e60\u89c4\u5219\u4e3a
    \n$$
    \n\\omega_j = \\omegaj + \\alpha{\\sum<\/em>{i=1}^m (y_i – g(\\omega^Tx_i)){x_i}^{(j)}}
    \n$$<\/p>\n
    \u7ebf\u6027\u56de\u5f52\u548c\u903b\u8f91\u56de\u5f52\u7684\u53c2\u6570\u66f4\u65b0<\/h3>\n
      \u7ebf\u6027\u56de\u5f52\u7684\u53c2\u6570\u5b66\u4e60\u516c\u5f0f\u4e3a
    \n$$
    \n\\omega_j = \\omegaj – \\alpha{\\sum<\/em>{i=1}^m (y_i – h(\\omega^Tx_i)){x_i}^{(j)}}
    \n$$
    \n  \u903b\u8f91\u56de\u5f52\u7684\u53c2\u6570\u5b66\u4e60\u516c\u5f0f\u4e3a
    \n$$
    \n\\omega_j = \\omegaj + \\alpha{\\sum<\/em>{i=1}^m (y_i – g(\\omega^Tx_i)){x_i}^{(j)}}
    \n$$
    \n  \u4ece\u4e0a\u8ff0\u4e24\u4e2a\u53c2\u6570\u5b66\u4e60\u516c\u5f0f\u53ef\u4ee5\u770b\u51fa\u7ebf\u6027\u56de\u5f52\u548c\u903b\u8f91\u56de\u5f52\u7684\u53c2\u6570\u66f4\u65b0\u65b9\u5f0f\u6709\u7740\u76f8\u540c\u7684\u516c\u5f0f\uff0c\u4f46\u662f\u7531\u4e8e\u7ebf\u6027\u56de\u5f52\u662f\u6700\u5c0f\u5316\u76ee\u6807\u51fd\u6570\uff0c\u800c\u903b\u8f91\u56de\u5f52\u662f\u6700\u5927\u5316\u4f3c\u7136\u51fd\u6570\u5373\u6700\u5927\u5316\u76ee\u6807\u51fd\u6570\uff0c\u56e0\u6b64\u7ebf\u6027\u56de\u5f52\u662f\u68af\u5ea6\u4e0b\u964d\u6cd5\u3001\u903b\u8f91\u56de\u5f52\u662f\u68af\u5ea6\u4e0a\u5347\u6cd5\uff0c\u66fe\u7ecf\u4e5f\u8bb2\u8fc7\u5176\u5b9e\u68af\u5ea6\u4e0b\u964d\u6cd5\u548c\u68af\u5ea6\u4e0a\u5347\u6cd5\u53ef\u4ee5\u8f6c\u6362\u3002<\/p>\n
    \u62df\u725b\u987f\u6cd5<\/h3>\n
      \u6536\u655b\u901f\u5ea6\u66f4\u5feb\uff0c\u4f46\u662f\u5982\u679c\u7279\u5f81\u7ef4\u5ea6\u8f83\u5927\u8ba1\u7b97\u65f6\u95f4\u6f2b\u957f\u3002<\/p>\n
    \u4e8c\u5143\u903b\u8f91\u56de\u5f52\u6a21\u578b<\/h2>\n
      \u5047\u8bbe\u6c42\u5f97\u903b\u8f91\u56de\u5f52\u53c2\u6570\u4e3a\u4e3a$\\omega^T$\uff0c\u5219\u4e8c\u5143\u903b\u8f91\u56de\u5f52\u6a21\u578b\u4e3a
    \n$$
    \n\\begin{align}
    \n& p(y=1|x) = {\\frac{e^{-\\omega^T{x}}}{1+e^{-\\omega^T{x}}}} \\
    \n& p(y=0|x) = {\\frac{1}{1+e^{-\\omega^T{x}}}}
    \n\\end{align}
    \n$$<\/p>\n
    \"\u5728\u8fd9\u91cc\u63d2\u5165\u56fe\u7247\u63cf\u8ff0\"<\/div><\/p>\n
    \u4e8c\u5143\u903b\u8f91\u56de\u5f52\u7684\u6b63\u5219\u5316<\/h2>\n
    L1\u6b63\u5219\u5316<\/h3>\n
      \u4e8c\u5143\u903b\u8f91\u56de\u5f52\u7684L1\u6b63\u5219\u5316\u4e0e\u666e\u901a\u7ebf\u6027\u56de\u5f52\u7684L1\u6b63\u5219\u5316\u7c7b\u4f3c\uff0c\u589e\u52a0\u4e86L1\u8303\u6570\u4f5c\u4e3a\u60e9\u7f5a\uff0c\u5373
    \n$$
    \nJ(\\omega) = \\sum_{i=1}^m [y_i(\\omega{x_i} – \\ln(1+exp(\\omega(x_i))] + \\lambda||\\omega||_1
    \n$$
    \n  \u4e8c\u5143\u903b\u8f91\u56de\u5f52L1\u6b63\u5219\u5316\u76ee\u6807\u51fd\u6570\u5e38\u7528\u7684\u4f18\u5316\u65b9\u6cd5\u6709\u5750\u6807\u8f74\u4e0b\u964d\u548c\u6700\u5c0f\u89d2\u56de\u5f52\u3002<\/p>\n
    L2\u6b63\u5219\u5316<\/h3>\n
      \u4e8c\u5143\u903b\u8f91\u56de\u5f52\u7684L2\u6b63\u5219\u5316\u4e0e\u666e\u901a\u7ebf\u6027\u56de\u5f52\u7684L2\u6b63\u5219\u5316\u7c7b\u4f3c\uff0c\u589e\u52a0\u4e86L2\u8303\u6570\u4f5c\u4e3a\u60e9\u7f5a\uff0c\u5373
    \n$$
    \nJ(\\omega) = \\sum_{i=1}^m [y_i(\\omega{x_i} – \\ln(1+exp(\\omega(x_i))] + {\\frac{1}{2}}\\lambda||\\omega||_2
    \n$$<\/p>\n
    \"\u5728\u8fd9\u91cc\u63d2\u5165\u56fe\u7247\u63cf\u8ff0\"<\/div><\/p>\n
    \u591a\u5143\u903b\u8f91\u56de\u5f52<\/h2>\n
    \"\u5728\u8fd9\u91cc\u63d2\u5165\u56fe\u7247\u63cf\u8ff0\"<\/div><\/p>\n
      \u4e0a\u9762\u4ecb\u7ecd\u7684\u903b\u8f91\u56de\u5f52\u90fd\u662f\u4e8c\u5143\u5206\u7c7b\u6a21\u578b\uff0c\u7528\u4e8e\u4e8c\u5206\u7c7b\u95ee\u9898\uff0c\u53ef\u4ee5\u4f7f\u7528\u4ee5\u4e0b\u4e09\u79cd\u65b9\u6cd5\u5c06\u5176\u63a8\u5e7f\u4e3a\u591a\u5143\u903b\u8f91\u56de\u5f52\u3002OvR\u548cMvM\u4e3b\u8981\u662f\u5bf9\u6570\u636e\u505a\u5904\u7406\uff0c\u8fd9\u91cc\u4e0d\u4ecb\u7ecd\u5176\u5177\u4f53\u8fc7\u7a0b\u3002\u800cSoftmax\u56de\u5f52\u5219\u662f\u4f18\u5316\u6a21\u578b\uff0c\u56e0\u6b64\u4e3b\u8981\u8bb2\u89e3Softmax\u56de\u5f52\u3002<\/p>\n
    OvR<\/h3>\n
      \u5047\u8bbe\u4e00\u4e2a\u6570\u636e\u96c6$D$\u6709$c_1,c_2,\\ldots,c_k$\u5171$k$\u4e2a\u7c7b\u522b\uff0c\u5219\u53ef\u4ee5\u628a$c_1$\u770b\u6210\u4e00\u4e2a\u7c7b\u522b\uff0c\u628a$c_2,c_3,\\ldots,c_k$\u770b\u6210\u53e6\u5916\u4e00\u4e2a\u7c7b\u522b\uff0c\u5373\u628a$D$\u5206\u4e3a\u4e24\u4e2a\u5b50\u96c6\uff0c\u628a\u591a\u5206\u7c7b\u95ee\u9898\u5219\u53d8\u6210\u4e86\u5173\u4e8e$c_1$\u548c$c_2,c_3,\\ldots,c_k$\u7684\u4e8c\u5206\u7c7b\u95ee\u9898\uff0c\u7136\u540e\u5bf9\u542b\u6709\u591a\u4e2a\u7c7b\u522b\u7684\u5b50\u96c6\u518d\u6b21\u4f7f\u7528OvR\u65b9\u6cd5\uff0c\u76f4\u81f3\u65e0\u6cd5\u5206\u7c7b\u4e3a\u6b62\u3002\u901a\u5e38\u8fd9\u79cd\u65b9\u6cd5\u79f0\u4e3aOvR\uff08One-vs-Rest\uff09\u3002<\/p>\n
    MvM<\/h3>\n
      \u5047\u8bbe\u4e00\u4e2a\u6570\u636e\u96c6$D$\u6709$c_1,c_2,\\ldots,c_k$\u5171$k$\u4e2a\u7c7b\u522b\uff0c\u5219\u53ef\u4ee5\u5148\u628a$c_1,c_2,\\ldots,c_i, \\quad i<k$\u770b\u6210\u4e00\u4e2a\u7c7b\u522b\uff0c\u628a$ci,c<\/em>{i+1},\\ldots,c_k$\u770b\u6210\u53e6\u5916\u4e00\u4e2a\u7c7b\u522b\uff0c\u5373\u628a$D$\u5206\u4e3a\u4e24\u4e2a\u5b50\u96c6\uff0c\u591a\u5206\u7c7b\u95ee\u9898\u5219\u53d8\u6210\u4e86\u5173\u4e8e$c_1,c_2,\\ldots,c_i$\u548c$ci,c<\/em>{i+1},\\ldots,c_k$\u7684\u4e8c\u5206\u7c7b\u95ee\u9898\uff0c\u7136\u540e\u5bf9\u4e24\u4e2a\u5b50\u96c6\u518d\u6b21\u4f7f\u7528MvM\u65b9\u6cd5\uff0c\u76f4\u81f3\u65e0\u6cd5\u5206\u7c7b\u4e3a\u6b62\u3002\u901a\u5e38\u8fd9\u79cd\u65b9\u6cd5\u79f0\u4e3aMvM\uff08Many-vs-Many\uff09\u3002<\/p>\n
      \u5982\u679c\u6bcf\u6b21\u53ea\u9009\u62e9\u4e24\u4e2a\u4e2a\u7c7b\u522b\u8fdb\u884c\u5206\u7c7b\u7684\u8bdd\uff0c\u5219\u8be5\u65b9\u6cd5\u79f0\u4e3aOvO\uff08One-vs-One\uff09\uff0c\u4e00\u822c\u60c5\u51b5\u4e0b\u9996\u5148\u8003\u8651\u4f7f\u7528OvO\u3002<\/p>\n
    \"\u5728\u8fd9\u91cc\u63d2\u5165\u56fe\u7247\u63cf\u8ff0\"<\/div><\/p>\n
    \u903b\u8f91\u56de\u5f52\u6d41\u7a0b<\/h1>\n
    \u8f93\u5165<\/h2>\n
      \u6709$m$\u4e2a\u5b9e\u4f8b$n$\u7ef4\u7279\u5f81\u7684\u6570\u636e\u96c6
    \n$$
    \nT={(x_1,y_1),(x_2,y_2),\\cdots,(x_m,y_m)}
    \n$$
    \n\u5176\u4e2d$x_i$\u662f\u5b9e\u4f8b\u7684\u7279\u5f81\u5411\u91cf\u5373$({x_i}^{(1)},{x_i}^{(2)},\\cdots,{x_i}^{(n)})$\u3002<\/p>\n
    \u8f93\u51fa<\/h2>\n
      $\\omega$\u548c\u4e8c\u5143\u903b\u8f91\u56de\u5f52\u6a21\u578b\u3002<\/p>\n
    \u6d41\u7a0b<\/h2>\n
      \n
    1. \u9009\u53d6\u521d\u503c$\\omega=0$<\/li>\n
    2. \u8bad\u7ec3\u96c6\u4e2d\u9009\u53d6\u6570\u636e$(x_i,y_i)$\uff0c\u5bf9$\\omega$\u4f7f\u7528\u68af\u5ea6\u4e0a\u5347\u66f4\u65b0
      \n$$
      \n\\omega = \\omega + \\alpha(y_i – g(\\omega^Tx_i)){x_i}^{(j)}
      \n$$<\/li>\n
    3. \u91cd\u590d\u6b65\u9aa42\uff0c\u76f4\u81f3$\\omega$\u6536\u655b\u505c\u6b62\u66f4\u65b0<\/li>\n
    4. \u5f97\u5230\u6700\u5c0f\u5316\u7684\u76ee\u6807\u51fd\u6570$J(\\omega)$\uff0c\u540c\u65f6\u53ef\u4ee5\u5f97\u5230\u6700\u4f18\u7684$\\omega^*$\uff0c\u4e8c\u5143\u903b\u8f91\u56de\u5f52\u6a21\u578b\u4e3a
      \n$$
      \n\\begin{align}
      \n& p(y=1|x) = {\\frac{e^{-\\omega^T{x}}}{1+e^{-\\omega^T{x}}}} \\
      \n& p(y=0|x) = {\\frac{1}{1+e^{-\\omega^T{x}}}}
      \n\\end{align}
      \n$$<\/li>\n<\/ol>\n
      \u903b\u8f91\u56de\u5f52\u4f18\u7f3a\u70b9<\/h1>\n
      \u4f18\u70b9<\/h2>\n
        \n
      1. \u7531\u4e8e\u8ba1\u7b97\u91cf\u53ea\u548c\u7279\u5f81\u7684\u6570\u76ee\u6709\u5173\uff0c\u8bad\u7ec3\u901f\u5ea6\u76f8\u8f83\u5176\u4ed6\u7684\u5206\u7c7b\u5668\u4f8b\u5982\u652f\u6301\u5411\u91cf\u673a\u5feb<\/li>\n
      2. \u5f62\u5f0f\u7b80\u5355\uff0c\u6a21\u578b\u7684\u53ef\u89e3\u91ca\u6027\u975e\u5e38\u597d\u3002\u4ece\u7279\u5f81\u7684\u6743\u91cd\u53ef\u4ee5\u770b\u5230\u4e0d\u540c\u7684\u7279\u5f81\u5bf9\u6700\u540e\u7ed3\u679c\u7684\u5f71\u54cd\uff0c\u67d0\u4e2a\u7279\u5f81\u7684\u6743\u91cd\u503c\u6bd4\u8f83\u9ad8\uff0c\u90a3\u4e48\u8fd9\u4e2a\u7279\u5f81\u6700\u540e\u5bf9\u7ed3\u679c\u7684\u5f71\u54cd\u4f1a\u6bd4\u8f83\u5927<\/li>\n
      3. \u53ef\u4ee5\u624b\u52a8\u8c03\u6574\u9608\u503c\uff08\u5e76\u4e0d\u4e00\u5b9a\u89810.5\uff09\uff0c\u8f93\u51fa\u7ed3\u679c\u53ef\u4ee5\u624b\u52a8\u8c03\u8282\u63a7\u5236\uff0c\u7075\u6d3b<\/li>\n<\/ol>\n
        \u7f3a\u70b9<\/h2>\n
          \n
        1. \u56e0\u4e3a\u7c7b\u4f3c\u7ebf\u6027\u56de\u5f52\u5f88\u5bb9\u6613\u6b20\u62df\u5408\uff0c\u5206\u7c7b\u7cbe\u5ea6\u4e0d\u9ad8<\/li>\n
        2. \u5047\u8bbe\u6570\u636e\u5206\u7c7b\u4e25\u91cd\u4e0d\u5e73\u8861\uff0c\u4f8b\u59821:10000\uff0c\u5219\u5206\u7c7b\u65f6\u4f1a\u6709\u503e\u5411\u95ee\u9898<\/li>\n
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          \u5c0f\u7ed3<\/h1>\n
            \u903b\u8f91\u56de\u5f52\u5f15\u5165\u65f6\u8bf4\u5230\u903b\u8f91\u56de\u5f52\u4e00\u5b9a\u7a0b\u5ea6\u4e0a\u4e5f\u662f\u57fa\u4e8e\u611f\u77e5\u673a\u6f14\u5316\u800c\u6765\uff0c\u5728\u66ff\u6362sigmoid\u51fd\u6570\u7684\u540c\u65f6\uff0c\u5c06sigmoid\u51fd\u6570\u5f97\u5230\u7684\u503c\u8f6c\u6362\u4e3a\u6982\u7387\u7684\u5f62\u5f0f\uff0c\u8fdb\u800c\u53ef\u4ee5\u6700\u5927\u5316\u4f3c\u7136\u51fd\u6570\u5f97\u5230\u6700\u4f18$w^*$\uff0c\u8fd9\u4e5f\u662f\u903b\u8f91\u56de\u5f52\u7684\u5de7\u5999\u4e4b\u5904\uff0c\u4f46\u662f\u4e24\u8005\u8fd8\u662f\u6362\u6c64\u4e0d\u6362\u836f\uff0c\u90fd\u662f\u57fa\u4e8e\u7279\u5f81\u7684\u7ebf\u6027\u6a21\u578b\u5206\u7c7b\u3002<\/p>\n
            \u7531\u4e8e\u611f\u77e5\u673a\u3001\u7ebf\u6027\u56de\u5f52\u548c\u903b\u8f91\u56de\u5f52\u90fd\u548c\u7ebf\u6027\u6a21\u578b\u6709\u4e00\u5b9a\u7684\u5173\u7cfb\uff0c\u56e0\u6b64\u653e\u5728\u4e00\u8d77\u8bb2\uff0c\u4e0b\u9762\u5c06\u4f1a\u5c06\u4e00\u4e2a\u5355\u72ec\u7684\u7b97\u6cd5\uff0c\u5b83\u4ece\u7406\u8bba\u4e0a\u800c\u8a00\u662f\u6700\u7b80\u5355\u6613\u61c2\u7684\u4e00\u4e2a\u7b97\u6cd5\uff0c\u5373k\u8fd1\u90bb\u7b97\u6cd5\u3002<\/p>\n
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