\u5728\u5f00\u59cb\u4e4b\u524d\uff0c\u4e5f\u6ca1\u60f3\u523099\u884c\u4ee3\u7801\u5c31\u591f\u4e86\uff0c\u539f\u4ee5\u4e3a\u91cc\u9762\u6709\u4e2a\u201c\u68af\u5ea6\u4e0b\u964d\u201d\uff0c\u4ee3\u7801\u884c\u6570\u5e94\u8be5\u662f\u6570\u767e\u7ea7\u522b\u5427\uff0c\u5b9e\u9645\u5b8c\u6210\u540e\uff0c\u53d1\u73b0\u52a0\u4e0a\u6ce8\u91ca\uff08\u7ea635%\uff09\u624d99\u884c\u3002<\/p>\n\n\n\n
\u5148\u6765\u770b\u201c\u6781\u7b80\u201d\u6709\u591a\u7b80\uff1a<\/p>\n\n\n\n
x -> w*x + b -> logistic function -> output\n ----------------------------------\n | |\n V V\n one neuron activation function<\/code><\/pre>\n\n\n\n\u5728\u540e\u7eed\u7684\u5b9e\u73b0\u4e2d\uff0c\u6211\u4eec\u6784\u9020\u4e86\u516d\u4e2a\u6837\u672c\u7528\u4e8e\u8be5\u795e\u7ecf\u7f51\u7edc\u7684\u8bad\u7ec3\u3002<\/p>\n\n\n\n
\u9274\u4e8e\u8fd9\u4e2a\u201c\u6781\u7b80\u201d\u795e\u7ecf\u7f51\u7edc\uff0c\u6ca1\u6709\u4efb\u4f55\u9690\u85cf\u5c42\uff0c\u6240\u4ee5\uff0c\u8fd9\u4e5f\u662f\u4e00\u4e2a\u5178\u578b\u7684\u201clogistic regression\u201d\u95ee\u9898\u3002<\/p>\n\n\n\n
\u524d\u7f6e\u77e5\u8bc6<\/h4>\n\n\n\n
\u4f60\u9700\u8981\u4e86\u89e3\u5982\u4e0b\u7684\u524d\u7f6e\u77e5\u8bc6\uff0c\u4ee5\u5f88\u597d\u7684\u7406\u89e3\u8be5\u795e\u7ecf\u7f51\u7edc\u7684\u5b9e\u73b0\u4e0e\u8bad\u7ec3\uff1a<\/p>\n\n\n\n
\n- \u4e86\u89e3\u795e\u7ecf\u7f51\u7edc\u7684\u57fa\u7840\uff1a\u6d45\u5c42\u795e\u7ecf\u7f51\u7edc<\/li>\n\n\n\n
- \u4e86\u89e3 \u68af\u5ea6\u4e0b\u964d\u7b97\u6cd5\uff0c\u4e86\u89e3\u57fa\u672c\u6700\u4f18\u5316\u7b97\u6cd5\u6982\u5ff5\uff0c\u4e86\u89e3\u94fe\u5f0f\u6cd5\u5219<\/li>\n\n\n\n
- \u4e86\u89e3 logistic function \u7684\u57fa\u672c\u7279\u6027<\/li>\n\n\n\n
- \u4e86\u89e3 Python \u548c NumPy \u7684\u57fa\u672c\u4f7f\u7528<\/li>\n<\/ul>\n\n\n\n
\u95ee\u9898\u63cf\u8ff0\u4e0e\u7b26\u53f7\u7ea6\u5b9a<\/h4>\n\n\n\n
\u7528\u4e8e\u8bad\u7ec3\u7684\u6837\u672c\u6570\u636e\u6709\\((x,\\hat{y}) \\)\uff1a \\( (1,0)\u3001(2,0)\u3001(3,0)\u3001(4,0)\u3001(5,1)\u3001(6,1) \\)\u3002<\/p>\n\n\n\n
\u4e00\u4e2a\u5177\u4f53\u7684\u6837\u672c\uff0c\u5728\u4e0b\u9762\u7684\u516c\u5f0f\u4e2d\u901a\u5e38\u4f7f\u7528 \\( (x^{(j)}, y^{(j)}) \\)\u8868\u793a\uff0c \u5176\u4e2d\uff0c\\( j = 1…m \\)\u3002<\/p>\n\n\n\n
\\( \\hat{y} \\)\u5219\u8868\u793a\u6839\u636e\u53c2\u6570\u8ba1\u7b97\u51fa\u7684\u9884\u6d4b\u503c\uff0c\u66f4\u4e3a\u5177\u4f53\u7684 \\( \\hat{y}^{(j)} \\)\u8868\u793a \\(x = x^{(j)} \\)\u65f6\u7684\u9884\u6d4b\u503c\u3002<\/p>\n\n\n\n
\u6784\u5efa\u76ee\u6807\u51fd\u6570<\/h4>\n\n\n\n
\u4ece\u6837\u672c\u6570\u636e\u53ef\u4ee5\u770b\u5230\uff0c\u8fd9\u662f\u4e00\u4e2a\u4e8c\u5206\u7c7b\u95ee\u9898\uff0c\u53ef\u4ee5\u4f7f\u7528logistic function<\/code>\u4f5c\u4e3a\u8f93\u51fa\u5c42\u7684\u6fc0\u6d3b\u51fd\u6570\uff0c\u5982\u679c\u8f93\u51fa\u503c\u5927\u4e8e0.5\uff0c\u5219\u9884\u6d4b\u4e3a1<\/code>\uff0c\u5426\u5219\u9884\u6d4b\u4e3a0<\/code>\u3002<\/p>\n\n\n\n\u5bf9\u4e8e\u4efb\u4f55\u4e00\u4e2a\u6837\u672c\uff0c\u5c31\u53ef\u4ee5\u5982\u4e0b\u51fd\u6570\u4f5c\u4e3alogistic function<\/code>\u7684\u635f\u5931\u51fd\u6570\\( L \\)\uff1a<\/p>\n\n\n$$
\nL(y,\\hat{y}) = – (yln(\\hat{y}) + (1-y)ln(1-\\hat{y}))
\n$$<\/p>\n\n\n\n
<\/p>\n\n\n\n
\u6240\u4ee5\uff0c\u5168\u5c40\u7684\u76ee\u6807\u51fd\u6570\u5c31\u662f\uff1a<\/p>\n\n\n
$$
\n\\begin{aligned}
\nJ(w,b) & = \\frac{1}{m} \\sum\\limits_{j=0}^{m} L(y^{(j)},\\hat{y}^{(j)}) \\\\
\n& = \\frac{1}{m} \\sum\\limits_{j=0}^{m} – (y^{(j)}ln(\\hat{y}^{(j)}) + (1-y^{(j)})ln(1-\\hat{y}^{(j)}))
\n\\end{aligned}
\n$$<\/p>\n\n\n\n
\u5176\u4e2d \\( m \\)\u8868\u793a\u603b\u6837\u672c\u6570\u91cf\uff0c\u8fd9\u91cc\u53d6\u503c\u662f6\u3002\u5728\u8fd9\u4e2a\u6781\u7b80\u7684\u795e\u7ecf\u7f51\u7edc\u4e2d \\( \\hat{y} \\)\u6709\u5982\u4e0b\u8ba1\u7b97\u8868\u8fbe\u5f0f\uff1a<\/p>\n\n\n
$$
\n\\hat{y} = \\frac{1}{1+e^{-(wx+b)}}
\n$$<\/p>\n\n\n\n
\u6700\u7ec8\uff0c\u8be5\u795e\u7ecf\u7f51\u7edc\u7684\u53c2\u6570\u6c42\u89e3\uff08\u4e5f\u5c31\u662f\u201c\u8bad\u7ec3\u201d\uff09\u8fc7\u7a0b\uff0c\u5c31\u662f\u6c42\u89e3\u5982\u4e0b\u7684\u6781\u503c\u95ee\u9898\uff1a<\/p>\n\n\n
$$
\n(w,b) = \\min_{w, b} J(w, b) = \\min_{w,b} \\frac{1}{m} \\sum\\limits_{j=0}^{m} L(y^{(j)},\\hat{y}^{(j)})
\n$$<\/p>\n\n\n\n
\u76ee\u6807\u51fd\u6570\u8ba1\u7b97\u7684\u5177\u4f53\u4ee3\u7801\u5982\u4e0b\uff1a<\/p>\n\n\n\n
def cost_function(w_p,b_p,x_p,y_p):\n c = 0\n for i in range(m):\n y = function_f(x_p[i],w_p,b_p)\n c += -y_p[i]*math.log(y) - (1-y_p[i])*math.log(1-y)\n return c<\/code><\/pre>\n\n\n\n<\/p>\n\n\n\n
\u68af\u5ea6\u8ba1\u7b97<\/h4>\n\n\n\n
\u524d\u9762\u4ecb\u7ecd\u4e86\u5f88\u591a\u68af\u5ea6\u7684\u5185\u5bb9\uff0c\u8fd9\u91cc\u4e0d\u518d\u8be6\u8ff0\u3002\u5728\u8fd9\u4e2a\u5177\u4f53\u7684\u95ee\u9898\u4e2d\uff0c\u9700\u8981\u6c42\u89e3\u7684\u68af\u5ea6\u4e3a\uff1a<\/p>\n\n\n
$$
\n(\\frac{\\partial J}{\\partial w},\\frac{\\partial J}{\\partial b})
\n$$<\/p>\n\n\n\n
\u5728\u8fd9\u91cc\uff0c\u7b80\u5355\u5c55\u793a\u8be5\u68af\u5ea6\u7684\u8ba1\u7b97\uff0c\u4e3b\u8981\u9700\u8981\u4f7f\u7528\u7684\u662f\u94fe\u5f0f\u6cd5\u5219\u548c\u57fa\u672c\u7684\u6c42\u5bfc\/\u5fae\u5206\u8fd0\u7b97\u3002<\/p>\n\n\n\n
\u9996\u5148\uff0c\u4e3a\u4e86\u4fbf\u4e8e\u8ba1\u7b97\uff0c\u8fd9\u91cc\u8bb0\uff1a<\/p>\n\n\n
$$
\n\\begin{aligned}
\n\\hat{y} & = \\frac{1}{1+e^{-z}} \\\\
\nz & = w*x + b
\n\\end{aligned}
\n$$<\/p>\n\n\n\n
\u6240\u4ee5\uff0c\u6839\u636e\u94fe\u5f0f\u6cd5\u5219\u5bb9\u6613\u6709\uff1a<\/p>\n\n\n
$$
\n\\frac{\\partial L}{\\partial w} = \\frac{\\partial L}{\\partial \\hat{y}} * \\frac{\\partial \\hat{y}}{\\partial z} * \\frac{\\partial z}{\\partial w} \\\\
\n\\frac{\\partial L}{\\partial b} = \\frac{\\partial L}{\\partial \\hat{y}} * \\frac{\\partial \\hat{y}}{\\partial z} * \\frac{\\partial z}{\\partial b}
\n$$<\/p>\n\n\n\n
\u8fd9\u5176\u4e2d\uff0c\\( \\frac{\\partial L}{\\partial \\hat{y}} \\) \u548c \\( \\frac{\\partial \\hat{y}}{\\partial z} \\)\u7565\u6709\u4e00\u4e9b\u8ba1\u7b97\u91cf\uff0c\\( \\frac{\\partial z}{\\partial w} \\) \u548c\\( \\frac{\\partial z}{\\partial b} \\)\u6bd4\u8f83\u7b80\u5355\uff0c\u5177\u4f53\u7684\uff1a<\/p>\n\n\n
$$
\n\\begin{aligned}
\n\\frac{\\partial L}{\\partial \\hat{y}} * \\frac{\\partial \\hat{y}}{\\partial z} & = \\hat{y} – y \\\\
\n\\frac{\\partial z}{\\partial w} & = x \\\\
\n\\frac{\\partial z}{\\partial b} & = 1
\n\\end{aligned}
\n$$<\/p>\n\n\n\n
<\/p>\n\n\n\n
\u6240\u4ee5\uff0c\u6700\u7ec8\u7684\u68af\u5ea6\u8ba1\u7b97\u516c\u5f0f\u5982\u4e0b\uff1a<\/p>\n\n\n
$$
\n\\begin{aligned}
\n\\frac{\\partial J}{\\partial w} & = \\frac{1}{m} \\sum\\limits_{j=1}^{m} (\\hat{y}^{(j)} – y^{(j)})*x^{(j)} \\\\
\n\\frac{\\partial J}{\\partial b} & = \\frac{1}{m} \\sum\\limits_{j=1}^{m} (\\hat{y}^{(j)} – y^{(j)})
\n\\end{aligned}
\n$$<\/p>\n\n\n\n
\u5728\u5b9e\u9645\u7684\u8ba1\u7b97\u4e2d\uff0c\u5148\u901a\u8fc7\u6b63\u5411\u4f20\u64ad\uff08Forward Propagation\uff09\u8ba1\u7b97\u51fa\\( \\hat{y}^{(j)} \\)\uff0c\u7136\u540e\u5728\u8ba1\u7b97\u51fa\u68af\u5ea6\u3002\u6b64\u5916\uff0c\u53ef\u4ee5\u4f7f\u7528NumPy<\/code>\u7684ndarray<\/code>\u7b80\u5316\u8868\u8fbe\uff0c\u540c\u65f6\u589e\u52a0\u8ba1\u7b97\u7684\u5e76\u884c\u6027\u3002\u8fd9\u91cc\u4e3a\u4fbf\u4e8e\u7406\u89e3\uff0c\u5168\u90e8\u90fd\u4f7f\u7528\u6807\u91cf\u8ba1\u7b97\uff0c\u5728\u6587\u7ae0\u7684\u6700\u540e\u4e5f\u63d0\u4f9b\u4e86NumPy\u7684\u5bf9\u5e94\u5b9e\u73b0\u3002<\/p>\n\n\n\n\u6b63\u5411\u4f20\u64ad\u8ba1\u7b97\u5982\u4e0b\uff1a<\/p>\n\n\n\n
# function_f: \n# x : scalar\n# w : scalar\n# b : scalar\ndef function_f(x,w,b): \n return 1\/(1+math.exp(-(x*w+b)))<\/code><\/pre>\n\n\n\n\u68af\u5ea6\uff08\u53cd\u5411\u4f20\u64ad\uff09\u8ba1\u7b97\u5982\u4e0b\uff1a<\/p>\n\n\n\n
# Gradient caculate \n# x_p: x_train\n# y_p: y_train\n# w_p: current w\n# b_p: current b\ndef gradient_caculate(x_p,y_p,w_p,b_p):\n gradient_w,gradient_b = (0.,0.)\n for i in range(m):\n gradient_w += x_p[i]*(function_f(x_p[i],w_p,b_p)-y_p[i])\n gradient_b += function_f(x_p[i],w_p,b_p)-y_p[i]\n return gradient_w,gradient_b<\/code><\/pre>\n\n\n\n\u68af\u5ea6\u4e0b\u964d\u8fed\u4ee3<\/h4>\n\n\n\n
\u8fd9\u91cc\u8bbe\u7f6e\u8fed\u4ee3\u6b21\u6570\u4e3a50000<\/code>\u6b21\uff0c\u5b66\u4e60\u7387\u8bbe\u7f6e\u4e3a0.01<\/code>\uff0c\u5f53\u8fed\u4ee3\u76ee\u6807\u51fd\u6570\u53d8\u5316\u503c\u5c0f\u4e8e0.000001<\/code>\u65f6\u4e5f\u505c\u6b62\u8fed\u4ee3\uff08\u8fd9\u5e76\u4e0d\u662f\u5fc5\u987b\u7684\uff09\u3002\u5177\u4f53\u7684\uff1a<\/p>\n\n\n\niteration_count = 50000\nlearning_rate = 0.01\ncost_reduce_threshold = 0.000001<\/code><\/pre>\n\n\n\n\u4e8e\u662f\u53c8\u5982\u4e0b\u68af\u5ea6\u4e0b\u964d\u8fed\u4ee3\u8fc7\u7a0b\u7684\u4ee3\u7801\uff1a<\/p>\n\n\n\n
cost_last = 0\nfor i in range(iteration_count):\n grad_w,grad_b = gradient_caculate(x_train,y_train,w,b)\n w = w - learning_rate*grad_w\n b = b - learning_rate*grad_b\n cost_current = cost_function(w,b,x_train,y_train)\n if i >= iteration_count\/2 and cost_last - cost_current<= cost_reduce_threshold:\n print(\"iteration: {:5d},cost_current:{:f},cost_last:{:f},cost reduce:{:f}\".format( i+1,cost_current,cost_last,cost_last-cost_current))\n break\n if (i+1)%(iteration_count\/10) == 0:\n print(\"iteration: {:5d},cost_current:{:f},cost_last:{:f},cost reduce:{:f}\".format( i+1,cost_current,cost_last,cost_last-cost_current))\n cost_last = cost_current<\/code><\/pre>\n\n\n\n\u9884\u6d4b<\/h4>\n\n\n\n
\u5b8c\u6210\u8bad\u7ec3\u540e\uff0c\u5219\u53ef\u4ee5\u5bf9\u8f93\u5165\u503c\u8fdb\u884c\u9884\u6d4b\u3002\u4ee3\u7801\u5982\u4e0b\uff1a<\/p>\n\n\n\n
print(\"after the training, parameter w = {:f} and b = {:f}\".format(w,b))\n\nfor i in range(m):\n y = function_f(x_train[i],w,b)\n p = 0\n if y>= 0.5: p = 1\n print(\"sample: x[{:d}]:{:d},y[{:d}]:{:d}; the prediction is {:d} with probability:{:4f}\".format(i,x_train[i],i,y_train[i],p,y))<\/code><\/pre>\n\n\n\n\u4e0a\u8ff0\u4ee3\u7801\u4f1a\u4ea7\u751f\u5982\u4e0b\u7684\u8f93\u51fa\uff1a<\/p>\n\n\n\n
after the training, parameter w = 5.056985 and b = -22.644516\nsample: x[0]:0,y[0]:0; the prediction is 0 with probability:0.000000\nsample: x[1]:1,y[1]:0; the prediction is 0 with probability:0.000000\nsample: x[2]:2,y[2]:0; the prediction is 0 with probability:0.000004\nsample: x[3]:3,y[3]:0; the prediction is 0 with probability:0.000568\nsample: x[4]:4,y[4]:0; the prediction is 0 with probability:0.081917\nsample: x[5]:5,y[5]:1; the prediction is 1 with probability:0.933417<\/code><\/pre>\n\n\n\n\u53ef\u4ee5\u770b\u5230\uff0c\u5728\u5b8c\u6210\u8bad\u7ec3\u540e\u7684\u8fd9\u4e2a\u6781\u7b80\u795e\u7ecf\u7f51\u7edc\u80fd\u591f\u8f83\u4e3a\u51c6\u786e\u7684\u9884\u6d4b\u6837\u672c\u4e2d\u7684\u6570\u636e\u3002<\/p>\n\n\n\n
\u5b8c\u6574\u7684\u4ee3\u7801<\/h4>\n\n\n\n