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个人觉得BP反向传播是深度学习的一个基础，所以很有必要把反向传播算法好好学一下

得益于这篇文章，给出一个例子来说明反向传播

不过是英文的，如果你感觉不好阅读的话，优秀的国人已经把它翻译出来了。

然后我使用了那个博客的图片。这次的目的主要是对那个博客的一个补充。但是首先我觉得先用面向过程的思想来实现一遍感觉会好一点。

随便把文中省略的公式给大家给写出来。大家可以先看那篇博文

1 import numpy as np 2  3 # "pd" 偏导 4 def sigmoid(x): 5     return 1 / (1 + np.exp(-x)) 6  7 def sigmoidDerivationx(y): 8     return y * (1 - y) 9 10 11 if __name__ == "__main__":12     #初始化13     bias = [0.35, 0.60]14     weight = [0.15, 0.2, 0.25, 0.3, 0.4, 0.45, 0.5, 0.55]15     output_layer_weights = [0.4, 0.45, 0.5, 0.55]16     i1 = 0.0517     i2 = 0.1018     target1 = 0.0119     target2 = 0.9920     alpha = 0.5 #学习速率21     numIter = 10000 #迭代次数22     for i in range(numIter):23         #正向传播24         neth1 = i1*weight[1-1] + i2*weight[2-1] + bias[0]25         neth2 = i1*weight[3-1] + i2*weight[4-1] + bias[0]26         outh1 = sigmoid(neth1)27         outh2 = sigmoid(neth2)28         neto1 = outh1*weight[5-1] + outh2*weight[6-1] + bias[1]29         neto2 = outh2*weight[7-1] + outh2*weight[8-1] + bias[1]30         outo1 = sigmoid(neto1)31         outo2 = sigmoid(neto2)32         print(str(i) + ", target1 : " + str(target1-outo1) + ", target2 : " + str(target2-outo2))33         if i == numIter-1:34             print("lastst result : " + str(outo1) + " " + str(outo2))35         #反向传播36         #计算w5-w8(输出层权重)的误差37         pdEOuto1 = - (target1 - outo1)38         pdOuto1Neto1 = sigmoidDerivationx(outo1)39         pdNeto1W5 = outh140         pdEW5 = pdEOuto1 * pdOuto1Neto1 * pdNeto1W541         pdNeto1W6 = outh242         pdEW6 = pdEOuto1 * pdOuto1Neto1 * pdNeto1W643         pdEOuto2 = - (target2 - outo2)44         pdOuto2Neto2 = sigmoidDerivationx(outo2)45         pdNeto1W7 = outh146         pdEW7 = pdEOuto2 * pdOuto2Neto2 * pdNeto1W747         pdNeto1W8 = outh248         pdEW8 = pdEOuto2 * pdOuto2Neto2 * pdNeto1W849 50         # 计算w1-w4(输出层权重)的误差51         pdEOuto1 = - (target1 - outo1) #之前算过52         pdEOuto2 = - (target2 - outo2)  #之前算过53         pdOuto1Neto1 = sigmoidDerivationx(outo1)    #之前算过54         pdOuto2Neto2 = sigmoidDerivationx(outo2)    #之前算过55         pdNeto1Outh1 = weight[5-1]56         pdNeto2Outh2 = weight[7-1]57 58         pdEOuth1 = pdEOuto1 * pdOuto1Neto1 * pdNeto1Outh1 + pdEOuto2 * pdOuto2Neto2 * pdNeto1Outh159         pdOuth1Neth1 = sigmoidDerivationx(outh1)60         pdNeth1W1 = i161         pdNeth1W2 = i262         pdEW1 = pdEOuth1 * pdOuth1Neth1 * pdNeth1W163         pdEW2 = pdEOuth1 * pdOuth1Neth1 * pdNeth1W264         pdNeto1Outh2 = weight[6-1]65         pdNeto2Outh2 = weight[8-1]66         pdOuth2Neth2 = sigmoidDerivationx(outh2)67         pdNeth2W3 = i168         pdNeth2W4 = i269         pdEOuth2 = pdEOuto1 * pdOuto1Neto1 * pdNeto1Outh2 + pdEOuto2 * pdOuto2Neto2 * pdNeto2Outh270         pdEW3 = pdEOuth2 * pdOuth2Neth2 * pdNeth2W371         pdEW4 = pdEOuth2 * pdOuth2Neth2 * pdNeth2W472         #权重更新73         weight[1-1] = weight[1-1] - alpha * pdEW174         weight[2-1] = weight[2-1] - alpha * pdEW275         weight[3-1] = weight[3-1] - alpha * pdEW376         weight[4-1] = weight[4-1] - alpha * pdEW477         weight[5-1] = weight[5-1] - alpha * pdEW578         weight[6-1] = weight[6-1] - alpha * pdEW679         weight[7-1] = weight[7-1] - alpha * pdEW780         weight[8-1] = weight[8-1] - alpha * pdEW881         # print(weight[1-1])82         # print(weight[2-1])83         # print(weight[3-1])84         # print(weight[4-1])85         # print(weight[5-1])86         # print(weight[6-1])87         # print(weight[7-1])88         # print(weight[8-1])

不知道你是否对此感到熟悉一点了呢？反正我按照公式实现一遍之后深有体会，然后用向量的又写了一次代码。

接下来我们要用向量来存储这些权重，输出结果等，因为如果我们不这样做，你看上面的例子就知道我们需要写很多w1,w2等，这要是参数一多就很可怕。

这些格式我是参考吴恩达的格式，相关课程资料->。

1 import numpy as np 2  3 def sigmoid(x): 4     return 1 / (1 + np.exp(-x)) 5 def sigmoidDerivationx(y): 6     return y * (1 - y) 7  8  9 if __name__ == '__main__':10     # 初始化一些参数11     alpha = 0.512     numIter = 1000000 #迭代次数13     w1 = [[0.15, 0.20], [0.25, 0.30]]  # Weight of input layer14     w2 = [[0.40, 0.45], [0.50, 0.55]]15     # print(np.array(w2).T)16     b1 = 0.3517     b2 = 0.6018     x = [0.05, 0.10]19     y = [0.01, 0.99]20     # 前向传播21     z1 = np.dot(w1, x) + b1     # dot函数是常规的矩阵相乘22     a1 = sigmoid(z1)23 24     z2 = np.dot(w2, a1) + b225     a2 = sigmoid(z2)26     for n in range(numIter):27         # 反向传播 使用代价函数为C=1 / (2n) * sum[(y-a2)^2]28         # 分为两次29         # 一次是最后一层对前面一层的错误30 31         delta2 = np.multiply(-(y-a2), np.multiply(a2, 1-a2))32         # for i in range(len(w2)):33         #     print(w2[i] - alpha * delta2[i] * a1)34         #计算非最后一层的错误35         # print(delta2)36         delta1 = np.multiply(np.dot(np.array(w2).T, delta2), np.multiply(a1, 1-a1))37         # print(delta1)38         # for i in range(len(w1)):39             # print(w1[i] - alpha * delta1[i] * np.array(x))40         #更新权重41         for i in range(len(w2)):42             w2[i] = w2[i] - alpha * delta2[i] * a143         for i in range(len(w1)):44             w1[i] = w1[i] - alpha * delta1[i] * np.array(x)45         #继续前向传播，算出误差值46         z1 = np.dot(w1, x) + b147         a1 = sigmoid(z1)48         z2 = np.dot(w2, a1) + b249         a2 = sigmoid(z2)50         print(str(n) + " result:" + str(a2[0]) + ", result:" +str(a2[1]))51         # print(str(n) + "  error1:" + str(y[0] - a2[0]) + ", error2:" +str(y[1] - a2[1]))

 

可以看到，用向量来表示的话代码就简短了非常多。但是用了向量化等的方法，如果不太熟，去看吴恩达深度学习的第一部分，再返过来看就能懂了。