【深度学习】BP反向传播算法Python简单实现

妖狐艹你老母 2022-06-05 11:09 195阅读 0赞

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[一步一步弄懂反向传播的例子][Link 1]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。  
  [一步一步弄懂反向传播的例子（中文翻译）][Link 2]  
![神经网络][SouthEast]  
  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。

∂Etotal∂w5=∂Etotal∂outo1×∂outo1∂neto1×∂neto1∂w5∂Etotal∂w6=∂Etotal∂outo1×∂outo1∂neto1×∂neto1∂w6∂Etotal∂w7=∂Etotal∂outo2×∂outo2∂neto2×∂neto2∂w7∂Etotal∂w8=∂Etotal∂outo2×∂outo2∂neto2×∂neto2∂w8∂Etotal∂w1=∂Etotal∂outh1×∂outh1∂neth1×∂neth1∂w1∂Etotal∂w2=∂Etotal∂outh1×∂outh1∂neth1×∂neth1∂w2∂Etotal∂w3=∂Etotal∂outh2×∂outh2∂neth2×∂neth2∂w3∂Etotal∂w4=∂Etotal∂outh2×∂outh2∂neth2×∂neth2∂w4(1)(2)(3)(4)(5)(6)(7)(8)  (1) ∂ E t o t a l ∂ w 5 = ∂ E t o t a l ∂ o u t o 1 × ∂ o u t o 1 ∂ n e t o 1 × ∂ n e t o 1 ∂ w 5 (2) ∂ E t o t a l ∂ w 6 = ∂ E t o t a l ∂ o u t o 1 × ∂ o u t o 1 ∂ n e t o 1 × ∂ n e t o 1 ∂ w 6 (3) ∂ E t o t a l ∂ w 7 = ∂ E t o t a l ∂ o u t o 2 × ∂ o u t o 2 ∂ n e t o 2 × ∂ n e t o 2 ∂ w 7 (4) ∂ E t o t a l ∂ w 8 = ∂ E t o t a l ∂ o u t o 2 × ∂ o u t o 2 ∂ n e t o 2 × ∂ n e t o 2 ∂ w 8 (5) ∂ E t o t a l ∂ w 1 = ∂ E t o t a l ∂ o u t h 1 × ∂ o u t h 1 ∂ n e t h 1 × ∂ n e t h 1 ∂ w 1 (6) ∂ E t o t a l ∂ w 2 = ∂ E t o t a l ∂ o u t h 1 × ∂ o u t h 1 ∂ n e t h 1 × ∂ n e t h 1 ∂ w 2 (7) ∂ E t o t a l ∂ w 3 = ∂ E t o t a l ∂ o u t h 2 × ∂ o u t h 2 ∂ n e t h 2 × ∂ n e t h 2 ∂ w 3 (8) ∂ E t o t a l ∂ w 4 = ∂ E t o t a l ∂ o u t h 2 × ∂ o u t h 2 ∂ n e t h 2 × ∂ n e t h 2 ∂ w 4

import numpy as np
    
    # "pd" 偏导
    def sigmoid(x):
        return 1 / (1 + np.exp(-x))
    def sigmoidDerivationx(y):
        return y * (1 - y)
    
    if __name__ == "__main__":
        #初始化
        bias = [0.35, 0.60]
        weight = [0.15, 0.2, 0.25, 0.3, 0.4, 0.45, 0.5, 0.55]
        output_layer_weights = [0.4, 0.45, 0.5, 0.55]
        i1 = 0.05
        i2 = 0.10
        target1 = 0.01
        target2 = 0.99
        alpha = 0.5 #学习速率
        numIter = 90000 #迭代次数
        for i in range(numIter):
            #正向传播
            neth1 = i1*weight[1-1] + i2*weight[2-1] + bias[0]
            neth2 = i1*weight[3-1] + i2*weight[4-1] + bias[0]
            outh1 = sigmoid(neth1)
            outh2 = sigmoid(neth2)
            neto1 = outh1*weight[5-1] + outh2*weight[6-1] + bias[1]
            neto2 = outh2*weight[7-1] + outh2*weight[8-1] + bias[1]
            outo1 = sigmoid(neto1)
            outo2 = sigmoid(neto2)
            print(str(i) + ", target1 : " + str(target1-outo1) + ", target2 : " + str(target2-outo2))
            if i == numIter-1:
                print("lastst result : " + str(outo1) + " " + str(outo2))
            #反向传播
            #计算w5-w8(输出层权重)的误差
            pdEOuto1 = - (target1 - outo1)
            pdOuto1Neto1 = sigmoidDerivationx(outo1)
            pdNeto1W5 = outh1
            pdEW5 = pdEOuto1 * pdOuto1Neto1 * pdNeto1W5
            pdNeto1W6 = outh2
            pdEW6 = pdEOuto1 * pdOuto1Neto1 * pdNeto1W6
            pdEOuto2 = - (target2 - outo2)
            pdOuto2Neto2 = sigmoidDerivationx(outo2)
            pdNeto1W7 = outh1
            pdEW7 = pdEOuto2 * pdOuto2Neto2 * pdNeto1W7
            pdNeto1W8 = outh2
            pdEW8 = pdEOuto2 * pdOuto2Neto2 * pdNeto1W8
            # 计算w1-w4(输出层权重)的误差
            pdEOuto1 = - (target1 - outo1) #之前算过
            pdEOuto2 = - (target2 - outo2)  #之前算过
            pdOuto1Neto1 = sigmoidDerivationx(outo1)    #之前算过
            pdOuto2Neto2 = sigmoidDerivationx(outo2)    #之前算过
            pdNeto1Outh1 = weight[5-1]
            pdNeto1Outh2 = weight[7-1]
            pdENeth1 = pdEOuto1 * pdOuto1Neto1 * pdNeto1Outh1 + pdEOuto2 * pdOuto2Neto2 * pdNeto1Outh2
            pdOuth1Neth1 = sigmoidDerivationx(outh1)
            pdNeth1W1 = i1
            pdNeth1W2 = i2
            pdEW1 = pdENeth1 * pdOuth1Neth1 * pdNeth1W1
            pdEW2 = pdENeth1 * pdOuth1Neth1 * pdNeth1W2
            pdNeto1Outh2 = weight[6-1]
            pdNeto2Outh2 = weight[8-1]
            pdOuth2Neth2 = sigmoidDerivationx(outh2)
            pdNeth1W3 = i1
            pdNeth1W4 = i2
            pdENeth2 = pdEOuto1 * pdOuto1Neto1 * pdNeto1Outh2 + pdEOuto2 * pdOuto2Neto2 * pdNeto2Outh2
            pdEW3 = pdENeth2 * pdOuth2Neth2 * pdNeth1W3
            pdEW4 = pdENeth2 * pdOuth2Neth2 * pdNeth1W4
            #权重更新
            weight[1-1] = weight[1-1] - alpha * pdEW1
            weight[2-1] = weight[2-1] - alpha * pdEW2
            weight[3-1] = weight[3-1] - alpha * pdEW3
            weight[4-1] = weight[4-1] - alpha * pdEW4
            weight[5-1] = weight[5-1] - alpha * pdEW5
            weight[6-1] = weight[6-1] - alpha * pdEW6
            weight[7-1] = weight[7-1] - alpha * pdEW7
            weight[8-1] = weight[8-1] - alpha * pdEW8
            # print(weight[1-1])
            # print(weight[2-1])
            # print(weight[3-1])
            # print(weight[4-1])
            # print(weight[5-1])
            # print(weight[6-1])
            # print(weight[7-1])
            # 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->[吴恩达深度学习视频][Link 3]。  
![神经网络][SouthEast 1]  
我将原文的图片的变量名改成如上  
然后正向传播的公式如下：

z\[1\]1=w\[1\]1T⋅x\+b1,a\[1\]1=σ(z\[1\]1)z\[1\]2=w\[1\]2T⋅x\+b1,a\[1\]2=σ(z\[1\]2)z\[2\]1=w\[2\]1T⋅a1\+b2,a\[2\]1=σ(z\[2\]1)z\[2\]2=w\[2\]2T⋅a1\+b2,a\[2\]2=σ(z\[2\]2)(9)(10)(11)(12)(13)  (9) z 1 \[ 1 \] = w 1 \[ 1 \] T ⋅ x + b 1 , a 1 \[ 1 \] = σ ( z 1 \[ 1 \] ) (10) z 2 \[ 1 \] = w 2 \[ 1 \] T ⋅ x + b 1 , a 2 \[ 1 \] = σ ( z 2 \[ 1 \] ) (11) z 1 \[ 2 \] = w 1 \[ 2 \] T ⋅ a 1 + b 2 , a 1 \[ 2 \] = σ ( z 1 \[ 2 \] ) (12) z 2 \[ 2 \] = w 2 \[ 2 \] T ⋅ a 1 + b 2 , a 2 \[ 2 \] = σ ( z 2 \[ 2 \] ) (13)

其中

w\[1\]1T=(w1,w2)w\[1\]2T=(w3,w4)w\[2\]1T=(w5,w6)w\[2\]2T=(w7,w8)(14)(15)(16)(17)  (14) w 1 \[ 1 \] T = ( w 1 , w 2 ) (15) w 2 \[ 1 \] T = ( w 3 , w 4 ) (16) w 1 \[ 2 \] T = ( w 5 , w 6 ) (17) w 2 \[ 2 \] T = ( w 7 , w 8 )

然后反向传播的公式如下：

∂E∂w\[2\]1=∂E∂a\[2\]1⋅∂a\[2\]1∂z\[2\]1⋅∂z\[2\]1∂w\[2\]1∂E∂w\[2\]1=∂E∂a\[2\]2⋅∂a\[2\]2∂z\[2\]2⋅∂z\[2\]2∂w\[2\]2∂E∂w\[1\]1=∂E∂a\[1\]1⋅∂a\[1\]1∂z\[1\]1⋅∂z\[1\]1∂w\[1\]1∂E∂w\[1\]2=∂E∂a\[1\]2⋅∂a\[1\]2∂z\[1\]2⋅∂z\[1\]2∂w\[1\]2(18)(19)(20)(21)  (18) ∂ E ∂ w 1 \[ 2 \] = ∂ E ∂ a 1 \[ 2 \] ⋅ ∂ a 1 \[ 2 \] ∂ z 1 \[ 2 \] ⋅ ∂ z 1 \[ 2 \] ∂ w 1 \[ 2 \] (19) ∂ E ∂ w 1 \[ 2 \] = ∂ E ∂ a 2 \[ 2 \] ⋅ ∂ a 2 \[ 2 \] ∂ z 2 \[ 2 \] ⋅ ∂ z 2 \[ 2 \] ∂ w 2 \[ 2 \] (20) ∂ E ∂ w 1 \[ 1 \] = ∂ E ∂ a 1 \[ 1 \] ⋅ ∂ a 1 \[ 1 \] ∂ z 1 \[ 1 \] ⋅ ∂ z 1 \[ 1 \] ∂ w 1 \[ 1 \] (21) ∂ E ∂ w 2 \[ 1 \] = ∂ E ∂ a 2 \[ 1 \] ⋅ ∂ a 2 \[ 1 \] ∂ z 2 \[ 1 \] ⋅ ∂ z 2 \[ 1 \] ∂ w 2 \[ 1 \]

具体地

∂E∂a\[2\]1=−(y1−a\[2\]1)∂a\[2\]1∂z\[2\]1=a\[2\]1(1−a\[2\]1)∂z\[2\]1∂w\[2\]1=a\[2\]1∂E∂a\[2\]2=−(y2−a\[2\]2)∂a\[2\]1∂z\[2\]1=a\[2\]2(1−a\[2\]2)∂z\[2\]1∂w\[2\]2=a\[2\]2∂E∂a\[1\]1=w\[2\]1Tδ2∂a\[1\]1∂z\[1\]1=a\[1\]1⋅(1−a\[1\]1)∂z\[1\]1∂w\[1\]1=a\[1\]1∂E∂a\[1\]1=w\[2\]2Tδ2∂a\[1\]2∂z\[1\]1=a\[1\]2⋅(1−a\[1\]2)∂z\[1\]1∂w\[1\]2=a\[1\]2(22)(23)(24)(25)(26)(27)(28)(29)(30)(31)(32)(33)  (22) ∂ E ∂ a 1 \[ 2 \] = − ( y 1 − a 1 \[ 2 \] ) (23) ∂ a 1 \[ 2 \] ∂ z 1 \[ 2 \] = a 1 \[ 2 \] ( 1 − a 1 \[ 2 \] ) (24) ∂ z 1 \[ 2 \] ∂ w 1 \[ 2 \] = a 1 \[ 2 \] (25) ∂ E ∂ a 2 \[ 2 \] = − ( y 2 − a 2 \[ 2 \] ) (26) ∂ a 1 \[ 2 \] ∂ z 1 \[ 2 \] = a 2 \[ 2 \] ( 1 − a 2 \[ 2 \] ) (27) ∂ z 1 \[ 2 \] ∂ w 2 \[ 2 \] = a 2 \[ 2 \] (28) ∂ E ∂ a 1 \[ 1 \] = w 1 \[ 2 \] T δ 2 (29) ∂ a 1 \[ 1 \] ∂ z 1 \[ 1 \] = a 1 \[ 1 \] ⋅ ( 1 − a 1 \[ 1 \] ) (30) ∂ z 1 \[ 1 \] ∂ w 1 \[ 1 \] = a 1 \[ 1 \] (31) ∂ E ∂ a 1 \[ 1 \] = w 2 \[ 2 \] T δ 2 (32) ∂ a 2 \[ 1 \] ∂ z 1 \[ 1 \] = a 2 \[ 1 \] ⋅ ( 1 − a 2 \[ 1 \] ) (33) ∂ z 1 \[ 1 \] ∂ w 2 \[ 1 \] = a 2 \[ 1 \]

其中

δ2=⎛⎝⎜⎜⎜⎜⎜⎜∂E∂a\[2\]1⋅∂a\[2\]1∂z\[2\]1∂E∂a\[2\]2⋅∂a\[2\]2∂z\[2\]2(34)(35)⎞⎠⎟⎟⎟⎟⎟⎟=(∂E∂a\[2\]⋅∂a\[2\]∂z\[2\])  δ 2 = ( (34) ∂ E ∂ a 1 \[ 2 \] ⋅ ∂ a 1 \[ 2 \] ∂ z 1 \[ 2 \] (35) ∂ E ∂ a 2 \[ 2 \] ⋅ ∂ a 2 \[ 2 \] ∂ z 2 \[ 2 \] ) = ( ∂ E ∂ a \[ 2 \] ⋅ ∂ a \[ 2 \] ∂ z \[ 2 \] )

为啥这样写呢，一开始我也没明白，后来看到  ∂E∂a\[2\]1⋅∂a\[2\]1∂z\[2\]1  ∂ E ∂ a 1 \[ 2 \] ⋅ ∂ a 1 \[ 2 \] ∂ z 1 \[ 2 \]  有好几次重复，且也便于梯度公式的书写。

import numpy as np
    
    def sigmoid(x):
        return 1 / (1 + np.exp(-x))
    def sigmoidDerivationx(y):
        return y * (1 - y)
    
    if __name__ == '__main__':
        #初始化一些参数
        alpha = 0.5
        w1 = [[0.15, 0.20], [0.25, 0.30]] #Weight of input layer
        w2 = [[0.40, 0.45], [0.50, 0.55]]
        b1 = 0.35
        b2 = 0.60
        x = [0.05, 0.10]
        y = [0.01, 0.99]
        #前向传播
        z1 = np.dot(w1, x) + b1
        a1 = sigmoid(z1)
        z2 = np.dot(w2, a1) + b2
        a2 = sigmoid(z2)
        for n in range(10000):
            #反向传播 使用代价函数为C=1 / (2n) * sum[(y-a2)^2]
            #分为两次
            # 一次是最后一层对前面一层的错误
            delta2 = np.multiply(-(y-a2), np.multiply(a2, 1-a2))
            # for i in range(len(w2)):
            #     print(w2[i] - alpha * delta2[i] * a1)
            #计算非最后一层的错误
            # print(delta2)
            delta1 = np.multiply(np.dot(w1, delta2), np.multiply(a1, 1-a1))
            # print(delta1)
            # for i in range(len(w1)):
                # print(w1[i] - alpha * delta1[i] * np.array(x))
            #更新权重
            for i in range(len(w2)):
                w2[i] = w2[i] - alpha * delta2[i] * a1
            for i in range(len(w1)):
                w1[i] - alpha * delta1[i] * np.array(x)
            #继续前向传播，算出误差值
            z1 = np.dot(w1, x) + b1
            a1 = sigmoid(z1)
            z2 = np.dot(w2, a1) + b2
            a2 = sigmoid(z2)
            print(str(n) + " result:" + str(a2[0]) + ", result:" +str(a2[1]))
            # 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。  
下面，来看一个例子。用神经网络实现XOR（01=1，10=1，00=0，11=0）。我们都知道感知机是没法实现异或的，原因是线性不可分。  
接下里的这个例子，我是用2个输入结点，3个隐层结点，1个输出结点来实现的。  
![异或][SouthEast 2]  
让我们以一个输入为例。  
前向传播：

⎛⎝⎜⎜⎜w\[1\]11w\[1\]12w\[1\]21w\[1\]22w\[1\]31w\[1\]32⎞⎠⎟⎟⎟⋅(x1x2)=⎛⎝⎜⎜⎜z\[1\]1z\[1\]2z\[1\]3⎞⎠⎟⎟⎟  ( w 11 \[ 1 \] w 12 \[ 1 \] w 21 \[ 1 \] w 22 \[ 1 \] w 31 \[ 1 \] w 32 \[ 1 \] ) ⋅ ( x 1 x 2 ) = ( z 1 \[ 1 \] z 2 \[ 1 \] z 3 \[ 1 \] )

⎛⎝⎜⎜⎜a\[1\]1a\[1\]2a\[1\]3⎞⎠⎟⎟⎟=⎛⎝⎜⎜⎜σ(z\[1\]1)σ(z\[1\]2)σ(z\[1\]3)⎞⎠⎟⎟⎟  ( a 1 \[ 1 \] a 2 \[ 1 \] a 3 \[ 1 \] ) = ( σ ( z 1 \[ 1 \] ) σ ( z 2 \[ 1 \] ) σ ( z 3 \[ 1 \] ) )

(w\[2\]11w\[2\]12w\[2\]13)⎛⎝⎜⎜⎜a\[1\]1a\[1\]2a\[1\]3⎞⎠⎟⎟⎟=(z\[2\]1)  ( w 11 \[ 2 \] w 12 \[ 2 \] w 13 \[ 2 \] ) ( a 1 \[ 1 \] a 2 \[ 1 \] a 3 \[ 1 \] ) = ( z 1 \[ 2 \] )

(a\[2\]1)=(σ(z\[2\]1))  ( a 1 \[ 2 \] ) = ( σ ( z 1 \[ 2 \] ) )

反向传播：  
主要是有2个公式比较重要

δL=∇aC⊙σ′(aL)=−(y−aL)⊙(aL(1−aL))  δ L = ∇ a C ⊙ σ ′ ( a L ) = − ( y − a L ) ⊙ ( a L ( 1 − a L ) )

原理同上

δl=((wl\+1)T)⊙σ′(al)  δ l = ( ( w l + 1 ) T ) ⊙ σ ′ ( a l )

wl=wl−ηδl(al−1)T  w l = w l − η δ l ( a l − 1 ) T

这次省略了偏导，代码如下

import numpy as np
    # sigmoid function
    def sigmoid(x):
        return 1 / (1 + np.exp(-x))
    def sigmoidDerivationx(y):
        return y * (1 - y)
    
    if __name__ == '__main__':
        alpha = 1
        input_dim = 2
        hidden_dim = 3
        output_dim = 1
        synapse_0 = 2 * np.random.random((hidden_dim, input_dim)) - 1  #(2, 3)
        # synapse_0 = np.ones((hidden_dim, input_dim)) * 0.5
        synapse_1 = 2 * np.random.random((output_dim, hidden_dim)) - 1  #(2, 2)
        # synapse_1 = np.ones((output_dim, hidden_dim)) * 0.5
        x = np.array([[0, 1], [1, 0], [0, 0], [1, 1]]).T #(2, 4)
        # x = np.array([[0, 1]]).T #(3, 1)
        y = np.array([[1], [1], [0], [0]]).T    #(1, 4)
        # y = np.array([[1]]).T    #(2, 1)
        for i in range(2000000):
            z1 = np.dot(synapse_0, x)   #(3, 4)
            a1 = sigmoid(z1)    #(3, 4)
            z2 = np.dot(synapse_1, a1)  #(1, 4)
            a2 = sigmoid(z2)    #(1, 4)
            error = -(y - a2) #(1, 4)
            delta2 = np.multiply(-(y - a2) / x.shape[1], sigmoidDerivationx(a2))  #(1, 4)
            delta1 = np.multiply(np.dot(synapse_1.T, delta2), sigmoidDerivationx(a1))  #(3, 4)
            synapse_1 = synapse_1 - alpha * np.dot(delta2, a1.T) #(1, 3)
            synapse_0 = synapse_0 - alpha * np.dot(delta1, x.T) #(3, 2)
            print(str(i) + ":", end=' ')
            print(a2)

[Link 1]: https://mattmazur.com/2015/03/17/a-step-by-step-backpropagation-example/
[Link 2]: https://www.cnblogs.com/charlotte77/p/5629865.html
[SouthEast]: /images/20220605/7159802a92e34f808b4abb6f18c729e1.png
[Link 3]: http://mooc.study.163.com/smartSpec/detail/1001319001.htm
[SouthEast 1]: /images/20220605/ca8e512b3d27410a8d0da3ade9729304.png
[SouthEast 2]: /images/20220605/8f23a636b4bd410d8b0a09335e9499ad.png