反向传播的目的是计算成本函数C对网络中任意w或b的偏导数。一旦我们有了这些偏导数，我们将通过一些常数 α的乘积和该数量相对于成本函数的偏导数来更新网络中的权重和偏差。这是流行的梯度下降算法。而偏导数给出了最大上升的方向。因此，关于反向传播算法，我们继续查看下文。

我们向相反的方向迈出了一小步——最大下降的方向，也就是将我们带到成本函数的局部最小值的方向。

图示演示:

反向传播算法中Sigmoid函数代码演示：

# 实现 sigmoid 函数return 1 / (1 + np.exp(-x))def sigmoid_derivative(x):# sigmoid 导数的计算return sigmoid(x)*(1-sigmoid(x))

反向传播算法中ReLU 函数导数函数代码演示：

def relu_derivative(x): # ReLU 函数的导数d = np.array(x, copy=True) # 用于保存梯度的张量d[x < 0] = 0 # 元素为负的导数为 0d[x >= 0] = 1 # 元素为正的导数为 1return d

实例扩展：

BP反向传播算法Python简单实现

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 = 10000 #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] pdNeto2Outh2 = weight[7-1] pdEOuth1 = pdEOuto1 * pdOuto1Neto1 * pdNeto1Outh1 + pdEOuto2 * pdOuto2Neto2 * pdNeto1Outh1 pdOuth1Neth1 = sigmoidDerivationx(outh1) pdNeth1W1 = i1 pdNeth1W2 = i2 pdEW1 = pdEOuth1 * pdOuth1Neth1 * pdNeth1W1 pdEW2 = pdEOuth1 * pdOuth1Neth1 * pdNeth1W2 pdNeto1Outh2 = weight[6-1] pdNeto2Outh2 = weight[8-1] pdOuth2Neth2 = sigmoidDerivationx(outh2) pdNeth2W3 = i1 pdNeth2W4 = i2 pdEOuth2 = pdEOuto1 * pdOuto1Neto1 * pdNeto1Outh2 + pdEOuto2 * pdOuto2Neto2 * pdNeto2Outh2 pdEW3 = pdEOuth2 * pdOuth2Neth2 * pdNeth2W3 pdEW4 = pdEOuth2 * pdOuth2Neth2 * pdNeth2W4 #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])

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