反向传播的重要性不必多说，手推也是必备基础，大厂面试要求用numpy实现一下BP也是经常的事。下面以一个简单的两层网络为例（简单到连bias都没有的那种），真·手推反向传播+numpy实现。
说明：
input：输入
net_h：隐层
S（x）：sigmoid简写
SD（x）：sigmoid导数的简写
out_h：隐层经过sigmoid激活的输出
net_o：输出层
out_o：经过经过sigmoid激活的输出层，也就是最后的输出
target（简写t1、t2）：Ground Truth
L：损失函数，一个简单的MSE
gw：L在w的偏导数
下面是代码：
1.对以上手推图片的实现：
import numpy as np def sigmoid(x): return 1 / (1 + np.exp(-x)) def sigmoid_derivationx(y): return y * (1 - y) def back(): input = np.array([0.9, 0.1], dtype='float') weight_1 = np.array([[0.15, 0.20], [0.25, 0.30]], dtype='float') weight_2 = np.array([[0.40, 0.45], [0.50, 0.55]], dtype='float') target_output = np.array([0.11, 0.89], dtype='float') learning_rate = 0.5 steps = 1000 while steps > 0: steps -= 1 # 正向传播 # layer-1 net_h1, net_h2 = np.dot(input, weight_1) # 第一层的计算结果 out_h1, out_h2 = [1 / (1 + np.exp(-net_h1)), 1 / (1 + np.exp(-net_h2))] # sigmoid激活后的输出 # layer-2 layer_2_input = np.array([out_h1, out_h2], dtype='float') net_o1, net_o2 = np.dot(layer_2_input, weight_2) # 第二层的计算结果 out_o1, out_o2 = [1 / (1 + np.exp(-net_o1)), 1 / (1 + np.exp(-net_o2))] # sigmoid激活后的输出 # 计算梯度,反向传播 total_loss = sum([pow(target_output-real_output, 2)/2 for target_output, real_output in zip(target_output, [out_o1, out_o2])]) print('网络的输出：step:%d loss=%f out_o1=%f out_o2=%f ' % (steps, total_loss, out_o1, out_o2)) # gradient for layer-2 gradient_w5 = -1 * (target_output[0] - out_o1) * out_o1 * (1 - out_o1) * out_h1 gradient_w6 = -1 * (target_output[0] - out_o1) * out_o1 * (1 - out_o1) * out_h2 gradient_w7 = -1 * (target_output[1] - out_o2) * out_o2 * (1 - out_o2) * out_h1 gradient_w8 = -1 * (target_output[1] - out_o2) * out_o2 * (1 - out_o2) * out_h2 # gradient for layer-1 gradient_w1 = -1 * (target_output[0] - out_o1) * out_o1 * (1 - out_o1) * weight_2[0][0] * out_h1 * \ (1 - out_h1)*input[0] - (target_output[1] - out_o2) * out_o2 * (1 - out_o2) * weight_2[1][0] * \ out_h1 * (1 - out_h1)*input[0] gradient_w2 = -1 * (target_output[0] - out_o1) * out_o1 * (1 - out_o1) * weight_2[0][0] * out_h1 * \ (1 - out_h1) * input[1] - (target_output[1] - out_o2) * out_o2 * (1 - out_o2) * weight_2[1][0] * \ out_h1 * (1 - out_h1)*input[1] gradient_w3 = -1 * (target_output[0] - out_o1) * out_o1 * (1 - out_o1) * weight_2[0][1] * out_h2 * \ (1 - out_h2) * input[0] - (target_output[1] - out_o2) * out_o2 * (1 - out_o2) * weight_2[1][1] * \ out_h2 * (1 - out_h2)*input[0] gradient_w4 = -1 * (target_output[0] - out_o1) * out_o1 * (1 - out_o1) * weight_2[0][1] * out_h2 * \ (1 - out_h2) * input[1] - (target_output[1] - out_o2) * out_o2 * (1 - out_o2) * weight_2[1][1] * \ out_h2 * (1 - out_h2)*input[1] # update weight # layer -1 weight_1[0][0] = weight_1[0][0] - learning_rate * gradient_w1 weight_1[1][0] = weight_1[1][0] - learning_rate * gradient_w2 weight_1[0][1] = weight_1[0][1] - learning_rate * gradient_w3 weight_1[1][1] = weight_1[1][1] - learning_rate * gradient_w4 back()
2.用矩阵相乘实现：
import numpy as np def sigmoid(x): return 1 / (1 + np.exp(-x)) def sigmoid_derivationx(y): return y * (1 - y) def back_matrix(): input = np.array([0.9, 0.1], dtype='float') weight_1 = np.array([[0.5, 0.5], [0.5, 0.5]], dtype='float') weight_2 = np.array([[0.5, 0.5], [0.5, 0.5]], dtype='float') target_output = np.array([0.11, 0.89], dtype='float') learning_rate = 0.5 steps = 1000 while steps > 0: steps -= 1 # 正向传播 # layer-1 net_h = np.dot(input, weight_1) # 第一层的计算结果 out_h = sigmoid(net_h) # sigmoid激活后的输出 # layer-2 layer_2_input = np.array(out_h, dtype='float') net_o = np.dot(layer_2_input, weight_2) # 第二层的计算结果 out_o = sigmoid(net_o) # sigmoid激活后的输出 # 计算梯度,反向传播 total_loss = sum([pow(target_output-real_output, 2)/2 for target_output, real_output in zip(target_output, out_o)]) print('网络的输出：step:%d loss=%f out_o1=%f out_o2=%f ' % (steps, total_loss, out_o[0], out_o[0])) gradient_weight_2 = np.zeros_like(weight_2) for i in range(gradient_weight_2.shape[1]): gradient_weight_2[:, i] = (out_o[i] - target_output[i]) * sigmoid_derivationx(out_o[i]) * out_h gradient_weight_1 = np.zeros_like(weight_1) for i in range(gradient_weight_1.shape[1]): dot1 = -1 * (target_output - out_o) * out_o * sigmoid_derivationx(out_h) dot2 = np.reshape(weight_2[:, i], (-1, 1)) gradient_weight_1[:, i] = np.dot([dot1], dot2) * input # update weight # layer -1 weight_1 = weight_1 - learning_rate * gradient_weight_1 # layer-2 weight_2 = weight_2 - learning_rate * gradient_weight_2 if steps == 1: print('网络的输出：loss=%f out_o1=%f out_o2=%f ' % (total_loss, out_o[0], out_o[1])) break back_matrix()
