backpropagation.py
# https://en.wikipedia.org/wiki/Backpropagation import numpy as np # https://en.wikipedia.org/wiki/Sigmoid_function def sigmoid(x): return 1.0 / (1 + np.exp(-x)) assert(sigmoid(0) == 0.5) def sigmoid_dx(x): return x * (1.0 - x) assert(sigmoid_dx(0.5) == 0.25) # data x = np.array([[0,0,1], [0,1,1], [1,0,1], [1,1,1]]) y = np.array([[0], [1], [1], [0]]) w_1 = np.random.rand(x.shape[1],4) w_2 = np.random.rand(4,1) output = np.zeros(y.shape) # feedforward def feedforward(x, w_1, w_2, output): layer_1 = sigmoid(np.dot(x, w_1)) output = sigmoid(np.dot(layer_1, w_2)) return layer_1, w_1, w_2, output # backpropagation def backpropagation(layer_1, w_1, w_2, x, y, output): w_2_dx = np.dot(np.transpose(layer_1), (2 * (y - output) * sigmoid_dx(output))) w_1_dx = np.dot(np.transpose(x), (np.dot(2 * (y - output) * sigmoid_dx(output), np.transpose(w_2)) * sigmoid_dx(layer_1))) # update weights w_1 += w_1_dx w_2 += w_2_dx return layer_1, w_1, w_2, x, y, output # run for i in range(1500): layer_1, w_1, w_2, output = feedforward(x, w_1, w_2, output) layer_1, w_1, w_2, x, y, output = backpropagation(layer_1, w_1, w_2, x, y, output) print(output) # [[0.01021434] # [0.97674973] # [0.97726171] # [0.02572076]]