I have adapted an example neural net written in Python to illustrate how the back-propagation algorithm works on a small toy example.

My modifications include printing, a learning rate and using the leaky ReLU activation function instead of sigmoid.

import numpy as np # seed random numbers to make calculation # deterministic (just a good practice) np.random.seed(1) # make printed output easier to read # fewer decimals and no scientific notation np.set_printoptions(precision=3, suppress=True) # learning rate lr = 1e-2 # sigmoid function def sigmoid(x,deriv=False): if deriv: result = x*(1-x) else: result = 1/(1+np.exp(-x)) return result # leaky ReLU function def prelu(x, deriv=False): c = np.zeros_like(x) slope = 1e-1 if deriv: c[x<=0] = slope c[x>0] = 1 else: c[x>0] = x[x>0] c[x<=0] = slope*x[x<=0] return c # non-linearity (activation function) nonlin = prelu # instead of sigmoid # initialize weights randomly with mean 0 W = 2*np.random.random((3,1)) - 1 # input dataset X = np.array([ [0,0,1], [0,1,1], [1,0,1], [1,1,1] ]) # output dataset y = np.array([[0,0,1,1]]).T print('X:\n', X) print('Y:\n', y) print() for iter in range(1000): # forward propagation l0 = X l1 = nonlin(np.dot(l0,W)) # how much did we miss? l1_error = y - l1 # compute gradient (slope of activation function at the values in l1) l1_gradient = nonlin(l1, True) # set delta to product of error, gradient and learning rate l1_delta = l1_error * l1_gradient * lr # update weights W += np.dot(l0.T,l1_delta) if iter % 100 == 0: print('pred:', l1.squeeze(), 'mse:', (l1_error**2).mean()) print ("Output After Training:") print ('l1:', np.around(l1)) |