Debugging backpropagation algorithm

Question

I am trying to implement the back-propagation algorithm using numpy in python. I have been using this site to implement the matrix form of back-propagation. While testing this code on XOR, my network does not converge even after multiple runs of thousands of iterations. I think there is some sort of logic error. I would be very grateful if anyone would be willing to look it over. Fully runnable code can be found at github

import numpy as np

def backpropagate(network, tests, iterations=50):

    #convert tests into numpy matrices
    tests = [(np.matrix(inputs, dtype=np.float64).reshape(len(inputs), 1),
            np.matrix(expected, dtype=np.float64).reshape(len(expected), 1))
            for inputs, expected in tests]

    for _ in range(iterations):

        #accumulate the weight and bias deltas
        weight_delta = [np.zeros(matrix.shape) for matrix in network.weights]
        bias_delta = [np.zeros(matrix.shape) for matrix in network.bias]

        #iterate over the tests
        for potentials, expected in tests:

            #input the potentials into the network
            #calling the network with trace == True returns a list of matrices,
            #representing the potentials of each layer 
            trace = network(potentials, trace=True)
            errors = [expected - trace[-1]]

            #iterate over the layers backwards
            for weight_matrix, layer in reversed(list(zip(network.weights, trace))):
                #compute the error vector for a layer
                errors.append(np.multiply(weight_matrix.transpose()*errors[-1],
                                          network.sigmoid.derivative(layer)))

            #remove the input layer
            errors.pop()
            errors.reverse()

            #compute the deltas for bias and weight
            for index, error in enumerate(errors):
                bias_delta[index] += error
                weight_delta[index] += error * trace[index].transpose()

        #apply the deltas
        for index, delta in enumerate(weight_delta):
            network.weights[index] += delta
        for index, delta in enumerate(bias_delta):
            network.bias[index] += delta

Additionally, here is the code that computes the output, and my sigmoid function. It is less likely that bug lies here; I was able to trained a network to simulate XOR using simulated annealing.

# the call function of the neural network
def __call__(self, potentials, trace=True):

    #ensure the input is properly formated
    potentials = np.matrix(potentials, dtype=np.float64).reshape(len(potentials), 1)

    #accumulate the trace
    trace = [potentials]

    #iterate over the weights
    for index, weight_matrix in enumerate(self.weights):
        potentials = weight_matrix * potentials + self.bias[index]
        potentials = self.sigmoid(potentials)
        trace.append(potentials)

    return trace

#The sigmoid function that is stored in the network
def sigmoid(x):
    return np.tanh(x)
sigmoid.derivative = lambda x : (1-np.square(x))

Answer 1

The problem is the missing step-size parameter. Gradient should be additionally scaled, not to make the whole step in the weights space at once. So instead of: network.weights[index] += delta and network.bias[index] += delta it should be:

def backpropagate(network, tests, stepSize = 0.01, iterations=50):

    #...

    network.weights[index] += stepSize * delta

    #...

    network.bias[index] += stepSize * delta