Project 2

The backpropagation learning algorithm can be divided into two phases: propagation and weight update.

1. Phase 1: Propagation
   Each propagation involves the following steps:
   1. Forward propagation of a training pattern's input through the neural network in order to generate the propagation's output activations.
   2. Backward propagation of the propagation's output activations through the neural network using the training pattern target in order to generate the deltas of all output and hidden neurons.

1. Phase 2: Weight update
   For each weight-synapse follow the following steps:
   1. Multiply its output delta and input activation to get the gradient of the weight.
   2. Subtract a ratio (percentage) of the gradient from the weight.
   This ratio (percentage) influences the speed and quality of learning; it is called the learning rate. The greater the ratio, the faster the neuron trains; the lower the ratio, the more accurate the training is. The sign of the gradient of a weight indicates where the error is increasing, this is why the weight must be updated in the opposite direction.
Repeat phase 1 and 2 until the performance of the network is satisfactory

import numpy as np

def sigmoid(x):
    return 1.0/(1.0 + np.exp(-x))

def sigmoid_prime(x):
    return sigmoid(x)*(1.0-sigmoid(x))

def tanh(x):
    return np.tanh(x)

def tanh_prime(x):
    return 1.0 - x**2


class NeuralNetwork:

    def __init__(self, layers, activation='tanh'):
        if activation == 'sigmoid':
            self.activation = sigmoid
            self.activation_prime = sigmoid_prime
        elif activation == 'tanh':
            self.activation = tanh
            self.activation_prime = tanh_prime

        # Set weights
        self.weights = []
        # layers = [2,2,1]
        # range of weight values (-1,1)
        # input and hidden layers - random((2+1, 2+1)) : 3 x 3
        for i in range(1, len(layers) - 1):
            r = 2*np.random.random((layers[i-1] + 1, layers[i] + 1)) -1
            self.weights.append(r)
        # output layer - random((2+1, 1)) : 3 x 1
        r = 2*np.random.random( (layers[i] + 1, layers[i+1])) - 1
        self.weights.append(r)

    def fit(self, X, y, learning_rate=0.2, epochs=100000):
        # Add column of ones to X
        # This is to add the bias unit to the input layer
        ones = np.atleast_2d(np.ones(X.shape[0]))
        X = np.concatenate((ones.T, X), axis=1)
         
        for k in range(epochs):
            if k % 10000 == 0: print 'epochs:', k
            
            i = np.random.randint(X.shape[0])
            a = [X[i]]

            for l in range(len(self.weights)):
                    dot_value = np.dot(a[l], self.weights[l])
                    activation = self.activation(dot_value)
                    a.append(activation)
            # output layer
            error = y[i] - a[-1]
            deltas = [error * self.activation_prime(a[-1])]

            # we need to begin at the second to last layer 
            # (a layer before the output layer)
            for l in range(len(a) - 2, 0, -1): 
                deltas.append(deltas[-1].dot(self.weights[l].T)*self.activation_prime(a[l]))

            # reverse
            # [level3(output)->level2(hidden)]  => [level2(hidden)->level3(output)]
            deltas.reverse()

            # backpropagation
            # 1. Multiply its output delta and input activation 
            #    to get the gradient of the weight.
            # 2. Subtract a ratio (percentage) of the gradient from the weight.
            for i in range(len(self.weights)):
                layer = np.atleast_2d(a[i])
                delta = np.atleast_2d(deltas[i])
                self.weights[i] += learning_rate * layer.T.dot(delta)

    def predict(self, x): 
        a = np.concatenate((np.ones(1).T, np.array(x)), axis=1)      
        for l in range(0, len(self.weights)):
            a = self.activation(np.dot(a, self.weights[l]))
        return a

if __name__ == '__main__':

    nn = NeuralNetwork([2,2,1])

    X = np.array([[0, 0],
                  [0, 1],
                  [1, 0],
                  [1, 1]])

    y = np.array([0, 1, 1, 0])

    nn.fit(X, y)

    for e in X:
        print(e,nn.predict(e))