# %load network.py

"""

network_mat.py

~~~~~~~~~~

IT WORKS

A module to implement the stochastic gradient descent learning

algorithm for a feedforward neural network. Gradients are calculated

using backpropagation. Note that I have focused on making the code

simple, easily readable, and easily modifiable. It is not optimized,

and omits many desirable features.

"""

#### Libraries

# Standard library

import random

# Third-party libraries

import numpy as np

class Network(object):

def __init__(self, sizes):

"""

The list ``sizes`` contains the number of neurons in the

respective layers of the network. For example, if the list

was [2, 3, 1] then it would be a three-layer network, with the

first layer containing 2 neurons, the second layer 3 neurons,

and the third layer 1 neuron. The biases and weights for the

network are initialized randomly, using a Gaussian

distribution with mean 0, and variance 1. Note that the first

layer is assumed to be an input layer, and by convention we

won't set any biases for those neurons, since biases are only

ever used in computing the outputs from later layers.

"""

self.num_layers = len(sizes)

self.sizes = sizes

self.biases = [np.random.randn(y, 1) for y in sizes[1:]]

self.weights = [np.random.randn(y, x)

for x, y in zip(sizes[:-1], sizes[1:])]

def feedforward(self, a):

"""Return the output of the network if ``a`` is input."""

for b, w in zip(self.biases, self.weights):

a = sigmoid(np.dot(w, a)+b)

return a

def SGD(self, training_data, epochs, mini_batch_size, eta,

test_data=None):

"""

Train the neural network using mini-batch stochastic

gradient descent. The ``training_data`` is a list of tuples

``(x, y)`` representing the training inputs and the desired

outputs. The other non-optional parameters are

self-explanatory. If ``test_data`` is provided then the

network will be evaluated against the test data after each

epoch, and partial progress printed out. This is useful for

tracking progress, but slows things down substantially.

"""

training_data = list(training_data)

n = len(training_data)

if test_data:

test_data = list(test_data)

n_test = len(test_data)

for j in range(epochs):

random.shuffle(training_data)

mini_batches = [

training_data[k:k+mini_batch_size]

for k in range(0, n, mini_batch_size)]

for mini_batch in mini_batches:

self.update_mini_batch(mini_batch, eta)

if test_data:

print("Epoch {} : {} / {}".format(j,self.evaluate(test_data),n_test));

else:

print("Epoch {} complete".format(j))

def update_mini_batch(self, mini_batch, eta):

"""

Update the network's weights and biases by applying

gradient descent using backpropagation to a single mini batch.

The ``mini_batch`` is a list of tuples ``(x, y)``, and ``eta``

is the learning rate.

"""

mini_batch_size = len(mini_batch) # size of mini_batch

x_len = len(mini_batch[0][0]) # length of x (axis=0)

y_len = len(mini_batch[0][1]) # length of y (axis=0)

# Creating empty numpy arrays for storing whole batch.

X = np.empty((mini_batch_size, x_len, 1), dtype='float32')

Y = np.empty((mini_batch_size, y_len, 1), dtype='float32')

# Copying data from mini_batch to X and Y

for i in range(mini_batch_size):

X[i] = mini_batch[i][0]

Y[i] = mini_batch[i][1]

# Removing last axes from X and Y which are of size 1

# and then transposing them to have input vectors as columns of X

# and output activations as columns of Y

X = np.reshape(X, (mini_batch_size, x_len)).transpose()

Y = np.reshape(Y, (mini_batch_size, y_len)).transpose()

# Calling backprop function for the mini_batch.

# In network.py backprop receives x and y of shapes

# (784, 1) and (10, 1) respectively. Here backprop receives

# X and Y of shapes (784, mini_batch_size) and (10, mini_batch_size)

# respectively. All operations remain valid with this input parameters.

# There is only one difference in returned values - each element of

# nabla_b is not of shape (*, 1), but (*, mini_batch_size). So each

# element needs to be summed over axis 1 to get sum over mini_batch.

nabla_b, nabla_w = self.backprop(X, Y)

self.weights = [w - (eta / len(mini_batch)) * nw

for w, nw in zip(self.weights, nabla_w)]

self.biases = [b - (eta / len(mini_batch)) * nb.sum(axis=1, keepdims=True)

for b, nb in zip(self.biases, nabla_b)]

def backprop(self, x, y):

"""

Return a tuple ``(nabla_b, nabla_w)`` representing the sum

of gradients for the cost function C_x over the mini batch. ``nabla_b``

and ``nabla_w`` are layer-by-layer lists of numpy arrays of shapes

(*, mini_batch_size).

"""

nabla_b = [np.zeros(b.shape) for b in self.biases]

nabla_w = [np.zeros(w.shape) for w in self.weights]

# feedforward

activation = x

activations = [x] # list to store all the activations for mini_batch, layer by layer

zs = [] # list to store all the z vectors for mini_batch, layer by layer

for b, w in zip(self.biases, self.weights):

z = np.dot(w, activation)+b

zs.append(z)

activation = sigmoid(z)

activations.append(activation)

# backward pass

delta = self.cost_derivative(activations[-1], y) * \

sigmoid_prime(zs[-1])

nabla_b[-1] = delta

nabla_w[-1] = np.dot(delta, activations[-2].transpose())

# Note that the variable l in the loop below is used a little

# differently to the notation in Chapter 2 of the book. Here,

# l = 1 means the last layer of neurons, l = 2 is the

# second-last layer, and so on. It's a renumbering of the

# scheme in the book, used here to take advantage of the fact

# that Python can use negative indices in lists.

for l in range(2, self.num_layers):

z = zs[-l]

sp = sigmoid_prime(z)

delta = np.dot(self.weights[-l+1].transpose(), delta) * sp

nabla_b[-l] = delta

nabla_w[-l] = np.dot(delta, activations[-l-1].transpose())

return (nabla_b, nabla_w)

def evaluate(self, test_data):

"""

Return the number of test inputs for which the neural

network outputs the correct result. Note that the neural

network's output is assumed to be the index of whichever

neuron in the final layer has the highest activation.

"""

test_results = [(np.argmax(self.feedforward(x)), y)

for (x, y) in test_data]

return sum(int(x == y) for (x, y) in test_results)

def cost_derivative(self, output_activations, y):

"""

Return the vector of partial derivatives \partial C_x /

\partial a for the output activations.

"""

return (output_activations-y)

#### Miscellaneous functions

def sigmoid(z):

"""The sigmoid function."""

return 1.0/(1.0+np.exp(-z))

def sigmoid_prime(z):

"""Derivative of the sigmoid function."""

return sigmoid(z)*(1-sigmoid(z))