153 lines
6.3 KiB
Python
153 lines
6.3 KiB
Python
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# %load network.py
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"""
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network.py
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~~~~~~~~~~
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IT WORKS
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A module to implement the stochastic gradient descent learning
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algorithm for a feedforward neural network. Gradients are calculated
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using backpropagation. Note that I have focused on making the code
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simple, easily readable, and easily modifiable. It is not optimized,
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and omits many desirable features.
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"""
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#### Libraries
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# Standard library
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import random
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# Third-party libraries
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import numpy as np
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class Network(object):
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def __init__(self, sizes):
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"""The list ``sizes`` contains the number of neurons in the
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respective layers of the network. For example, if the list
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was [2, 3, 1] then it would be a three-layer network, with the
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first layer containing 2 neurons, the second layer 3 neurons,
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and the third layer 1 neuron. The biases and weights for the
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network are initialized randomly, using a Gaussian
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distribution with mean 0, and variance 1. Note that the first
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layer is assumed to be an input layer, and by convention we
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won't set any biases for those neurons, since biases are only
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ever used in computing the outputs from later layers."""
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self.num_layers = len(sizes)
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self.sizes = sizes
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self.biases = [np.random.randn(y, 1) for y in sizes[1:]]
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self.weights = [np.random.randn(y, x)
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for x, y in zip(sizes[:-1], sizes[1:])]
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def feedforward(self, a):
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"""Return the output of the network if ``a`` is input."""
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for b, w in zip(self.biases, self.weights):
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a = sigmoid(np.dot(w, a)+b)
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return a
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def SGD(self, training_data, epochs, mini_batch_size, eta,
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test_data=None):
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"""Train the neural network using mini-batch stochastic
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gradient descent. The ``training_data`` is a list of tuples
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``(x, y)`` representing the training inputs and the desired
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outputs. The other non-optional parameters are
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self-explanatory. If ``test_data`` is provided then the
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network will be evaluated against the test data after each
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epoch, and partial progress printed out. This is useful for
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tracking progress, but slows things down substantially."""
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training_data = list(training_data)
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n = len(training_data)
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if test_data:
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test_data = list(test_data)
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n_test = len(test_data)
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for j in range(epochs):
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random.shuffle(training_data)
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mini_batches = [
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training_data[k:k+mini_batch_size]
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for k in range(0, n, mini_batch_size)]
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for mini_batch in mini_batches:
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self.update_mini_batch(mini_batch, eta)
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if test_data:
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print("Epoch {} : {} / {}".format(j,self.evaluate(test_data),n_test))
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else:
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print("Epoch {} complete".format(j))
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def update_mini_batch(self, mini_batch, eta):
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"""Update the network's weights and biases by applying
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gradient descent using backpropagation to a single mini batch.
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The ``mini_batch`` is a list of tuples ``(x, y)``, and ``eta``
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is the learning rate."""
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nabla_b = [np.zeros(b.shape) for b in self.biases]
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nabla_w = [np.zeros(w.shape) for w in self.weights]
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for x, y in mini_batch:
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delta_nabla_b, delta_nabla_w = self.backprop(x, y)
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nabla_b = [nb+dnb for nb, dnb in zip(nabla_b, delta_nabla_b)]
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nabla_w = [nw+dnw for nw, dnw in zip(nabla_w, delta_nabla_w)]
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self.weights = [w-(eta/len(mini_batch))*nw
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for w, nw in zip(self.weights, nabla_w)]
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self.biases = [b-(eta/len(mini_batch))*nb
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for b, nb in zip(self.biases, nabla_b)]
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def backprop(self, x, y):
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"""Return a tuple ``(nabla_b, nabla_w)`` representing the
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gradient for the cost function C_x. ``nabla_b`` and
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``nabla_w`` are layer-by-layer lists of numpy arrays, similar
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to ``self.biases`` and ``self.weights``."""
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nabla_b = [np.zeros(b.shape) for b in self.biases]
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nabla_w = [np.zeros(w.shape) for w in self.weights]
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# feedforward
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activation = x
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activations = [x] # list to store all the activations, layer by layer
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zs = [] # list to store all the z vectors, layer by layer
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for b, w in zip(self.biases, self.weights):
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z = np.dot(w, activation)+b
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zs.append(z)
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activation = sigmoid(z)
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activations.append(activation)
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# backward pass
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delta = self.cost_derivative(activations[-1], y) * \
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sigmoid_prime(zs[-1])
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nabla_b[-1] = delta
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nabla_w[-1] = np.dot(delta, activations[-2].transpose())
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# Note that the variable l in the loop below is used a little
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# differently to the notation in Chapter 2 of the book. Here,
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# l = 1 means the last layer of neurons, l = 2 is the
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# second-last layer, and so on. It's a renumbering of the
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# scheme in the book, used here to take advantage of the fact
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# that Python can use negative indices in lists.
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for l in range(2, self.num_layers):
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z = zs[-l]
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sp = sigmoid_prime(z)
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delta = np.dot(self.weights[-l+1].transpose(), delta) * sp
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nabla_b[-l] = delta
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nabla_w[-l] = np.dot(delta, activations[-l-1].transpose())
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return (nabla_b, nabla_w)
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def evaluate(self, test_data):
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"""Return the number of test inputs for which the neural
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network outputs the correct result. Note that the neural
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network's output is assumed to be the index of whichever
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neuron in the final layer has the highest activation."""
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test_results = [(np.argmax(self.feedforward(x)), y)
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for (x, y) in test_data]
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return sum(int(x == y) for (x, y) in test_results)
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def cost_derivative(self, output_activations, y):
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"""Return the vector of partial derivatives \partial C_x /
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\partial a for the output activations."""
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return (output_activations-y)
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#### Miscellaneous functions
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def sigmoid(z):
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"""The sigmoid function."""
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return 1.0/(1.0+np.exp(-z))
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def sigmoid_prime(z):
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"""Derivative of the sigmoid function."""
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return sigmoid(z)*(1-sigmoid(z))
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