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"""
network.py
~~~~~~~~~~
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.
"""
import random
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."""
if test_data: n_test = len(test_data)
n = len(training_data)
num_batches = n/mini_batch_size
for j in xrange(epochs):
random.shuffle(training_data)
for k in xrange(0,num_batches):
mini_batch = training_data[k*mini_batch_size : (k+1)*mini_batch_size]
self.update_mini_batch(mini_batch, eta)
if test_data:
print "Epoch {0}: {1} / {2}".format(j, self.evaluate(test_data), n_test)
else:
print "Epoch {0} complete".format(j)
def calculate_sum_derivatives_of_mini_batch(self,mini_batch):
"""
计算m个样本的总梯度和。
利用反向传播计算每一个样本(x,y)对应的梯度。
"""
nabla_b = [np.zeros(b.shape) for b in self.biases]
nabla_w = [np.zeros(w.shape) for w in self.weights]
for x, y in mini_batch:
delta_nabla_b, delta_nabla_w = self.backprop(x, y)
nabla_b = [nb+dnb for nb, dnb in zip(nabla_b, delta_nabla_b)]
nabla_w = [nw+dnw for nw, dnw in zip(nabla_w, delta_nabla_w)]
return nabla_b,nabla_w
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."""
m = len(mini_batch)
nabla_b,nabla_w = self.calculate_sum_derivatives_of_mini_batch(mini_batch)
self.weights = [w-(eta/m)*nw for w, nw in zip(self.weights, nabla_w)]
self.biases = [b-(eta/m)*nb for b, nb in zip(self.biases, nabla_b)]
def backprop(self, x, y):
"""Return a tuple "(nabla_b, nabla_w)" representing the
gradient for the cost function C_x. "nabla_b" and
"nabla_w" are layer-by-layer lists of numpy arrays, similar
to "self.biases" and "self.weights"."""
nabla_b = [np.zeros(b.shape) for b in self.biases]
nabla_w = [np.zeros(w.shape) for w in self.weights]
activation = x
activations = [x]
zs = []
for b, w in zip(self.biases, self.weights):
z = np.dot(w, activation)+b
zs.append(z)
activation = sigmoid(z)
activations.append(activation)
l = -1
delta = self.cost_derivative_of_a_L(activations[l], y) * sigmoid_prime(zs[l])
deltas = [delta]
for i in xrange(2, self.num_layers):
l = -i
delta = np.dot(self.weights[l+1].transpose(), deltas[l+1]) * sigmoid_prime(zs[l])
deltas.insert(0,delta)
for i in xrange(1, self.num_layers):
l = -i
nabla_b[l] = deltas[l]
nabla_w[l] = np.dot(deltas[l], 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."""
"""
l = [0,1,0,0,0,0,0,0,0,0]
a = np.array(l).reshape(10,1)
np.argmax(a) #输出向量对应的数字１
test_results = [(1,1),(2,2),(3,3),(1,9)]
[int(x == y) for (x, y) in test_results]
#[1, 1, 1, 0]
sum([int(x == y) for (x, y) in test_results])
#3
"""
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_of_a_L(self, output_activations, y):
"""Return the vector of partial derivatives \partial C_x /
\partial a for the output activations."""
return (output_activations-y)
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))
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