# network1.py

## network1.py

#!/usr/bin/python
# -*- coding: utf-8 -*-
"""
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.
"""

#### 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."""
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:
# 给定一个样本X,利用反向传播算法计算对应w,b的梯度
delta_nabla_b, delta_nabla_w = self.backprop(x, y)
# 对m个样本的梯度进行累计求和
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"."""
# 初始化nb,nw,结构和b,w一样
nabla_b = [np.zeros(b.shape) for b in self.biases]
nabla_w = [np.zeros(w.shape) for w in self.weights]

# feedforward
# 执行算法的feedforward阶段
#　(1)初始化x作为a_1
activation = x
activations = [x] # list to store all the activations, layer by layer
zs = [] # list to store all the z vectors, layer by layer
# (2)l=2,....L层，分别计算z_l,a_l并且保存下来。
for b, w in zip(self.biases, self.weights):
z = np.dot(w, activation)+b
zs.append(z)
activation = sigmoid(z)
activations.append(activation)

#========================================================================
# 先计算所有的误差delta，最后计算所有层的梯度nb,nw，代码可读性更高一些
#========================================================================
# method2
# backward pass
# 执行算法的backward阶段
# (3)初始化第L层的误差,delta_L　= cost(a_L,y) * sigmoid_prime(z_L)
l = -1
delta = self.cost_derivative_of_a_L(activations[l], y) * sigmoid_prime(zs[l])
deltas = [delta] # list to store all the errors,layer by layer
# (4)初始化l=L-1,....2层的误差,delta_l = np.dot(w_l+1^T,delta_l+1)* sigmoid_prime(z_l)
for i in xrange(2, self.num_layers):
l = -i #(-２代表L-1,-3代表L-2,-(L-1)代表2)
delta = np.dot(self.weights[l+1].transpose(), deltas[l+1]) * sigmoid_prime(zs[l])
deltas.insert(0,delta) # 确保误差的顺序，从后往前计算，所以需要insert在数组的最前面

#(5)l=L,L-1,....2层，计算所有的梯度向量nb,nw
for i in xrange(1, self.num_layers):
l = -i #(-1,-2,....-(L-1))
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)

#### 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))


"""
~~~~~~~~~~~~

A library to load the MNIST image data.  For details of the data
structures that are returned, see the doc strings for load_data
and load_data_wrapper.  In practice, load_data_wrapper is the
function usually called by our neural network code.
"""

#### Libraries
# Standard library
import cPickle
import gzip

# Third-party libraries
import numpy as np

"""Return the MNIST data as a tuple containing the training data,
the validation data, and the test data.

The training_data is returned as a tuple with two entries.
The first entry contains the actual training images.  This is a
numpy ndarray with 50,000 entries.  Each entry is, in turn, a
numpy ndarray with 784 values, representing the 28 * 28 = 784
pixels in a single MNIST image.

The second entry in the training_data tuple is a numpy ndarray
containing 50,000 entries.  Those entries are just the digit
values (0...9) for the corresponding images contained in the first
entry of the tuple.

The validation_data and test_data are similar, except
each contains only 10,000 images.

This is a nice data format, but for use in neural networks it's
helpful to modify the format of the training_data a little.
That's done in the wrapper function load_data_wrapper(), see
below.
"""
f = gzip.open('../data/mnist.pkl.gz', 'rb')
f.close()
return (training_data, validation_data, test_data)

"""Return a tuple containing (training_data, validation_data,
test_data). Based on load_data, but the format is more
convenient for use in our implementation of neural networks.

In particular, training_data is a list containing 50,000
2-tuples (x, y).  x is a 784-dimensional numpy.ndarray
containing the input image.  y is a 10-dimensional
numpy.ndarray representing the unit vector corresponding to the
correct digit for x.

validation_data and test_data are lists containing 10,000
2-tuples (x, y).  In each case, x is a 784-dimensional
numpy.ndarry containing the input image, and y is the
corresponding classification, i.e., the digit values (integers)
corresponding to x.

Obviously, this means we're using slightly different formats for
the training data and the validation / test data.  These formats
turn out to be the most convenient for use in our neural network
code."""
# train
training_inputs = [np.reshape(x, (784, 1)) for x in tr_d]
# vector train_y, while val and test y are integers.
training_results = [vectorized_result(y) for y in tr_d]
training_data = zip(training_inputs, training_results)

# val
validation_inputs = [np.reshape(x, (784, 1)) for x in va_d]
validation_data = zip(validation_inputs, va_d)
# test
test_inputs = [np.reshape(x, (784, 1)) for x in te_d]
test_data = zip(test_inputs, te_d)

return (training_data, validation_data, test_data)

def vectorized_result(j):
"""Return a 10-dimensional unit vector with a 1.0 in the jth
position and zeroes elsewhere.  This is used to convert a digit
(0...9) into a corresponding desired output from the neural
network."""
e = np.zeros((10, 1))
e[j] = 1.0
return e


# test

#!/usr/bin/python
# -*- coding: utf-8 -*-

from ke_network import Network

def test():
epoch = 30
mini_batch_size  = 10
eta = 3.0
net = Network([784,30,10])
net.SGD(train,epoch,mini_batch_size,eta,test_data=test)

def main():
test()

if __name__=="__main__":
main()

Epoch 0: 9135 / 10000
Epoch 1: 9238 / 10000
Epoch 2: 9337 / 10000
Epoch 3: 9345 / 10000
Epoch 4: 9393 / 10000
Epoch 5: 9389 / 10000
Epoch 6: 9419 / 10000
Epoch 7: 9410 / 10000


## History

• 20180807: created.

Author: kezunlin
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