深度学习：多层感知机MLP数字识别的代码实现

深度学习我看的是neural network and deep learning 这本书，这本书写的真的非常好，是我的导师推荐的。这篇博客里的代码也是来自于这，我最近是在学习Pytorch，学习的过程我觉得还是有必要把代码自己敲一敲，就像当初学习机器学习一样。也是希望通过这个代码能够加深对反向传播原路的认识。

在下面的代码中，比较复杂的部分就是mini_batch部分了，我们一定要有清晰的认识，我们在上一篇博客中给出了反向传播方法四公式，而且在文章的末尾又给出了使用这四个公式的方法。

所以牢牢记住这个个流程，一共分为：
1.输入训练集
2.前向传播
3.计算输出层产生的错误
4.反向传播的错误
5.使用梯度下降，训练参数

我们将在下面的代码中用以上的几个数字（1到5）来表示当前代码是哪个过程

import random
import mnist_loader

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:])]

前向传播方法
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
随机梯度方法
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)

打乱数据，然后按照mini_batch大小取数据，在这里有update_mini_batch方法，这里面就是反向传播的核心功能。还有evaluate方法，这里包含feedforward方法。
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))
在这个代码里最核心的部分便是
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.”""
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)]
下面这个代码是属于过程5，使用梯度下降，训练参数
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
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)
    下面这个代码是计算输出层误差
    delta = self.cost_derivative(activations[-1], y) * \
        sigmoid_prime(zs[-1])
    下面这个公式属于反向传播公式3
    nabla_b[-1] = delta
    下面这个公式属于反向创博公式4
    nabla_w[-1] = np.dot(delta, activations[-2].transpose())
    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)

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))

if name == ‘main’:
training_data, validation_data, test_data = mnist_loader.load_data_wrapper()
net = Network([784, 30, 10])
net.SGD(training_data, 30, 10, 100.0, test_data=test_data)