反向传播（BP）网络的mnist实例（提升篇）

提高神经学习的学习效率

并行计算

可以使用GPU进行并行计算，以此提高学习效率。

梯度消失问题

梯度消失问题的存在使得学习效率会变慢，出现梯度消失的原因如下：


其中，对sigmoid函数求导的时候，在[-4, 4]的范围外会发生导数很小，接近于0的情况，进而导致学习的梯度消失。

改进的思路

归一化

因为数据度量的量纲可能不同，所以需要对数据进行归一化处理。

归一化的效果如图：

参数初始化问题

上图的参数初始化方法是业界比较认可的一种初始化方式。

参数初始化的代码如下：

def default_weight_initializer(self):
    # 初始化每层的偏置
    self.biases = [np.random.randn(y, 1) for y in self.sizes[1:]]
    # 初始化每层的权重
    self.weights = [np.random.randn(y, x)/np.sqrt(x) for x, y in zip(self.sizes[:-1], self.sizes[1:])]

正则化

L1正则化：

L1正则化项的导数：

L2正则化：

L2正则化项的导数：

代码需要修改如下：

self.weights = [(1-eta*(lmbda/n))*w - (eta / len(mini_batch)) * nw for w, nw in zip(self.weights, nabla_w)]

学习率

交叉熵

交叉熵定义

交叉熵求导

交叉熵代码：

class CrossEntropyCost(object):
    '''
    a = np.array([[np.nan, np.inf],
                  [-np.nan, -np.inf]])
    np.nan_to_sum(a)

    array([[0.00000000e+000, 1.79769313e+308],
           [0.00000000e+000, -1.79769313e+308]])
    '''
    @staticmethod
    def fn(a, y):
        return np.sum(np.nan_to_num(-y * np.log(a) - (1-y) * np.log(1-a)))

    @staticmethod
    def delta(z, a, y):
        return (a-y)

模型的保存与加载

模型保存与加载代码如下：

# 保存模型
    def save(self, filename):
        data = {"sizes": self.sizes,
                "weights": [w.tolist() for w in self.weights],
                "biases": [b.tolist() for b in self.biases],
                "cost": str(self.cost.__name__)
                }
        f = open(filename, "w")
        json.dump(data, f)  # json把字典类型转换成字符串
        f.close()

# 加载模型
def load(filename):
    f = open(filename, "r")
    data = json.load(f)
    f.close()
    cost = getattr(sys.modules[__name__], data["cost"])
    net = Network(data["sizes"], cost=cost)
    net.weights = [np.array(w) for w in data["weights"]]
    net.biases = [np.array(b) for b in data["biases"]]
    return net

最后在MNIST数据集上训练改进的模型，同时加上准确率等度量方法。

完整代码如下：

#!/user/bin/env python3
# -*- coding: utf-8 -*-
import random
import json
import sys

import numpy as np

#定义神经网络结构
class QuadraticCost(object):
    @staticmethod
    def fn(a, y):
        return 0.5 * np.linalg.norm(a-y) ** 2
    
    @staticmethod
    def delta(z, a, y):
        return (a-y) * sigmoid_prime(z)

class CrossEntropyCost(object):
    '''
    a = np.array([[np.nan, np.inf],
                  [-np.nan, -np.inf]])
    np.nan_to_sum(a)

    array([[0.00000000e+000, 1.79769313e+308],
           [0.00000000e+000, -1.79769313e+308]])
    '''
    @staticmethod
    def fn(a, y):
        return np.sum(np.nan_to_num(-y * np.log(a) - (1-y) * np.log(1-a)))

    @staticmethod
    def delta(z, a, y):
        return (a-y)

class Network(object):
    def __init__(self, sizes, cost=CrossEntropyCost):
        # 网络层数
        self.num_layers = len(sizes)
        # 每层神经元的个数
        self.sizes = sizes
        # 初始化每层的偏置和权重
        self.default_weight_initializer()
        # 损失函数
        self.cost = cost

    def default_weight_initializer(self):
        # 初始化每层的偏置
        self.biases = [np.random.randn(y, 1) for y in self.sizes[1:]]
        # 初始化每层的权重
        self.weights = [np.random.randn(y, x)/np.sqrt(x) for x, y in zip(self.sizes[:-1], self.sizes[1:])]

    def large_weight_initializer(self):
        # 初始化每层的偏置
        self.biases = [np.random.randn(y, 1) for y in self.sizes[1:]]
        # 初始化每层的权重
        self.weights = [np.random.randn(y, x) for x, y in zip(self.sizes[:-1], self.sizes[1:])]

    def feedforward(self, a):
        for b, w in zip(self.biases, self.weights):
            a = sigmoid(np.dot(w, a) + b)
        return a

    # 梯度下降
    def SGD(self, training_data, epochs, mini_batch_size, eta, lmbda=0.0, test_data=None):
        if test_data:
            n_test = len(test_data)
        # 训练数据总个数
        n = len(training_data)

        # 开始训练，循环每一个epochs
        for j in range(epochs):  # 在python2.7中为xrange
            # 洗牌 打乱训练数据
            random.shuffle(training_data)

            # mini_batch
            mini_batches = [training_data[k:k+mini_batch_size] for k in range(0, n, mini_batch_size)]

            # 训练mini_batch
            for mini_batch in mini_batches:
                self.update_mini_batch(mini_batch, eta, lmbda, n)

            print("Epoch {0} complete".format(j))

            cost = self.total_cost(training_data, lmbda)
            print("Cost on training data: {}".format(cost))
            accuracy = self.accuracy(training_data, convert=True)
            print("Accuracy on training data: {} / {}".format(accuracy, n))

            if test_data:
                cost = self.total_cost(test_data, lmbda, convert=True)
                print("Cost on test data: {}".format(cost))
                accuracy = self.accuracy(test_data)
                print("Accuracy on test data: {} / {}".format(accuracy, len(test_data)))

    def update_mini_batch(self, mini_batch, eta, lmbda, n):
        # 保存每层偏导
        nabla_b = [np.zeros(b.shape) for b in self.biases]
        nabla_w = [np.zeros(w.shape) for w in self.weights]

        # 训练一个mini_batch
        for x, y in mini_batch:
            delta_nabla_b, delta_nabla_w = self.update(x, y)

            # 保存一次训练网络中每层的偏导
            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)]

        # 更新权重和偏置 Wn+1 = Wn - eta * nw
        self.weights = [(1-eta*(lmbda/n))*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)]

    # 前向传播
    def update(self, x, y):
        # 保存每层偏导
        nabla_b = [np.zeros(b.shape) for b in self.biases]
        nabla_w = [np.zeros(w.shape) for w in self.weights]
        activation = x  # 保存的输入（训练数据）

        # 保存每一层的激励值a=sigmoid(z)
        activations = [x]

        # 保存每一层的z=wx+b
        zs = []
        # 前向传播
        for b, w in zip(self.biases, self.weights):
            # 计算每层的z
            z = np.dot(w, activation) + b

            # 保存每层的z
            zs.append(z)

            # 计算每层的a
            activation = sigmoid(z)

            # 保存每一层的a
            activations.append(activation)
        # 反向更新
        # 计算最后一层的误差
        # delta = self.cost_derivative(activations[-1], y) * sigmoid_prime(zs[-1])
        delta = (self.cost).delta(zs[-1], activations[-1], y)

        # 最后一层权重和偏置的导数
        nabla_b[-1] = delta
        nabla_w[-1] = np.dot(delta, activations[-2].T)

        # 倒数第二层一直到第一层 权重和偏置的导数
        for l in range(2, self.num_layers):
            z = zs[-l]

            sp = sigmoid_prime(z)

            # 当前层的误差
            delta = np.dot(self.weights[-l+1].T, delta) * sp

            # 当前层的偏置和权重的导数
            nabla_b[-l] = delta
            nabla_w[-l] = np.dot(delta, activations[-l-1].T)

        return (nabla_b, nabla_w)

    def accuracy(self, data, convert=False):
        if convert:
            # 如果是训练集数据，得到的结果是非0即1的one-hot编码方式
            results = [(np.argmax(self.feedforward(x)), np.argmax(y)) for (x, y) in data]
        else:
            # 如果是测试集数据，得到的结果就是0-9之间的数字分类
            results = [(np.argmax(self.feedforward(x)), y) for (x, y) in data]
        return sum(int(x == y) for (x, y) in results)

    def total_cost(self, data, lmbda, convert=False):
        cost = 0.0
        for x, y in data:
            a = self.feedforward(x)
            if convert:
                # convert为真，表示是测试数据集，将y转换为一个one-hot编码的十维向量
                y = mnist_loader.vectorized_result(y)
            cost += self.cost.fn(a, y) / len(data)
        cost += 0.5*(lmbda/len(data))*sum(np.linalg.norm(w)**2 for w in self.weights)
        return cost

    def cost_derivative(self, output_activation, y):
        return (output_activation - y)

    # 保存模型
    def save(self, filename):
        data = {"sizes": self.sizes,
                "weights": [w.tolist() for w in self.weights],
                "biases": [b.tolist() for b in self.biases],
                "cost": str(self.cost.__name__)
                }
        f = open(filename, "w")
        json.dump(data, f)  # json把字典类型转换成字符串
        f.close()

# 加载模型
def load(filename):
    f = open(filename, "r")
    data = json.load(f)
    f.close()
    cost = getattr(sys.modules[__name__], data["cost"])
    net = Network(data["sizes"], cost=cost)
    net.weights = [np.array(w) for w in data["weights"]]
    net.biases = [np.array(b) for b in data["biases"]]
    return net

# sigmoid激励函数
def sigmoid(z):
    return 1.0 / (1.0 + np.exp(-z))

def sigmoid_prime(z):
    return sigmoid(z) * (1-sigmoid(z))

if __name__ == "__main__":
    import mnist_loader

    training_data, validation_data, test_data = mnist_loader.load_data_wrapper()

    # 28*28=784个像素， 可以定义30个神经元， 共有10种分类
    net = Network([784, 30, 10])
    net.SGD(training_data, 30, 10, 0.5, test_data=test_data)

运行结果如下图所示：

可以看到，模型的效果显然比不采取提升学习效率的措施时效果要好很多