一、问题描述

设计神经网络，利用反向传播算法，基于MNIST数据集做手写识别，并在神经元个数或隐含层个数上进行改变，探究其性能差别。

二、算法核心思想分析

利用sigmoid神经元构建神经网络，使用前馈神经网络实现mini-batch随机梯度下降学习算法，使用反向传播计算梯度,更新权重(weights)和偏置(biases)。利用均值为0方差为1的高斯分布随机初始化网络。

三、题目分析

首先读取MNIST数据集，利用训练集进行训练，完成后利用测试集进行测试，从而得出准确率。

四、代码及运行结果

mnist_loader.py

loader对应格式的数据：https://github.com/mnielsen/neural-networks-and-deep-learning/tree/master/data

import pickle
import gzip
import numpy as np


def load_data():
    f = gzip.open('mnist.pkl.gz', 'rb')
    training_data, validation_data, test_data = pickle.load(f, encoding='bytes')
    f.close()
    return training_data, validation_data, test_data


def load_data_wrapper():
    tr_d, va_d, te_d = load_data()
    training_inputs = [np.reshape(x, (784, 1)) for x in tr_d[0]]
    training_results = [vectorized_result(y) for y in tr_d[1]]
    training_data = zip(training_inputs, training_results)
    validation_inputs = [np.reshape(x, (784, 1)) for x in va_d[0]]
    validation_data = zip(validation_inputs, va_d[1])
    test_inputs = [np.reshape(x, (784, 1)) for x in te_d[0]]
    test_data = zip(test_inputs, te_d[1])
    return training_data, validation_data, test_data


def vectorized_result(j):
    e = np.zeros((10, 1))
    e[j] = 1.0
    return e

network.py

import random
import numpy as np


class Network(object):

    def __init__(self, sizes):
        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):
        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):
        if test_data:
            test_data = list(test_data)
            n_test = len(test_data)
        training_data = list(training_data)
        n = len(training_data)
        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 {0}: {1} / {2}".format(j, self.evaluate(test_data), n_test))
            else:
                print("Epoch {0} complete".format(j))

    def update_mini_batch(self, mini_batch, eta):
        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)]
        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):
        nabla_b = [np.zeros(b.shape) for b in self.biases]
        nabla_w = [np.zeros(w.shape) for w in self.weights]
        # feedforward
        activation = x
        activations = [x]  # list to store all the activations, layer by layer
        zs = []  # list to store all the z vectors, layer by layer
        for b, w in zip(self.biases, self.weights):
            z = np.dot(w, activation)+b
            zs.append(z)
            activation = sigmoid(z)
            activations.append(activation)
        # backward pass
        delta = self.cost_derivative(activations[-1], y) * \
            sigmoid_prime(zs[-1])
        nabla_b[-1] = delta
        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):
        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 output_activations-y


def sigmoid(z):
    return 1.0/(1.0+np.exp(-z))


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

main.py

import mnist_loader
import network

training_data, validation_data, test_data = mnist_loader.load_data_wrapper()

net = network.Network([784, 30, 10])
net.SGD(training_data, 30, 10, 3.0, test_data=test_data)

使用[784,30,10]的网路接口，即输入层784个神经元，隐含层30个神经元，输出层10个神经元，结果如下：

使用[784, 10]的网路接口，即输入层784个神经元，隐含层30个神经元，输出层10个神经元，结果如下：

结果显而易见，第一个网络要比第二个网络工作得更好。

五、总结

设计合适的神经网络结构和选择合理的学习率至关重要。

如有错误请指正