9.以MNIST数据集为例实现神经网络学习算法

1.2层神经网络的构建：

import sys, os
# 为了导入父目录的文件而进行的设定
sys.path.append(os.pardir)
# 将之前的基本函数都放到一个functions.py文件中，方便以后直接调用
from common.functions import *
# 计算梯度的函数也是如此操作
from common.gradient import numerical_gradient

# 定义一个类
class TwoLayerNet:

    def __init__(self, input_size, hidden_size, output_size, weight_init_std=0.01):
        # 初始化权重参数
        self.params = {}
        self.params['W1'] = weight_init_std * np.random.randn(input_size, hidden_size)
        self.params['b1'] = np.zeros(hidden_size)
        self.params['W2'] = weight_init_std * np.random.randn(hidden_size, output_size)
        self.params['b2'] = np.zeros(output_size)

    # 前向计算函数
	def predict(self, x):
        W1, W2 = self.params['W1'], self.params['W2']
        b1, b2 = self.params['b1'], self.params['b2']
    
        a1 = np.dot(x, W1) + b1
        z1 = sigmoid(a1)
        a2 = np.dot(z1, W2) + b2
        y = softmax(a2)
        return y
        
    # 损失函数，其中，x:输入数据, t:监督数据
    def loss(self, x, t):
        y = self.predict(x)
        return cross_entropy_error(y, t)
    
	# 计算准确率的函数
    def accuracy(self, x, t):
        y = self.predict(x)
        y = np.argmax(y, axis=1)
        t = np.argmax(t, axis=1)
        accuracy = np.sum(y == t) / float(x.shape[0])
        return accuracy
        
    
    def numerical_gradient(self, x, t):
        loss_W = lambda W: self.loss(x, t)
        
        grads = {}
        grads['W1'] = numerical_gradient(loss_W, self.params['W1'])
        grads['b1'] = numerical_gradient(loss_W, self.params['b1'])
        grads['W2'] = numerical_gradient(loss_W, self.params['W2'])
        grads['b2'] = numerical_gradient(loss_W, self.params['b2'])
        
        return grads
        
    # 梯度计算函数加速版，会比之前定义的梯度函数计算速度更快
	def gradient(self, x, t):
        W1, W2 = self.params['W1'], self.params['W2']
        b1, b2 = self.params['b1'], self.params['b2']
        grads = {}
        
        batch_num = x.shape[0]
        
        # forward
        a1 = np.dot(x, W1) + b1
        z1 = sigmoid(a1)
        a2 = np.dot(z1, W2) + b2
        y = softmax(a2)
        
        # backward
        dy = (y - t) / batch_num
        grads['W2'] = np.dot(z1.T, dy)
        grads['b2'] = np.sum(dy, axis=0)
        
        da1 = np.dot(dy, W2.T)
        dz1 = sigmoid_grad(a1) * da1
        grads['W1'] = np.dot(x.T, dz1)
        grads['b1'] = np.sum(dz1, axis=0)

2.mini-batch版学习过程实现：

import sys, os
sys.path.append(os.pardir)
import numpy as np
from dataset.mnist import load_mnist
from two_layer_net import TwoLayerNet


# 读入数据
(x_train, t_train), (x_test, t_test) = load_mnist(normalize=True, one_hot_label=True)

# 损失值列表
train_loss_list = []

# 超参数
iters_num = 10000
train_size = x_train.shape[0]
batch_size = 100
learning_rate = 0.1


# 定义网络
network = TwoLayerNet(input_size=784, hidden_size=50, output_size=10)

for i in range(iters_num):
	# 获取mini_batch
	batch_mask = np.random.choice(train_size,batch_size)
	x_batch = x_train[batch_mask]
	t_batch = t_train[batch_mask]

	# 计算梯度
	# grad = network.numerical_gradient(x_batch,t_batch)
	grad = network.gradient(x_batch, t_batch)

	# 更新参数
	for key in ('W1', 'b1', 'W2', 'b2'):
		network.params[key] -= learning_rate*grad[key]

	# 记录学习过程
	loss = network.loss(x_batch,t_batch)
	train_loss_list.append(loss)

	print(loss)

运行可以看到随着学习的进行，损失函数的值在不断减小，说明神经网络的确在学习改进。

3.基于测试数据的评价：

因为神经网络追求的是其泛化能力，所以需要在训练集以外的数据上进行测试，查看神经网络效果。

import sys, os
sys.path.append(os.pardir)
import numpy as np
import matplotlib.pyplot as plt
from dataset.mnist import load_mnist
from two_layer_net import TwoLayerNet

# 读入数据
(x_train, t_train), (x_test, t_test) = load_mnist(normalize=True, one_hot_label=True)

network = TwoLayerNet(input_size=784, hidden_size=50, output_size=10)

iters_num = 10000  # 适当设定循环的次数
train_size = x_train.shape[0]
batch_size = 100
learning_rate = 0.1

train_loss_list = []
train_acc_list = []
test_acc_list = []

iter_per_epoch = max(train_size / batch_size, 1)

for i in range(iters_num):
    batch_mask = np.random.choice(train_size, batch_size)
    x_batch = x_train[batch_mask]
    t_batch = t_train[batch_mask]
    
    # 计算梯度
    #grad = network.numerical_gradient(x_batch, t_batch)
    grad = network.gradient(x_batch, t_batch)
    
    # 更新参数
    for key in ('W1', 'b1', 'W2', 'b2'):
        network.params[key] -= learning_rate * grad[key]
    
    loss = network.loss(x_batch, t_batch)
    train_loss_list.append(loss)
    
    if i % iter_per_epoch == 0:
        train_acc = network.accuracy(x_train, t_train)
        test_acc = network.accuracy(x_test, t_test)
        train_acc_list.append(train_acc)
        test_acc_list.append(test_acc)
        print("train acc, test acc | " + str(train_acc) + ", " + str(test_acc))

# 绘制图形
markers = {'train': 'o', 'test': 's'}
x = np.arange(len(train_acc_list))
plt.plot(x, train_acc_list, label='train acc')
plt.plot(x, test_acc_list, label='test acc', linestyle='--')
plt.xlabel("epochs")
plt.ylabel("accuracy")
plt.ylim(0, 1.0)
plt.legend(loc='lower right')
plt.show()

运行得到

train acc, test acc | 0.10056666666666667, 0.1032
train acc, test acc | 0.7883, 0.7913
train acc, test acc | 0.87705, 0.8806
train acc, test acc | 0.8985333333333333, 0.9028
train acc, test acc | 0.9092833333333333, 0.9109
train acc, test acc | 0.91495, 0.9167
train acc, test acc | 0.9198166666666666, 0.9208
train acc, test acc | 0.92455, 0.925
train acc, test acc | 0.92695, 0.9281
train acc, test acc | 0.9307833333333333, 0.9306
train acc, test acc | 0.93345, 0.9345
train acc, test acc | 0.9363666666666667, 0.9347
train acc, test acc | 0.93795, 0.9365
train acc, test acc | 0.9408666666666666, 0.9384
train acc, test acc | 0.9427, 0.9411
train acc, test acc | 0.9452833333333334, 0.9433
train acc, test acc | 0.94675, 0.9457

可以看出，随着学习的进行，使用训练数据和测试数据评价的识别精度都提高了，并且两者基本重叠在一起，说明这次学习过程没有发生过拟合现象。

Reference：
《Deep Learning from Scratch》