纯numpy数值微分法实现手写数字识别

手写数字识别作为深度学习入门经典的识别案例，各种深度学习框架都有这个例子的实现方法。我这里将不用任何深度学习现有框架，例如TensorFlow、Keras、pytorch，直接使用Python语言的numpy实现各种激活函数、损失函数、梯度下降的方法。

程序分为两部分，首先是手写数字数据的准备，直接使用如下mnist.py文件中的方法load_minist即可。文件代码如下：

# coding: utf-8
try:
    import urllib.request
except ImportError:
    raise ImportError(\'You should use Python 3.x\')
import os.path
import gzip
import pickle
import os
import numpy as np


url_base = \'http://yann.lecun.com/exdb/mnist/\'
key_file = {
    \'train_img\':\'train-images-idx3-ubyte.gz\',
    \'train_label\':\'train-labels-idx1-ubyte.gz\',
    \'test_img\':\'t10k-images-idx3-ubyte.gz\',
    \'test_label\':\'t10k-labels-idx1-ubyte.gz\'
}

dataset_dir = os.path.dirname(os.path.abspath(__file__))
save_file = dataset_dir + \"/mnist.pkl\"

train_num = 60000
test_num = 10000
img_dim = (1, 28, 28)
img_size = 784


def _download(file_name):
    file_path = dataset_dir + \"/\" + file_name
    
    if os.path.exists(file_path):
        return

    print(\"Downloading \" + file_name + \" ... \")
    urllib.request.urlretrieve(url_base + file_name, file_path)
    print(\"Done\")
    
def download_mnist():
    for v in key_file.values():
       _download(v)
        
def _load_label(file_name):
    file_path = dataset_dir + \"/\" + file_name
    
    print(\"Converting \" + file_name + \" to NumPy Array ...\")
    with gzip.open(file_path, \'rb\') as f:
            labels = np.frombuffer(f.read(), np.uint8, offset=8)
    print(\"Done\")
    
    return labels

def _load_img(file_name):
    file_path = dataset_dir + \"/\" + file_name
    
    print(\"Converting \" + file_name + \" to NumPy Array ...\")    
    with gzip.open(file_path, \'rb\') as f:
            data = np.frombuffer(f.read(), np.uint8, offset=16)
    data = data.reshape(-1, img_size)
    print(\"Done\")
    
    return data
    
def _convert_numpy():
    dataset = {}
    dataset[\'train_img\'] =  _load_img(key_file[\'train_img\'])
    dataset[\'train_label\'] = _load_label(key_file[\'train_label\'])    
    dataset[\'test_img\'] = _load_img(key_file[\'test_img\'])
    dataset[\'test_label\'] = _load_label(key_file[\'test_label\'])
    
    return dataset

def init_mnist():
    download_mnist()
    dataset = _convert_numpy()
    print(\"Creating pickle file ...\")
    with open(save_file, \'wb\') as f:
        pickle.dump(dataset, f, -1)
    print(\"Done!\")

def _change_one_hot_label(X):
    T = np.zeros((X.size, 10))
    for idx, row in enumerate(T):
        row[X[idx]] = 1
        
    return T
    

def load_mnist(normalize=True, flatten=True, one_hot_label=False):
    \"\"\"读入MNIST数据集
    
    Parameters
    ----------
    normalize : 将图像的像素值正规化为0.0~1.0
    one_hot_label : 
        one_hot_label为True的情况下，标签作为one-hot数组返回
        one-hot数组是指[0,0,1,0,0,0,0,0,0,0]这样的数组
    flatten : 是否将图像展开为一维数组
    
    Returns
    -------
    (训练图像, 训练标签), (测试图像, 测试标签)
    \"\"\"
    if not os.path.exists(save_file):
        init_mnist()
        
    with open(save_file, \'rb\') as f:
        dataset = pickle.load(f)
    
    if normalize:
        for key in (\'train_img\', \'test_img\'):
            dataset[key] = dataset[key].astype(np.float32)
            dataset[key] /= 255.0
            
    if one_hot_label:
        dataset[\'train_label\'] = _change_one_hot_label(dataset[\'train_label\'])
        dataset[\'test_label\'] = _change_one_hot_label(dataset[\'test_label\'])
    
    if not flatten:
         for key in (\'train_img\', \'test_img\'):
            dataset[key] = dataset[key].reshape(-1, 1, 28, 28)

    return (dataset[\'train_img\'], dataset[\'train_label\']), (dataset[\'test_img\'], dataset[\'test_label\']) 


if __name__ == \'__main__\':
    init_mnist()

使用上述文件中的函数就可以直接得到手写数字的训练数据、训练标签，测试样本以及测试标签。
接下里使用如下代码就可以进行手写数字的训练，代码如下：

import numpy as np
from numpy.lib.function_base import select
from dataset.mnist import load_mnist
import matplotlib.pylab as plt


def sigmoid(x):
    return 1 / (1 + np.exp(-x))  

def sigmoid_grad(x):
    return (1.0 - sigmoid(x)) * sigmoid(x)

def softmax(x):
    if x.ndim == 2:
        x = x.T
        x = x - np.max(x, axis=0)
        y = np.exp(x) / np.sum(np.exp(x), axis=0)
        return y.T 

    x = x - np.max(x) # 溢出对策
    return np.exp(x) / np.sum(np.exp(x))

def cross_entropy_error(y, t):
    if y.ndim == 1:
        t = t.reshape(1, t.size)
        y = y.reshape(1, y.size)
        
    # 监督数据是one-hot-vector的情况下，转换为正确解标签的索引
    if t.size == y.size:
        t = t.argmax(axis=1)
             
    batch_size = y.shape[0]
    return -np.sum(np.log(y[np.arange(batch_size), t] + 1e-7)) / batch_size

def numerical_gradient(f, x):
    h = 1e-4 # 0.0001
    grad = np.zeros_like(x)
    
    it = np.nditer(x, flags=[\'multi_index\'], op_flags=[\'readwrite\'])
    while not it.finished:
        idx = it.multi_index
        tmp_val = x[idx]
        x[idx] = float(tmp_val) + h
        fxh1 = f(x) # f(x+h)
        
        x[idx] = tmp_val - h 
        fxh2 = f(x) # f(x-h)
        grad[idx] = (fxh1 - fxh2) / (2*h)
        
        x[idx] = tmp_val # 还原值
        it.iternext()   
        
    return grad

#(x_train,t_train),(x_test,t_test)=load_mnist(normalize=True,one_hot_label=True)
#两层神经网络的类
class TwoLayerNet:
    def __init__(self,input_size,hidden_size,output_size,weight_init_std=0.01):
        #初始化权重
        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)
    
    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
    def loss(self,x,t):
        y=self.predict(x)
        return cross_entropy_error(y,t)
    #数值微分法
    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

    #误差反向传播法
    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)

        return grads
    #准确率
    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

if __name__==\'__main__\':
    (x_train,t_train),(x_test,t_test)=load_mnist(normalize=True,one_hot_label=True)
    net=TwoLayerNet(input_size=784,hidden_size=50,output_size=10)

    train_loss_list=[]

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

    #记录准确率
    train_acc_list=[]
    test_acc_list=[]
    #平均每个epoch的重复次数
    iter_per_epoch=max(train_size/batch_size,1)

    for i in range(iter_nums):
        #小批量数据
        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
        #数值微分 计算很慢
        #grad=net.numerical_gradient(x_batch,t_batch)
        #误差反向传播法 计算很快
        grad=net.gradient(x_batch,t_batch)

        #更新参数 权重W和偏重b
        for key in [\'W1\',\'b1\',\'W2\',\'b2\']:
            net.params[key]-=learning_rate*grad[key]
        
        #记录学习过程
        loss=net.loss(x_batch,t_batch)
        print(\'训练次数:\'+str(i)+\'    loss:\'+str(loss))
        train_loss_list.append(loss)

        #计算每个epoch的识别精度
        if i%iter_per_epoch==0:
            #测试在所有训练数据和测试数据上的准确率
            train_acc=net.accuracy(x_train,t_train)
            test_acc=net.accuracy(x_test,t_test)
            train_acc_list.append(train_acc)
            test_acc_list.append(test_acc)
            print(\'train acc:\'+str(train_acc)+\'   test acc:\'+str(test_acc))
    
    print(train_acc_list)
    print(test_acc_list)

    # 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()

训练完成后，查看绘制准确率的图片，可以获取到成功实现了手写数字识别。

随着训练批次的增加，准确率逐渐增大接近于1，说明训练过程按着正确拟合的方向前进。