手寫數(shù)字識別作為深度學習入門經(jīng)典的識別案例，各種深度學習框架都有這個例子的實現(xiàn)方法。我這里將不用任何深度學習現(xiàn)有框架，例如TensorFlow、Keras、pytorch，直接使用Python語言的numpy實現(xiàn)各種激活函數(shù)、損失函數(shù)、梯度下降的方法。

程序分為兩部分，首先是手寫數(shù)字數(shù)據(jù)的準備，直接使用如下mnist.py文件中的方法load_minist即可。文件代碼如下：

# 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):
    """讀入MNIST數(shù)據(jù)集
    
    Parameters
    ----------
    normalize : 將圖像的像素值正規(guī)化為0.0~1.0
    one_hot_label : 
        one_hot_label為True的情況下，標簽作為one-hot數(shù)組返回
        one-hot數(shù)組是指[0,0,1,0,0,0,0,0,0,0]這樣的數(shù)組
    flatten : 是否將圖像展開為一維數(shù)組
    
    Returns
    -------
    (訓練圖像, 訓練標簽), (測試圖像, 測試標簽)
    """
    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數(shù)就可以直接得到手寫數(shù)字的訓練數(shù)據(jù)、訓練標簽，測試樣本以及測試標簽。
接下里使用如下代碼就可以進行手寫數(shù)字的訓練，代碼如下：

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) # 溢出對策
    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)
        
    # 監(jiān)督數(shù)據(jù)是one-hot-vector的情況下，轉(zhuǎn)換為正確解標簽的索引
    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 # 還原值
        it.iternext()   
        
    return grad

#(x_train,t_train),(x_test,t_test)=load_mnist(normalize=True,one_hot_label=True)
#兩層神經(jīng)網(wǎng)絡(luò)的類
class TwoLayerNet:
    def __init__(self,input_size,hidden_size,output_size,weight_init_std=0.01):
        #初始化權(quán)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
    #損失函數(shù)
    def loss(self,x,t):
        y=self.predict(x)
        return cross_entropy_error(y,t)
    #數(shù)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

    #誤差反向傳播法
    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
    #準確率
    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=[]

    #超參數(shù)
    iter_nums=10000
    train_size=x_train.shape[0]
    batch_size=100
    learning_rate=0.1

    #記錄準確率
    train_acc_list=[]
    test_acc_list=[]
    #平均每個epoch的重復次數(shù)
    iter_per_epoch=max(train_size/batch_size,1)

    for i in range(iter_nums):
        #小批量數(shù)據(jù)
        batch_mask=np.random.choice(train_size,batch_size)
        x_batch=x_train[batch_mask]
        t_batch=t_train[batch_mask]

        #計算梯度
        #數(shù)值微分 計算很慢
        #grad=net.numerical_gradient(x_batch,t_batch)
        #誤差反向傳播法 計算很快
        grad=net.gradient(x_batch,t_batch)

        #更新參數(shù) 權(quán)重W和偏重b
        for key in ['W1','b1','W2','b2']:
            net.params[key]-=learning_rate*grad[key]
        
        #記錄學習過程
        loss=net.loss(x_batch,t_batch)
        print('訓練次數(shù):'+str(i)+'    loss:'+str(loss))
        train_loss_list.append(loss)

        #計算每個epoch的識別精度
        if i%iter_per_epoch==0:
            #測試在所有訓練數(shù)據(jù)和測試數(shù)據(jù)上的準確率
            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)

    # 繪制圖形
    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()

訓練完成后，查看繪制準確率的圖片，可以獲取到成功實現(xiàn)了手寫數(shù)字識別。

在這里插入圖片描述