纯numpy数值微分法实现手写数字识别纯numpy数值微分法实现手写数字

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-8try:import urllib.requestexcept ImportError:raise ImportError('You should use Python 3.x')import os.pathimport gzipimport pickleimport osimport numpy as npurl_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 = 60000test_num = 10000img_dim = (1, 28, 28)img_size = 784def _download(file_name):file_path = dataset_dir + "/" + file_nameif os.path.exists(file_path):returnprint("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_nameprint("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 labelsdef _load_img(file_name):file_path = dataset_dir + "/" + file_nameprint("Converting " + file_name + " to NumPy Array ...")with gzip.open(file_path, 'rb') as f:data = https://www.it610.com/article/np.frombuffer(f.read(), np.uint8, offset=16)data = data.reshape(-1, img_size)print("Done")return datadef _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 datasetdef 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]] = 1return Tdef 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0.0~1.0one_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] /= 255.0if 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 npfrom numpy.lib.function_base import selectfrom dataset.mnist import load_mnistimport matplotlib.pylab as pltdef 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.Tx = 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_sizedef numerical_gradient(f, x):h = 1e-4 # 0.0001grad = np.zeros_like(x)it = np.nditer(x, flags=['multi_index'], op_flags=['readwrite'])while not it.finished:idx = it.multi_indextmp_val = x[idx]x[idx] = float(tmp_val) + hfxh1 = 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)+b1z1=sigmoid(a1)a2=np.dot(z1,W2)+b2y=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]# forwarda1 = np.dot(x, W1) + b1z1 = sigmoid(a1)a2 = np.dot(z1, W2) + b2y = softmax(a2)# backwarddy = (y - t) / batch_numgrads['W2'] = np.dot(z1.T, dy)grads['b2'] = np.sum(dy, axis=0)da1 = np.dot(dy, W2.T)dz1 = sigmoid_grad(a1) * da1grads['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 accuracyif __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=10000train_size=x_train.shape[0]batch_size=100learning_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bfor 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。

文章图片

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。
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