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。
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