Homepage of Jinghua Xu

3.1 线性回归

%matplotlib inline
import math
import time
import numpy as np
import torch
from d2l import torch as d2l

n=10000
a=torch.ones([n])
b=torch.ones([n])

#定义计时器
class Timer: #@save
    #记录多次运行时间
    def __init__(self):
        self.times=[]
        self.start()
        
    def start(self):
        #启动计时器
        self.tik=time.time()
    
    def stop(self):
        #停止计时器并将时间记录在列表中
        self.times.append(time.time()-self.tik)
        return self.times[-1]
    
    def avg(self):
        #返回平均时间
        return sum(self.times)/len(self.times)
    
    def sum(self):
        #返回时间总和
        return sum(self.times)
    
    def cumsum(self):
        #返回累计时间
        return np.array(self.times).cumsum().tolist()
    

c=torch.zeros(n)
timer=Timer()
for i in range(n):
    c[i]=a[i]+b[i]
f'{timer.stop():.5f}sec'

'0.14660sec'

timer.start()
d=a+b
f'{timer.stop():.5f}sec'

'0.00000sec'

def normal(x,mu,sigma):
    p=1/math.sqrt(2*math.pi*sigma**2)
    return p*np.exp(-0.5/sigma**2*(x-mu)**2)

#再次使用numpy进行可视化
x=np.arange(-7,7,0.01)

#均值和标准差对
params=[(0,1),(0,2),(3,1)]
d2l.plot(x,[normal(x,mu,sigma)for mu,sigma in params],xlabel='x',ylabel='p(x)',figsize=(4.5,2.5),legend=[f'mean{mu},std{sigma}'for mu,sigma in params])

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3.2 线性回归的从零开始实现

%matplotlib inline
import random
import torch
from d2l import torch as d2l

#生成数据集
def synthetic_data(w,b,num_examples): #@save
    #生成y=Xw+b+噪声
    X=torch.normal(0,1,(num_examples,len(w)))
    y=torch.matmul(X,w)+b
    y+=torch.normal(0,0.01,y.shape)
    return X,y.reshape((-1,1))

true_w=torch.tensor([2,-3.4])
true_b=4.2
features,labels=synthetic_data(true_w,true_b,1000)

print('features:',features[0],'\nlabel:',labels[0])

features: tensor([ 0.7328, -0.5520]) 
label: tensor([7.5513])

d2l.set_figsize()
d2l.plt.scatter(features[:,1].detach().numpy(),labels.detach().numpy(),1);

​

​

#读取数据集
def data_iter(batch_size,features,labels):
    num_examples=len(features)
    indices=list(range(num_examples))
    #这些样本是随机读取的，没有特定顺序
    random.shuffle(indices)
    for i in range(0,num_examples,batch_size):
        batch_indices=torch.tensor(indices[i:min(i+batch_size,num_examples)])
        yield features[batch_indices],labels[batch_indices]
        
batch_size=10

for X,y in data_iter(batch_size,features,labels):
    print(X,'\n',y)
    break

tensor([[ 0.9175, -0.1441],
        [-0.3328, -0.4237],
        [-0.1287,  1.6801],
        [ 0.8705, -0.9030],
        [-0.4966,  1.4015],
        [ 1.3378, -1.8026],
        [ 0.5129,  1.2806],
        [ 1.1026,  1.2080],
        [ 0.6151,  0.6337],
        [-0.4683, -0.4388]]) 
 tensor([[ 6.5336],
        [ 4.9801],
        [-1.7645],
        [ 9.0089],
        [-1.5652],
        [12.9979],
        [ 0.8680],
        [ 2.2893],
        [ 3.2902],
        [ 4.7695]])

#初始化模型参数
w=torch.normal(0,0.01,size=(2,1),requires_grad=True)
b=torch.zeros(1,requires_grad=True)

#定义模型
def linreg(X,w,b): #@save
    #线性回归模型
    return torch.matmul(X,w)+b

#定义损失函数
def squared_loss(y_hat,y): #@save
    #均方损失
    return(y_hat-y.reshape(y_hat.shape))**2/2

#定义优化算法——小批量随机梯度下降
#lr:学习速率(梯度下降步长)；batch_size:批量大小
def sgd(params,lr,batch_size): #@save
    #小批量随机梯度下降
    with torch.no_grad():
        for param in params:
            param-=lr*param.grad/batch_size
            param.grad.zero_()
            
#训练
#设置超参数
lr=0.03#学习率
num_epochs=3#迭代周期个数
net=linreg
loss=squared_loss

for epoch in range(num_epochs):
    for X,y in data_iter(batch_size,features,labels):
        l=loss(net(X,w,b),y)#X和y的小批量损失
        #因为l的形状是（batch_size,1），而不是一个标量。l中的所有元素被加到一起，并以此计算关于[w,b]的梯度
        l.sum().backward()
        sgd([w,b],lr,batch_size)#使用参数的梯度以更新参数
    with torch.no_grad():
        train_l=loss(net(features,w,b),labels)
        print(f'epoch{epoch+1},loss{float(train_l.mean()):f}')

epoch1,loss0.036941
epoch2,loss0.000134
epoch3,loss0.000049

print(f'w的估计误差：{true_w-w.reshape(true_w.shape)}')
print(f'b的估计误差：{true_b-b}')

w的估计误差：tensor([ 0.0002, -0.0002], grad_fn=<SubBackward0>)
b的估计误差：tensor([0.0002], grad_fn=<RsubBackward1>)

3.3 线性回归的简洁实现

#生成数据集
import numpy as np
import torch
from torch.utils import data
from d2l import torch as d2l

true_w=torch.tensor([2,-3.4])
true_b=4.2
features,labels=d2l.synthetic_data(true_w,true_b,1000)

#读取数据集
def load_array(data_arrays,batch_size,is_train=True): #@save
    #构造一个pytorch数据迭代器
    dataset=data.TensorDataset(*data_arrays)
    return data.DataLoader(dataset,batch_size,shuffle=is_train)

batch_size=10
data_iter=load_array((features,labels),batch_size)

next(iter(data_iter))

[tensor([[-0.6422, -0.7470],
         [-2.1785,  0.3340],
         [ 1.4011, -1.1104],
         [-0.8083, -0.3035],
         [ 0.1077, -0.1201],
         [-0.4151, -0.1079],
         [ 1.8074,  0.0904],
         [-0.3707,  0.6197],
         [ 0.3739,  0.2972],
         [ 0.2383,  1.1791]]),
 tensor([[ 5.4457],
         [-1.3122],
         [10.7837],
         [ 3.6281],
         [ 4.8105],
         [ 3.7284],
         [ 7.5017],
         [ 1.3465],
         [ 3.9247],
         [ 0.6800]])]

#定义模型
from torch import nn#nn:神经网络
net=nn.Sequential(nn.Linear(2,1))

#初始化模型参数
net[0].weight.data.normal_(0,0.01),net[0].bias.data.fill_(0)

(tensor([[ 0.0106, -0.0055]]), tensor([0.]))

#定义损失函数
loss=nn.MSELoss()#均方误差：MSELoss类，平方L2范数
#定义优化算法
trainer=torch.optim.SGD(net.parameters(),lr=0.03)

num_epochs=3#迭代周期个数
for epoch in range(num_epochs):
    for X,y in data_iter:
        l=loss(net(X),y)#X和y的小批量损失
        trainer.zero_grad()
        l.backward()
        trainer.step()#使用参数的梯度以更新参数
    l=loss(net(features),labels)
    print(f'epoch{epoch+1},loss{l:f}')

epoch1,loss0.000223
epoch2,loss0.000112
epoch3,loss0.000112

w=net[0].weight.data
print('w的估计误差：',true_w-w.reshape(true_w.shape))
b=net[0].bias.data
print('b的估计误差：',true_b-b)

w的估计误差： tensor([ 0.0014, -0.0004])
b的估计误差： tensor([0.0008])