发表更新动手学深度学习20 分钟读完 (大约3052个字)

3.线性神经网络

3.1 线性回归

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
%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()
    
1
2
3
4
5
c=torch.zeros(n)
timer=Timer()
for i in range(n):
    c[i]=a[i]+b[i]
f'{timer.stop():.5f}sec'
'0.14660sec'

1
2
3
timer.start()
d=a+b
f'{timer.stop():.5f}sec'
'0.00000sec'

1
2
3
4
5
6
7
8
9
10
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])

​
svg
​

3.2 线性回归的从零开始实现

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
%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])

1
2
d2l.set_figsize()
d2l.plt.scatter(features[:,1].detach().numpy(),labels.detach().numpy(),1);

​
svg
​ 

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
#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]])

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
#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

1
2
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

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
#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]])]

1
2
3
4
5
6
#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.]))

1
2
3
4
#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)
1
2
3
4
5
6
7
8
9
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

1
2
3
4
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])

3.5 图像分类数据集

1
2
3
4
5
6
7
8
%matplotlib inline
import torch
import torchvision
from torch.utils import data
from torchvision import transforms
from d2l import torch as d2l

d2l.use_svg_display()
1
2
3
4
5
#通过框架内内置函数将Fashion-MNIST数据集下载并读取到内存中
#通过ToTensor实例将图像数据由PIL类型变换为32位浮点数格式，并除以255使得所有像素数值均在0-1之间
trans=transforms.ToTensor()
mnist_train=torchvision.datasets.FashionMNIST(root="../data",train=True,transform=trans,download=True)
mnist_test=torchvision.datasets.FashionMNIST(root="../data",train=False,transform=trans,download=True)
1
len(mnist_train),len(mnist_test)
(60000, 10000)

1
mnist_train[0][0].shape
torch.Size([1, 28, 28])

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
#在数字标签索引与文本名称之间进行转换
def get_fashion_mnist_labels(labels): #@save
    #返回Fashion-MNIST数据集的文本标签
    text_labels=['t-shirt','trouser','pullover','dress','coat','sandal','shirt','sneaker','bag','ankle boot']
    return [text_labels[int(i)] for i in labels]

#可视化样本
def show_images(imgs,num_rows,num_cols,titles=None,scale=1.5): #@save
    #绘制图像列表
    figsize=(num_cols*scale,num_rows*scale)
    _,axes=d2l.plt.subplots(num_rows,num_cols,figsize=figsize)
    axes=axes.flatten()
    for i,(ax,img) in enumerate(zip(axes,imgs)):
        if torch.is_tensor(img):
            #图片张量
            ax.imshow(img.numpy())
        else:
            #PIL图片
            ax.imshow(img)
        ax.axes.get_xaxis().set_visible(False)
        ax.axes.get_yaxis().set_visible(False)
        if titles:
            ax.set_title(titles[i])
    return axes

X,y=next(iter(data.DataLoader(mnist_train,batch_size=50)))
show_images(X.reshape(50,28,28),10,5,titles=get_fashion_mnist_labels(y));

​
svg
​ 

1
2
3
4
5
6
7
8
9
10
11
12
13
14
#读取小批量
batch_size=256 #批量大小

def get_dataloader_workers():  #@save
    #使用4个进程来读取数据
    return 4

train_iter=data.DataLoader(mnist_train,batch_size,shuffle=True,num_workers=get_dataloader_workers())

#读取训练数据所需时间
timer=d2l.Timer()
for X,y in train_iter:
    continue
f'{timer.stop():.2f}sec'
'2.11sec'

1
2
3
4
5
6
7
8
9
10
11
12
13
14
def load_data_fashion_mnist(batch_size,resize=None): #@save
    #下载Fashion-MNIST数据集，然后将其加载到内存中
    trans=[transforms.ToTensor()]
    if resize:
        trans.insert(0,transforms.Resize(resize))
    trans=transforms.Compose(trans)
    mnist_train=torchvision.datasets.FashionMNIST(root="../data",train=True,transform=trans,download=True)
    mnist_test=torchvision.datasets.FashionMNIST(root="../data",train=False,transform=trans,download=True)
    return(data.DataLoader(mnist_train,batch_size,shuffle=True,num_workers=get_dataloader_workers()),data.DataLoader(mnist_test,batch_size,shuffle=False,num_workers=get_dataloader_workers()))

train_iter,test_iter=load_data_fashion_mnist(32,resize=64)
for X,y in train_iter:
    print(X.shape,X.dtype,y.shape,y.dtype)
    break
torch.Size([32, 1, 64, 64]) torch.float32 torch.Size([32]) torch.int64

3.6 softmax回归的从零开始实现

1
2
3
4
5
6
import torch
from IPython import display
from d2l import torch as d2l

batch_size=256
train_iter,test_iter=d2l.load_data_fashion_mnist(batch_size)
1
2
3
4
5
6
#初始化模型参数
num_inputs=784
num_outputs=10

W=torch.normal(0,0.01,size=(num_inputs,num_outputs),requires_grad=True)#用正态分布初始化权重W
b=torch.zeros(num_outputs,requires_grad=True)#偏置b初始化为0
1
2
X=torch.tensor([[1.0,2.0,3.0],[4.0,5.0,6.0]])
X.sum(0,keepdim=True),X.sum(1,keepdim=True)
(tensor([[5., 7., 9.]]),
 tensor([[ 6.],
         [15.]]))

1
2
3
4
5
6
7
8
9
#定义softmax操作
def softmax(X):
    X_exp=torch.exp(X)
    partition=X_exp.sum(1,keepdim=True)
    return X_exp/partition #应用广播机制

X=torch.normal(0,1,(2,5))
X_prob=softmax(X)
X_prob,X_prob.sum(1)
(tensor([[0.2143, 0.0127, 0.1268, 0.2248, 0.4214],
         [0.3360, 0.2826, 0.0913, 0.1454, 0.1446]]),
 tensor([1.0000, 1.0000]))

1
2
3
#定义模型
def net(X):
    return softmax(torch.matmul(X.reshape((-1,W.shape[0])),W)+b)   #使用reshape函数将每张原始图像展平为向量
1
2
3
y=torch.tensor([0,2])
y_hat=torch.tensor([[0.1,0.3,0.6],[0.3,0.2,0.5]])
y_hat[[0,1],y]
tensor([0.1000, 0.5000])

1
2
3
4
5
#定义交叉熵损失函数
def cross_entropy(y_hat,y):
    return - torch.log(y_hat[range(len(y_hat)),y])

cross_entropy(y_hat,y)
tensor([2.3026, 0.6931])

1
2
3
4
5
6
7
8
def accuracy(y_hat,y): #@save
    #计算正确预测的数量
    if len(y_hat.shape)>1 and y_hat.shape[1]>1:
        y_hat=y_hat.argmax(axis=1)
    cmp=y_hat.type(y.dtype)==y
    return float(cmp.type(y.dtype).sum())

accuracy(y_hat,y)/len(y)  #正确预测的概率（分类精度率）
0.5

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
def evaluate_accuracy(net,data_iter): #@save
    #计算在指定数据集上模型的精度
    if isinstance(net,torch.nn.Module):
        net.eval() #将模型设置为评估模式
    metric=Accumulator(2) #正确预测数、预测总数的叠加
    with torch.no_grad():
        for X,y in data_iter:
            metric.add(accuracy(net(X),y),y.numel())  #accracy(net(X),y):正确预测数；y.numel()预测总数
    return metric[0]/metric[1]

class Accumulator: #@save
    #在n个变量上累加
    def __init__(self,n):
        self.data=[0.0]*n
    
    def add(self,*args):
        self.data=[a+float(b) for a,b in zip(self.data,args)]
        
    def reset(self):
        self.data=[0.0]*len(self.data)
        
    def __getitem__(self,idx):
        return self.data[idx]
    
evaluate_accuracy(net,test_iter)
0.1326

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
#训练
def train_epoch_ch3(net,train_iter,loss,updater): #@save
    #训练模型一个迭代周期
    #将模型设置为训练模型
    if isinstance(net,torch.nn.Module):
        net.train()
    #训练损失总和、训练准确度总和、样本数
    metric=Accumulator(3)
    for X,y in train_iter:
        #计算梯度并更新参数
        y_hat=net(X)
        l=loss(y_hat,y)
        if isinstance(updater,torch.optim.Optimizer):
            #使用PyTorch内置的优化器和损失函数
            updater.zero_grad()
            l.mean().backward()
            updater.step()
        else:
            #使用定制的优化器和损失函数
            l.sum().backward()
            updater(X.shape[0])
        metric.add(float(l.sum()),accuracy(y_hat,y),y.numel())
    #返回训练损失和训练精度
    return metric[0]/metric[2],metric[1]/metric[2]

class Animator:  #@save
    #在动画中绘制数据
    def __init__(self, xlabel=None, ylabel=None, legend=None, xlim=None,ylim=None, xscale='linear', yscale='linear',fmts=('-', 'm--', 'g-.', 'r:'), nrows=1, ncols=1,figsize=(3.5, 2.5)):
        # 增量地绘制多条线
        if legend is None:
            legend = []
        d2l.use_svg_display()
        self.fig, self.axes = d2l.plt.subplots(nrows, ncols, figsize=figsize)
        if nrows * ncols == 1:
            self.axes = [self.axes, ]
        # 使用lambda函数捕获参数
        self.config_axes = lambda: d2l.set_axes(
            self.axes[0], xlabel, ylabel, xlim, ylim, xscale, yscale, legend)
        self.X, self.Y, self.fmts = None, None, fmts

    def add(self, x, y):
        # 向图表中添加多个数据点
        if not hasattr(y, "__len__"):
            y = [y]
        n = len(y)
        if not hasattr(x, "__len__"):
            x = [x] * n
        if not self.X:
            self.X = [[] for _ in range(n)]
        if not self.Y:
            self.Y = [[] for _ in range(n)]
        for i, (a, b) in enumerate(zip(x, y)):
            if a is not None and b is not None:
                self.X[i].append(a)
                self.Y[i].append(b)
        self.axes[0].cla()
        for x, y, fmt in zip(self.X, self.Y, self.fmts):
            self.axes[0].plot(x, y, fmt)
        self.config_axes()
        display.display(self.fig)
        display.clear_output(wait=True)
        
def train_ch3(net,train_iter,test_iter,loss,num_epochs,updater): #@save
    #训练模型
    animator=Animator(xlabel='epoch',xlim=[1,num_epochs],ylim=[0.3,0.9],legend=['train loss','train acc','test acc'])
    for epoch in range(num_epochs):
        train_metrics=train_epoch_ch3(net,train_iter,loss,updater)
        test_acc=evaluate_accuracy(net,test_iter)
        animator.add(epoch+1,train_metrics+(test_acc,))
    train_loss,train_acc=train_metrics
    assert train_loss<0.5,train_loss
    assert train_acc<=1 and train_acc>0.7,train_acc
    assert test_acc<=1 and test_acc>0.7,test_acc
    
lr=0.1 #学习率
def updater(batch_size):
    return d2l.sgd([W,b],lr,batch_size)#小批量随机梯度下降

num_epochs=10 #迭代周期
train_ch3(net,train_iter,test_iter,cross_entropy,num_epochs,updater)

​
svg
​ 

1
2
3
4
5
6
7
8
9
10
11
#预测
def predict_ch3(net,test_iter,n=6): #@save
    #预测标签
    for X,y in test_iter:
        break
    trues=d2l.get_fashion_mnist_labels(y)
    preds=d2l.get_fashion_mnist_labels(net(X).argmax(axis=1))
    titles=[true+'\n'+pred for true,pred in zip(trues,preds)]
    d2l.show_images(X[0:n].reshape((n,28,28)),1,n,titles=titles[0:n])
    
predict_ch3(net,test_iter)

​
svg
​

3.7 softmax回归的简洁实现

1
2
3
4
5
6
import torch
from torch import nn
from d2l import torch as d2l

batch_size=256
train_iter,test_iter=d2l.load_data_fashion_mnist(batch_size)
1
2
3
4
5
6
7
8
9
#初始化模型参数
#PyTorch不会隐式地调整输入的形状，因此，我们在线性层前定义了展平层（flatten),来调整网络输入的形状
net=nn.Sequential(nn.Flatten(),nn.Linear(784,10))

def init_weights(m):
    if type(m)==nn.Linear:
        nn.init.normal_(m.weight,std=0.01)

net.apply(init_weights);
1
loss=nn.CrossEntropyLoss(reduction='none') #保留softmax函数，但在计算交叉熵损失函数时传递未规范化的预测并同时计算softmax及其对数以防止数值上溢或下溢
1
2
#优化算法：学习度为0.1的小批量随机梯度下降
trainer=torch.optim.SGD(net.parameters(),lr=0.1)
1
2
3
#训练
num_epochs=10
d2l.train_ch3(net,train_iter,test_iter,loss,num_epochs,trainer)

​
svg
​

3.线性神经网络

http://asgard-tim.github.io/2023/10/05/3.线性神经网络/

作者

Tim

发布于

2023-10-05

更新于

2025-04-30

许可协议

#

喜欢这篇文章？打赏一下作者吧

评论

目录

关注我

最新文章

分类

归档

标签

单片机5
STM325
VLA4
具身智能4
多模态大模型4
C++4
Python4
斯特林发动机4
MATLAB3
Matlab3
机器学习3
动手学深度学习2
学习笔记2
小车2
信号与系统2
频谱分析2
斯特林循环2
动力学仿真2
有限元仿真2
算法与数据结构2
3D-VLA1
PaLM-E1
路径规划1
RANSAC1
ROS1
×