线性回归的从零开始：

对应目录：notebooks/chapter_linear-networks/linear-regression-scratch.ipynb

练习一：

import random
import torch
from d2l import torch as d2l

def synthetic_data(w,b,num_examples):
    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)

#生成数据集
W_true = torch.tensor([2, -3.4])
b_true = 4.2

features, labels = synthetic_data(W_true, b_true,100)

def data_iter(batch_size, features, labels):
    num_examples = len(features)
    indices = list(range(num_examples))
    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]

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

def linreg(X,w,b):
    return torch.matmul(X, w) + b

def squared_loss(y_hat, y):
    return (y_hat - y.reshape(y_hat.shape)) ** 2 / 2

def sgd(params, lr, batch_size):
    with torch.no_grad():
        for param in params:
            param -= lr * param.grad / batch_size
            param.grad.zero_()
            
lr = 0.03
num_epochs = 3
net = linreg
loss = squared_loss
batch_size = 2
for epoch in range(num_epochs):
    for X, y in data_iter(batch_size, features, labels):
        los = loss(net(X,w,b), y)
        los.sum().backward()
        sgd([w,b], lr, batch_size)
    
    with torch.no_grad():
        train_l = loss(net(features,w,b), labels)
        print(f'epoch {epoch + 1}, loss{float(train_l.mean()):f}')
        
print(f'w的估计误差：{W_true - w.reshape(W_true.shape)}')  
print(f'b的估计误差：{b_true - b}')       
        
    

可以正常运行，但是因为这是单层线性模型，在多层神经网络中，这种初始化方法是有问题的。

练习二：

 可以，U = IR，构建线性模型，设定R为需要学习的参数即可：

 import random
import torch
from d2l import torch as d2l


def synthetic_data(R,num_examples):
    X = torch.normal(0,1,(num_examples, 1))
    Y = torch.matmul(X,R) 
    Y += torch.normal(0,0.01, Y.shape)
    return X, Y.reshape(-1,1)

#生成数据集
R_true = torch.tensor([[10.0]])

features, labels = synthetic_data(R_true,1000)

def data_iter(batch_size, features, labels):
    num_examples = len(features)
    indices = list(range(num_examples))
    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]

R = torch.normal(0,0.01,size = (1,1), requires_grad = True)

def linreg(X,R):
    return torch.matmul(X, R)

def squared_loss(y_hat, y):
    return (y_hat - y.reshape(y_hat.shape)) ** 2 / 2

def sgd(params, lr, batch_size):
    with torch.no_grad():
        for param in params:
            param -= lr * param.grad / batch_size
            param.grad.zero_()
            
lr = 0.03
num_epochs = 3
net = linreg
loss = squared_loss
batch_size = 2
for epoch in range(num_epochs):
    for X, y in data_iter(batch_size, features, labels):
        los = loss(net(X,R), y)
        los.sum().backward()
        sgd([R], lr, batch_size)
    
    with torch.no_grad():
        train_l = loss(net(features,R), labels)
        print(f'epoch {epoch + 1}, loss{float(train_l.mean()):f}')
        
print(f'R的估计误差：{R_true - R}')  
     

#运行结果
epoch 1, loss0.000050
epoch 2, loss0.000053
epoch 3, loss0.000051
R的估计误差：tensor([[0.0012]], grad_fn=<SubBackward0>)

练习三：

可以，但是没写了。

练习四：

二阶导数需要在一阶导数的基础上求导，第一次反向传播之后会释放计算图，再次.backward()会报错，可以在loss.backward(create_graph = True)从而保留图结构以支持二阶求导。

练习五：

为了统一y_hat和y的形状，使其可以按元素对应计算误差。

练习六：

学习率太大会震荡，学习率太小会使得优化过程慢。

练习七：

不影响，因为data_iter中 indices[i:min(i+batch_size, num_examples)]，即使不能整除，最后一个batch仍然会正常运行，只是最后一个batch的样本数少于batch_size。

 线性回归的简洁实现

对应地址：notebooks/chapter_linear-networks/linear-regression-concise.ipynb

练习一：

从.sum替换为.mean梯度大小由原先的大小缩小为原先的1/n倍，所以梯度需要乘以批次数。（原先更新公式可以视为*batch_size,求导之后batch_size依然存在，现在.mean，相当于失去了batch_size项，为了保证更新梯度和原先相同，所以需要*batch_size）

练习二：

HuberLoss — PyTorch 2.2 documentation

损失函数对应文档。页面左上角是对应的pytorch版本，需要和自己使用的Pytorch版本相符合

import numpy as np
import torch
from torch.utils import data
from d2l import torch as d2l
from torch import nn

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

def load_array(data_arrays, batch_size, is_train = True): #@save
    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)

net = nn.Sequential(nn.Linear(2,1))

net[0].weight.data.normal_(0, 0.01)
net[0].bias.data.fill_(0)

loss = nn.HuberLoss(reduction='mean', delta=1.0) #更换为HUber损失函数


trainer = torch.optim.SGD(net.parameters(), lr = 0.03)

num_epochs = 3
for epoch in range(num_epochs):
    for X,y in data_iter:
        l = loss(net(X), y)
        trainer.zero_grad()
        l.backward()
        trainer.step()
    l = loss(net(features), labels)
    print(f"epoch {epoch + 1}, loss:{l:f}")

###运行结果###
epoch 1, loss:2.200701
epoch 2, loss:0.523420
epoch 3, loss:0.003929

练习三：

net[0].weight.data #权重数值
net[0].weight.grad #权重梯度

net[0].bias.data #权重数值
net[0].bias.grad #权重梯度