1. 线性回归
2. 基础优化方法

1. 线性回归
总结

• 线性回归是对n维输入的加权，外加偏差
• 使用平方损失来衡量预测值和真实值的差异
• 线性回归有显示解
• 线性回归可以看作是单层神经网络

2. 基础优化方法
梯度下降

小批量随机梯度下降

3. 总结

• 梯度下降通过不断沿着反梯度方向更新参数求解
• 小批量随机梯度下降是深度学习默认的求解算法
• 两个重要的超参数是批量大小和学习率

4. 线性回归的从零开始实现

import torch
import random
def synthetic_data(w, b, num):
    "生成 y = Xw + b + 噪声"
    ''' 根据带有噪声的线性模型构造一个人造数据集。 我们使用线性模型参数𝐰 = [2,−3.4]⊤
    w=[2,−3.4]⊤、𝑏 = 4.2和噪声项𝜖ϵ生成数据集及其标签：𝐲 = 𝐗𝐰 + 𝑏 + 𝜖'''
    X = torch.normal(0, 1, (num, 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
num_examples = 1000
batch_size = 10
features, labels = synthetic_data(true_w, true_b, num_examples)

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):
        num_indices = torch.tensor(indices[i: min(i +
                                                  batch_size, num_examples)])
        yield features[num_indices], labels[num_indices]

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_()

w = torch.normal(0, 0.01, size=(2, 1), requires_grad=True)
b = torch.zeros(1, requires_grad=True)

loss = squared_loss
net = linreg
lr = 0.03
epoches = 3

for epoch in range(epoches):
    for X, y in data_iter(batch_size, features, labels):
        l = loss(net(X, w, b), y)
        l.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}')

5. 线性回归的简洁实现

import torch
from d2l import torch as d2l
from torch.utils import data

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

def load_array(batch_size, data_array, is_train=True):
    dataset = data.TensorDataset(*data_array)
    return data.DataLoader(dataset, batch_size, shuffle=is_train)

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

from torch import nn

net = nn.Sequential(nn.Linear(2, 1))
net[0].weight.data.normal_(0, 0.01)
net[0].bias.data.fill_(0)

loss = nn.MSELoss()

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

num_epoches = 3

iter_temp = iter(data_iter)

for epoch in range(num_epoches):
    for X, y in next(iter_temp):
        l = loss(net(X), y)
        l.backward()
        trainer.step()
        trainer.zero_grad()
    l = loss(net(features), labels)
    print(f'epoch {epoch + 1}, loss {l:f}')