在本次实验中,使用人造数据集为:
$$y=0.05\sum_{i=1}^{d} 0.01x_i + \epsilon ,其中\epsilon \sim N(0,0.01^2)$$

import torch
from torch import nn
from d2l import torch as d2l

# 生成数据集
n_train, n_test, num_inputs, batch_size = 20, 100, 200, 5
true_w, true_b = torch.ones((num_inputs, 1)) * 0.01, 0.05
train_data = d2l.synthetic_data(true_w, true_b, n_train)  # 合成训练数据
train_iter = d2l.load_array(train_data, batch_size)
test_data = d2l.synthetic_data(true_w, true_b, n_test)  # 合成测试数据
test_iter = d2l.load_array(test_data, batch_size, is_train=False)


# 初始化模型参数
def init_params():
    w = torch.normal(0, 1, (num_inputs, 1), requires_grad=True)
    b = torch.zeros(1, requires_grad=True)
    return [w, b]


# 定义L2范数惩罚
def l2_penalty(w):
    return torch.sum(w.pow(2)) / 2


# 定义训练代码实现
def train(lambd):
    w, b = init_params()
    net, loss = lambda X: d2l.linreg(X, w, b), d2l.squared_loss
    num_epochs, lr = 100, 0.003
    animator = d2l.Animator(xlabel='epochs', ylabel='loss', yscale='log', xlim=[5, num_epochs], legend=['train', 'test'])
    for epoch in range(num_epochs):
        for X, y in train_iter:
            # 增加了L2范数惩罚项
            # 广播机制使得l2_penalty(w)成为一个长度为batch_size的向量
            l = loss(net(X), y) + lambd * l2_penalty(w)
            l.sum().backward()
            d2l.sgd([w, b], lr, batch_size)
        if (epoch + 1) % 5 == 0:
            animator.add(epoch + 1, (d2l.evaluate_loss(net, train_iter, loss),
                                     d2l.evaluate_loss(net, test_iter, loss)))
    print('w的L2范数是：', torch.norm(w).item())

开始训练！

# 忽略正则化直接训练
train(lambd=0)
d2l.plt.show()

可以看到，发生了毁灭性的overfitting

# 使用权重衰减
train(lambd=3)
d2l.plt.show()

正常拟合!