8.通过高维线性回归实验观察权重衰减

import d2lzh as d2l
from mxnet import autograd, gluon, init, nd
from mxnet.gluon import data as gdata, loss as gloss, nn

# 生成数据集
n_train, n_test, num_inputs = 20, 100, 200
true_w, true_b = nd.ones((num_inputs, 1)) * 0.01, 0.05

features = nd.random.normal(shape=(n_train + n_test, num_inputs))
labels = nd.dot(features, true_w) + true_b
labels += nd.random.normal(scale=0.1, shape=labels.shape)
train_features, test_features = features[:n_train, :], features[n_train:, :]
train_labels, test_labels = labels[:n_train], labels[n_train:]


# 初始化权重参数
def init_params():
	w = nd.random.normal(scale=1, shape=(num_inputs, 1))
	b = nd.zeros(shape=(1,))
	w.attach_grad()
	b.attach_grad()
	return [w, b]


# 定义L2范数惩罚项
def L2_penalty(w):
	return (w ** 2).sum() / 2


# 定义训练和测试
batch_size, num_epochs, lr = 1, 100, 0.003
net, loss = d2l.linreg, d2l.squared_loss
train_iter = gdata.DataLoader(gdata.ArrayDataset(train_features, train_labels), batch_size, shuffle=True)


def fit_and_plot(lambd):
	w, b = init_params()
	train_ls, test_ls = [], []
	for _ in range(num_epochs):
		for X, y in train_iter:
			with autograd.record():
				l = loss(net(X, w, b), y) + lambd * L2_penalty(w)  # 添加L2范数惩罚项
			l.backward()
			d2l.sgd([w, b], lr, batch_size)
		train_ls.append(loss(net(train_features, w, b), train_labels).mean().asscalar())
		test_ls.append(loss(net(test_features, w, b), test_labels).mean().asscalar())

	d2l.semilogy(range(1, num_epochs + 1), train_ls, 'epochs', 'loss', range(1, num_epochs + 1), test_ls,
				 ['train', 'test'])
	print('L2 norm of w:', w.norm().asscalar())

1.观察过拟合：

# 当lambd=0时，没有使用权重衰减，可以看到训练误差远小于测试集上的误差
fit_and_plot(lambd=0)

2.使用权重衰减：

# 可以看到，训练集上误差虽然有所提高，但是测试集上的误差有所下降，过拟合现象得到了一定程度的缓解
fit_and_plot(lambd=3)

Reference:
《动手学深度学习》