12.权重衰退（与课程对应）
目录
一、权重衰退
1、使用均方范数作为硬性限制
（1）通过限制参数值的选择范围来控制模型容量
subject to
通常不限制便宜b（限不限制都差不多）
小的意味着更强的正则项
2、使用均方范数作为柔性限制（通常这么做）
（1）对每个，都可以找到
使得之前的目标函数等价于下面
min
可以通过拉格朗日乘子来证明
（2）超参数控制了正则项的重要程度
=0：无作用
,
3、演示对最优解的影响
4、参数更新法则
（1）计算梯度
（2）时间t更新参数
通常，在深度学习中通常叫做权重衰退
5、总结
（1）权重衰退通过L2正则项使得模型参数不会过大，从而控制模型复杂度
（2）正则项权重是控制模型复杂度的超参数
二、代码实现+从零实现
1、生成数据集：训练集越小，越容易过拟合；特征维度200
python
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)
train_iter = d2l.load_array(train_data, batch_size)
test_data = d2l.synthetic_data(true_w, true_b, n_test)
test_iter = d2l.load_array(test_data, batch_size, is_train=False)2、初始化模型参数
python
def init_params():
w = torch.normal(0, 1, size=(num_inputs, 1), requires_grad=True)
b = torch.zeros(1, requires_grad=True)
return [w, b]3、定义L2范数惩罚
python
def l2_penalty(w):
return torch.sum(w.pow(2)) / 24、定义训练代码实现
python
def train(lambd):
w, b = init_params() # 初始化权重
net, loss = lambda X: d2l.linreg(X, w, b), d2l.squared_loss # 模型，损失
num_epochs, lr = 100, 0.003
animator = d2l.Animator(xlabel='epochs', ylabel='loss', yscale='log',
xlim=[5, num_epochs], legend=['train', 'test']) # 绘制
for epoch in range(num_epochs):
for X, y in train_iter:
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())5、忽略正则化直接训练
python
train(lambd=0)
d2l.plt.show()6、使用权重衰减
python
train(lambd=3)
d2l.plt.show()7、完整代码
python
import torch
from torch import nn
from d2l import torch as d2l
# 权重衰退：从零实现
# 1、生成数据集：训练集越小，越容易过拟合；特征维度200
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)
train_iter = d2l.load_array(train_data, batch_size)
test_data = d2l.synthetic_data(true_w, true_b, n_test)
test_iter = d2l.load_array(test_data, batch_size, is_train=False)
# 2、初始化模型参数
def init_params():
w = torch.normal(0, 1, size=(num_inputs, 1), requires_grad=True)
b = torch.zeros(1, requires_grad=True)
return [w, b]
# 3、定义L2范数惩罚
def l2_penalty(w):
return torch.sum(w.pow(2)) / 2
# 4、定义训练代码实现
def train(lambd):
w, b = init_params() # 初始化权重
net, loss = lambda X: d2l.linreg(X, w, b), d2l.squared_loss # 模型，损失
num_epochs, lr = 100, 0.003
animator = d2l.Animator(xlabel='epochs', ylabel='loss', yscale='log',
xlim=[5, num_epochs], legend=['train', 'test']) # 绘制
for epoch in range(num_epochs):
for X, y in train_iter:
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())
# 忽略正则化直接训练
train(lambd=0)
d2l.plt.show()
# 使用权重衰减
train(lambd=3)
d2l.plt.show()三、代码实现+简介实现
1、生成数据集：训练集越小，越容易过拟合；特征维度200
python
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)
train_iter = d2l.load_array(train_data, batch_size)
test_data = d2l.synthetic_data(true_w, true_b, n_test)
test_iter = d2l.load_array(test_data, batch_size, is_train=False)2、权重衰退+简洁实现
python
def train_concise(wd):
net = nn.Sequential(nn.Linear(num_inputs, 1))
for param in net.parameters():
param.data.normal_()
loss = nn.MSELoss(reduction='none')
num_epochs, lr = 100, 0.003
# 偏置参数没有衰减
trainer = torch.optim.SGD([
{"params":net[0].weight, 'weight_decay': wd},
{"params":net[0].bias}], lr=lr)
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:
trainer.zero_grad()
l = loss(net(X), y)
l.mean().backward()
trainer.step()
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：', net[0].weight.norm().item())3、忽略正则化直接训练
python
train_concise(0)
d2l.plt.show()4、使用权重衰减
python
train_concise(3)
d2l.plt.show()5、完整代码
python
import torch
from matplotlib.pyplot import xlabel
from torch import nn
from d2l import torch as d2l
# 生成数据集：训练集越小，越容易过拟合；特征维度200
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)
train_iter = d2l.load_array(train_data, batch_size)
test_data = d2l.synthetic_data(true_w, true_b, n_test)
test_iter = d2l.load_array(test_data, batch_size, is_train=False)
# 权重衰退+简洁实现
def train_concise(wd):
net = nn.Sequential(nn.Linear(num_inputs, 1))
for param in net.parameters():
param.data.normal_()
loss = nn.MSELoss(reduction='none')
num_epochs, lr = 100, 0.003
# 偏置参数没有衰减
trainer = torch.optim.SGD([
{"params":net[0].weight, 'weight_decay': wd},
{"params":net[0].bias}], lr=lr)
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:
trainer.zero_grad()
l = loss(net(X), y)
l.mean().backward()
trainer.step()
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：', net[0].weight.norm().item())
# 忽略正则化直接训练
train_concise(0)
d2l.plt.show()
# 使用权重衰减
train_concise(3)
d2l.plt.show()如果此文章对您有所帮助，那就请点个赞吧，收藏+关注 那就更棒啦，十分感谢！！！