使用PyTorch实现L1, L2和Elastic Net正则化

在机器学习中，L1正则化、L2正则化和Elastic Net正则化是用来避免过拟合的技术，它们通过在损失函数中添加一个惩罚项来实现。

正则化介绍

L1 正则化（Lasso回归）：

L1 正则化通过向损失函数添加参数的绝对值的和来实施惩罚，公式可以表示为：

其中 L0 是原始的损失函数，λ 是正则化强度，wi是模型参数。

L1 正则化的特点是它可以产生稀疏模型，即许多模型参数会被设置为零。这种特性使得L1正则化不仅可以防止过拟合，还可以进行特征选择。

L2 正则化（Ridge回归）：

L2 正则化通过添加参数的平方和来施加惩罚，公式为：

λ 控制着正则化的强度。

L2 正则化倾向于让参数值趋近于零但不会完全为零，这有助于处理参数值过大的问题，从而减少模型在训练数据上的过拟合。

Elastic Net 正则化：

Elastic Net 正则化是L1和L2正则化的组合，它在损失函数中同时添加了L1和L2惩罚项，公式为：

这种方法结合了L1和L2的优点，既可以产生稀疏模型，也可以平滑模型参数。

在实际应用中，Elastic Net特别适合于那些特征数量多于样本数量，或者特征之间高度相关的情况。

在sklearn中，我们可以使用内置的回归函数来实现

Lasso回归是应用L1正则化的典型模型。它可以通过Lasso类实现；Ridge回归使用L2正则化。它可以通过Ridge类来实现；Elastic Net回归结合了L1和L2正则化。它通过ElasticNet类实现

Pytorch代码实现

但是这些都是最简单的线性回归的扩展，通过上面的介绍，我们看到这些正则化的方式都是通过修改模型本身的权重来实现的，所以我们可以在MLP上也使用这些正则化的方法，下面我们将使用Pytorch来演示这个步骤

首先我们看下L1

import os
import torch
from torch import nn
from torchvision.datasets import MNIST
from torch.utils.data import DataLoader
from torchvision import transforms
class MLP(nn.Module):
'''
Multilayer Perceptron.
'''
def __init__(self):
super().__init__()
self.layers = nn.Sequential(
nn.Flatten(),
nn.Linear(28 * 28 * 1, 64),
nn.ReLU(),
nn.Linear(64, 32),
nn.ReLU(),
nn.Linear(32, 10)
)
def forward(self, x):
'''Forward pass'''
return self.layers(x)
def compute_l1_loss(self, w):
return torch.abs(w).sum()
if __name__ == '__main__':
# Set fixed random number seed
torch.manual_seed(42)
# Prepare CIFAR-10 dataset
dataset = MNIST(os.getcwd(), download=True, transform=transforms.ToTensor())
trainloader = torch.utils.data.DataLoader(dataset, batch_size=10, shuffle=True, num_workers=1)
# Initialize the MLP
mlp = MLP()
# Define the loss function and optimizer
loss_function = nn.CrossEntropyLoss()
optimizer = torch.optim.Adam(mlp.parameters(), lr=1e-4)
# Run the training loop
for epoch in range(0, 5): # 5 epochs at maximum
# Print epoch
print(f'Starting epoch {epoch+1}')
# Iterate over the DataLoader for training data
for i, data in enumerate(trainloader, 0):
# Get inputs
inputs, targets = data
# Zero the gradients
optimizer.zero_grad()
# Perform forward pass
outputs = mlp(inputs)
# Compute loss
loss = loss_function(outputs, targets)
# Compute L1 loss component
l1_weight = 1.0
l1_parameters = []
for parameter in mlp.parameters():
l1_parameters.append(parameter.view(-1))
l1 = l1_weight * mlp.compute_l1_loss(torch.cat(l1_parameters))
# Add L1 loss component
loss += l1
# Perform backward pass
loss.backward()
# Perform optimization
optimizer.step()
# Print statistics
minibatch_loss = loss.item()
if i % 500 == 499:
print('Loss after mini-batch %5d: %.5f (of which %.5f L1 loss)' %
(i + 1, minibatch_loss, l1))
current_loss = 0.0
# Process is complete.
print('Training process has finished.')

我们在本身的一个简单的MLP中增加了一个compute_l1_loss方法，在我们计算完基本的损失后，还会计算模型参数的L1 损失，然后与基本损失相加，最后使用这个最终损失来进行反向传播。

L2正则化也很容易。我们不取权重值的绝对值，而是取它们的平方。

import os
import torch
from torch import nn
from torchvision.datasets import MNIST
from torch.utils.data import DataLoader
from torchvision import transforms
class MLP(nn.Module):
'''
Multilayer Perceptron.
'''
def __init__(self):
super().__init__()
self.layers = nn.Sequential(
nn.Flatten(),
nn.Linear(28 * 28 * 1, 64),
nn.ReLU(),
nn.Linear(64, 32),
nn.ReLU(),
nn.Linear(32, 10)
)
def forward(self, x):
'''Forward pass'''
return self.layers(x)
def compute_l2_loss(self, w):
return torch.square(w).sum()
if __name__ == '__main__':
# Set fixed random number seed
torch.manual_seed(42)
# Prepare CIFAR-10 dataset
dataset = MNIST(os.getcwd(), download=True, transform=transforms.ToTensor())
trainloader = torch.utils.data.DataLoader(dataset, batch_size=10, shuffle=True, num_workers=1)
# Initialize the MLP
mlp = MLP()
# Define the loss function and optimizer
loss_function = nn.CrossEntropyLoss()
optimizer = torch.optim.Adam(mlp.parameters(), lr=1e-4)
# Run the training loop
for epoch in range(0, 5): # 5 epochs at maximum
# Print epoch
print(f'Starting epoch {epoch+1}')
# Iterate over the DataLoader for training data
for i, data in enumerate(trainloader, 0):
# Get inputs
inputs, targets = data
# Zero the gradients
optimizer.zero_grad()
# Perform forward pass
outputs = mlp(inputs)
# Compute loss
loss = loss_function(outputs, targets)
# Compute l2 loss component
l2_weight = 1.0
l2_parameters = []
for parameter in mlp.parameters():
l2_parameters.append(parameter.view(-1))
l2 = l2_weight * mlp.compute_l2_loss(torch.cat(l2_parameters))
# Add l2 loss component
loss += l2
# Perform backward pass
loss.backward()
# Perform optimization
optimizer.step()
# Print statistics
minibatch_loss = loss.item()
if i % 500 == 499:
print('Loss after mini-batch %5d: %.5f (of which %.5f l2 loss)' %
(i + 1, minibatch_loss, l2))
current_loss = 0.0
# Process is complete.
print('Training process has finished.')

最终的计算过程和L1正则化一样，只不过是计算附加损失的方法不同。

对于L2的正则化Pytorch的Adam优化器有一个官方的参数，叫做权重衰减weight_decay

optimizer = torch.optim.Adam(mlp.parameters(), lr=1e-4, weight_decay=1.0)

你可能不知道他和L2的关系，但是你一定用到过，所以我们这样一解释就非常明白了对吧

最后就是Elastic Net (L1 + L2)

class MLP(nn.Module):
'''
Multilayer Perceptron.
'''
def __init__(self):
super().__init__()
self.layers = nn.Sequential(
nn.Flatten(),
nn.Linear(28 * 28 * 1, 64),
nn.ReLU(),
nn.Linear(64, 32),
nn.ReLU(),
nn.Linear(32, 10)
)
def forward(self, x):
'''Forward pass'''
return self.layers(x)
def compute_l1_loss(self, w):
return torch.abs(w).sum()
def compute_l2_loss(self, w):
return torch.square(w).sum()
if __name__ == '__main__':
# Set fixed random number seed
torch.manual_seed(42)
# Prepare CIFAR-10 dataset
dataset = MNIST(os.getcwd(), download=True, transform=transforms.ToTensor())
trainloader = torch.utils.data.DataLoader(dataset, batch_size=10, shuffle=True, num_workers=1)
# Initialize the MLP
mlp = MLP()
# Define the loss function and optimizer
loss_function = nn.CrossEntropyLoss()
optimizer = torch.optim.Adam(mlp.parameters(), lr=1e-4)
# Run the training loop
for epoch in range(0, 5): # 5 epochs at maximum
# Print epoch
print(f'Starting epoch {epoch+1}')
# Iterate over the DataLoader for training data
for i, data in enumerate(trainloader, 0):
# Get inputs
inputs, targets = data
# Zero the gradients
optimizer.zero_grad()
# Perform forward pass
outputs = mlp(inputs)
# Compute loss
loss = loss_function(outputs, targets)
# Specify L1 and L2 weights
l1_weight = 0.3
l2_weight = 0.7
# Compute L1 and L2 loss component
parameters = []
for parameter in mlp.parameters():
parameters.append(parameter.view(-1))
l1 = l1_weight * mlp.compute_l1_loss(torch.cat(parameters))
l2 = l2_weight * mlp.compute_l2_loss(torch.cat(parameters))
# Add L1 and L2 loss components
loss += l1
loss += l2
# Perform backward pass
loss.backward()
# Perform optimization
optimizer.step()
# Print statistics
minibatch_loss = loss.item()
if i % 500 == 499:
print('Loss after mini-batch %5d: %.5f (of which %.5f L1 loss; %0.5f L2 loss)' %
(i + 1, minibatch_loss, l1, l2))
current_loss = 0.0
# Process is complete.
print('Training process has finished.')

也非常的简单，并且我们可以设置两个权重，就是L1和L2的占比，使用不同的加权，可以获得更好的结果。

总结

这篇文章是要是为了介绍L1, L2和Elastic Net (L1+L2)正则化在理论上是如何工作的。并且我们也在PyTorch中使用了L1, L2和Elastic Net (L1+L2)正则化。这三种正则化方法在不同的情况和数据集上有不同的效果，选择哪种正则化方法取决于具体的应用场景和数据特性。

https://avoid.overfit.cn/post/c99ec105e41c4a71a0a1a29735245944