深度学习 —— 个人学习笔记6（权重衰减）

声明

本文章为个人学习使用，版面观感若有不适请谅解，文中知识仅代表个人观点，若出现错误，欢迎各位批评指正。

十三、权重衰减

使用以下公式为例做演示：

y = 0.05 + ∑ i = 1 d 0.01 x i + ε w h e r e ε ～ N ( 0 , 0.0 1 2 ) y = 0.05 + \sum_{i=1}^{d} 0.01x_i + \varepsilon \quad where \quad \varepsilon \; ～ \; N ( 0 , 0.01^2 ) y=0.05+i=1∑d0.01xi+εwhereε～N(0,0.012)

权重衰减的实现

import torch
from torch import nn
from d2l import torch as d2l
from IPython import display

def synthetic_data(w, b, num_examples):
"""生成 y = Xw + b + 噪声。"""
X = torch.normal(0, 1, (num_examples, len(w))).cuda() # 均值为 0，方差为 1，有 num_examples 个样本，列数为 w 长度
y = torch.matmul(X, w).cuda() + b # y = Xw + b
y += torch.normal(0, 0.01, y.shape).cuda() # 随机噪音
return X, y.reshape((-1, 1)) # x，y作为列向量返回

class Animator: # 定义一个在动画中绘制数据的实用程序类 Animator
"""在动画中绘制数据"""
def init(self, xlabel=None, ylabel=None, legend=None, xlim=None,
ylim=None, xscale='linear', yscale='linear',
fmts=('-', 'm--', 'g-.', 'r:'), nrows=1, ncols=1,
figsize=(3.5, 2.5)):
# 增量地绘制多条线
if legend is None:
legend = []
d2l.use_svg_display()
self.fig, self.axes = d2l.plt.subplots(nrows, ncols, figsize=figsize)
if nrows * ncols == 1:
self.axes = [self.axes, ]
# 使用lambda函数捕获参数
self.config_axes = lambda: d2l.set_axes(
self.axes[0], xlabel, ylabel, xlim, ylim, xscale, yscale, legend)
self.X, self.Y, self.fmts = None, None, fmts
```
  def add(self, x, y):
      # Add multiple data points into the figure
      if not hasattr(y, "__len__"):
          y = [y]
      n = len(y)
      if not hasattr(x, "__len__"):
          x = [x] * n
      if not self.X:
          self.X = [[] for _ in range(n)]
      if not self.Y:
          self.Y = [[] for _ in range(n)]
      for i, (a, b) in enumerate(zip(x, y)):
          if a is not None and b is not None:
              self.X[i].append(a)
              self.Y[i].append(b)
      self.axes[0].cla()
      for x, y, fmt in zip(self.X, self.Y, self.fmts):
          self.axes[0].plot(x, y, fmt)
      self.config_axes()
      display.display(self.fig)
      # 通过以下两行代码实现了在PyCharm中显示动图
      d2l.plt.draw()
      d2l.plt.pause(interval=0.001)
      display.clear_output(wait=True)
      d2l.plt.rcParams['font.sans-serif'] = ['Microsoft YaHei']
```
n_train, n_test, num_inputs, batch_size = 20, 100, 200, 5
true_w, true_b = torch.ones((num_inputs, 1)).cuda() * 0.01, 0.05
train_data = synthetic_data(true_w, true_b, n_train)
train_iter = d2l.load_array(train_data, batch_size)
test_data = synthetic_data(true_w, true_b, n_test)
test_iter = d2l.load_array(test_data, batch_size, is_train=False)

############## 权重衰减的实现 #############
def init_params():
""" 初始化参数 """
w = torch.normal(0, 1, size=(num_inputs, 1)).cuda()
b = torch.zeros(1).cuda()
w.requires_grad_(True)
b.requires_grad_(True)
return [w, b]

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

def train(lambd):
flag_button = "使用"
w, b = init_params()
net, loss = lambda X: d2l.linreg(X, w, b), d2l.squared_loss
num_epochs, lr = 150, 0.005
animator = 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 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())
if lambd == 0:flag_button = "禁用"
d2l.plt.title(f"{flag_button}权重衰减 (lambda = {lambd})\nw 的 L2 范数是：{torch.norm(w).item()}")
d2l.plt.show()

train(lambd=0)

train(lambd=15)

权重衰减的简洁实现

import torch
from torch import nn
from d2l import torch as d2l
from IPython import display

def synthetic_data(w, b, num_examples):
"""生成 y = Xw + b + 噪声。"""
X = torch.normal(0, 1, (num_examples, len(w))).cuda() # 均值为 0，方差为 1，有 num_examples 个样本，列数为 w 长度
y = torch.matmul(X, w).cuda() + b # y = Xw + b
y += torch.normal(0, 0.01, y.shape).cuda() # 随机噪音
return X, y.reshape((-1, 1)) # x，y作为列向量返回

class Animator: # 定义一个在动画中绘制数据的实用程序类 Animator
"""在动画中绘制数据"""
def init(self, xlabel=None, ylabel=None, legend=None, xlim=None,
ylim=None, xscale='linear', yscale='linear',
fmts=('-', 'm--', 'g-.', 'r:'), nrows=1, ncols=1,
figsize=(3.5, 2.5)):
# 增量地绘制多条线
if legend is None:
legend = []
d2l.use_svg_display()
self.fig, self.axes = d2l.plt.subplots(nrows, ncols, figsize=figsize)
if nrows * ncols == 1:
self.axes = [self.axes, ]
# 使用lambda函数捕获参数
self.config_axes = lambda: d2l.set_axes(
self.axes[0], xlabel, ylabel, xlim, ylim, xscale, yscale, legend)
self.X, self.Y, self.fmts = None, None, fmts
```
  def add(self, x, y):
      # Add multiple data points into the figure
      if not hasattr(y, "__len__"):
          y = [y]
      n = len(y)
      if not hasattr(x, "__len__"):
          x = [x] * n
      if not self.X:
          self.X = [[] for _ in range(n)]
      if not self.Y:
          self.Y = [[] for _ in range(n)]
      for i, (a, b) in enumerate(zip(x, y)):
          if a is not None and b is not None:
              self.X[i].append(a)
              self.Y[i].append(b)
      self.axes[0].cla()
      for x, y, fmt in zip(self.X, self.Y, self.fmts):
          self.axes[0].plot(x, y, fmt)
      self.config_axes()
      display.display(self.fig)
      # 通过以下两行代码实现了在PyCharm中显示动图
      d2l.plt.draw()
      d2l.plt.pause(interval=0.001)
      display.clear_output(wait=True)
      d2l.plt.rcParams['font.sans-serif'] = ['Microsoft YaHei']
```
n_train, n_test, num_inputs, batch_size = 20, 100, 200, 5
true_w, true_b = torch.ones((num_inputs, 1)).cuda() * 0.01, 0.05
train_data = synthetic_data(true_w, true_b, n_train)
train_iter = d2l.load_array(train_data, batch_size)
test_data = synthetic_data(true_w, true_b, n_test)
test_iter = d2l.load_array(test_data, batch_size, is_train=False)

############## 权重衰减的简洁实现 #############

def train_concise(wd):
flag_button = "使用"
net = nn.Sequential(nn.Linear(num_inputs, 1)).cuda()
for param in net.parameters():
param.data.normal_().cuda()
loss = nn.MSELoss(reduction='none').cuda()
num_epochs, lr = 150, 0.005
# 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 = 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())
if wd == 0:flag_button = "禁用"
d2l.plt.title(f"{flag_button}权重衰减 (lambda = {wd})\nw 的 L2 范数是：{net[0].weight.norm().item()}")
d2l.plt.show()

train_concise(0)

train_concise(-2)

文中部分知识参考：B 站 ------ 跟李沐学AI；百度百科