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十三、权重衰减

  使用以下公式为例做演示：

$$y=0.05+\sum _{i=1}^{d}0.01x_i+\epsilon where\epsilon ～N(0,0.01^2)$$

• 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