PyTorch学习笔记（25） 正则化 weight_decay

Regularzation 正则化

Regularzation:减小方差的策略
误差可分解为:偏差，方差与噪声之和。即 误差 = 偏差 + 方差 + 噪声 之和
偏差度量了学习算法的期望预测与真实结果的偏离程度，即刻画了学习算法本身的拟合能力
方差度量了同样大小的训练集的变动所导致的学习性能的变化，即刻画了数据扰动所造成的影响
噪声则表达了在当前任务上任何学习算法所能达到的期望泛化误差的下界

损失函数:衡量模型输出与真实标签的差异

损失函数(Loss Function):

损失函数:衡量模型输出与真实标签的差异
$\text { Loss }=f\left(y^{\wedge}, y\right)$
代价函数(Cost Function):
$Cost=\frac{1}{N} \sum_{i}^{N} f\left(y_{i}^{\wedge}, y_{i}\right)$
目标函数(Objective Function):
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L1 Regularzation Term :

$\sum_{i}^{N}\left|\boldsymbol{w}_{i}\right|$

L2 Regularzation Term :

$\sum_{i}^{N} w_{i}^{2}$

L2 Regularzation = weight decay(权值衰减)

将目标函数中的Cost 替换为Loss

O b j=C o s t+$ Regularization Term

$O b j=L o s s+\frac{\lambda}{2} * \sum_{i}^{N} w_{i}^{2}$
$w_{i+1}=w_{i}-\frac{\partial o b j}{\partial w_{i}}=w_{i}-\frac{\partial L o s s}{\partial w_{i}}$
加入正则项之后
$\boldsymbol{w}_{i+1}=\boldsymbol{w}_{i}-\frac{\partial O b j}{\partial w_{i}}=\boldsymbol{w}_{i}-\left(\frac{\partial L o s s}{\partial w_{i}}+\lambda \star \boldsymbol{w}_{i}\right)$
$=w_{i}(1-\lambda)-\frac{\partial L o s s}{\partial w_{i}}$

# -*- coding:utf-8 -*-

import torch
import torch.nn as nn
import matplotlib.pyplot as plt
from tools.common_tools import set_seed
from torch.utils.tensorboard import SummaryWriter

set_seed(1)  # 设置随机种子
n_hidden = 200
max_iter = 2000
disp_interval = 200
lr_init = 0.01


# ============================ step 1/5 数据 ============================
def gen_data(num_data=10, x_range=(-1, 1)):
    # 构建数据
    w = 1.5
    train_x = torch.linspace(*x_range, num_data).unsqueeze_(1)
    train_y = w*train_x + torch.normal(0, 0.5, size=train_x.size())
    test_x = torch.linspace(*x_range, num_data).unsqueeze_(1)
    test_y = w*test_x + torch.normal(0, 0.3, size=test_x.size())

    return train_x, train_y, test_x, test_y


train_x, train_y, test_x, test_y = gen_data(x_range=(-1, 1))


# ============================ step 2/5 模型 ============================
class MLP(nn.Module):
    def __init__(self, neural_num):
        super(MLP, self).__init__()
        # 采用三层全连接网络
        self.linears = nn.Sequential(
            nn.Linear(1, neural_num),
            nn.ReLU(inplace=True),
            nn.Linear(neural_num, neural_num),
            nn.ReLU(inplace=True),
            nn.Linear(neural_num, neural_num),
            nn.ReLU(inplace=True),
            nn.Linear(neural_num, 1),
        )

    def forward(self, x):
        return self.linears(x)

# 实例化两个模型 一个采用weight_decay 一个不采用
net_normal = MLP(neural_num=n_hidden)
net_weight_decay = MLP(neural_num=n_hidden)

# ============================ step 3/5 优化器 ============================
# weight_decay 是在优化器中实现的
# SGD随机梯度下降优化器
optim_normal = torch.optim.SGD(net_normal.parameters(), lr=lr_init, momentum=0.9)
# 设置超参1e-2
optim_wdecay = torch.optim.SGD(net_weight_decay.parameters(), lr=lr_init, momentum=0.9, weight_decay=1e-2)

# ============================ step 4/5 损失函数 ============================
loss_func = torch.nn.MSELoss()

# ============================ step 5/5 迭代训练 ============================

writer = SummaryWriter(comment='_test_tensorboard', filename_suffix="12345678")
for epoch in range(max_iter):

    # forward
    pred_normal, pred_wdecay = net_normal(train_x), net_weight_decay(train_x)
    loss_normal, loss_wdecay = loss_func(pred_normal, train_y), loss_func(pred_wdecay, train_y)

    optim_normal.zero_grad()
    optim_wdecay.zero_grad()

    loss_normal.backward()
    loss_wdecay.backward()

    optim_normal.step()
    optim_wdecay.step()

    if (epoch+1) % disp_interval == 0:

        # 可视化
        for name, layer in net_normal.named_parameters():
            writer.add_histogram(name + '_grad_normal', layer.grad, epoch)
            writer.add_histogram(name + '_data_normal', layer, epoch)

        for name, layer in net_weight_decay.named_parameters():
            writer.add_histogram(name + '_grad_weight_decay', layer.grad, epoch)
            writer.add_histogram(name + '_data_weight_decay', layer, epoch)

        test_pred_normal, test_pred_wdecay = net_normal(test_x), net_weight_decay(test_x)

        # 绘图
        plt.scatter(train_x.data.numpy(), train_y.data.numpy(), c='blue', s=50, alpha=0.3, label='train')
        plt.scatter(test_x.data.numpy(), test_y.data.numpy(), c='red', s=50, alpha=0.3, label='test')
        plt.plot(test_x.data.numpy(), test_pred_normal.data.numpy(), 'r-', lw=3, label='no weight decay')
        plt.plot(test_x.data.numpy(), test_pred_wdecay.data.numpy(), 'b--', lw=3, label='weight decay')
        plt.text(-0.25, -1.5, 'no weight decay loss={:.6f}'.format(loss_normal.item()), fontdict={'size': 15, 'color': 'red'})
        plt.text(-0.25, -2, 'weight decay loss={:.6f}'.format(loss_wdecay.item()), fontdict={'size': 15, 'color': 'red'})

        plt.ylim((-2.5, 2.5))
        plt.legend(loc='upper left')
        plt.title("Epoch: {}".format(epoch+1))
        plt.show()
        plt.close()