一、正則化與偏差—方差分解

Regularization: 減小方差的策略

誤差可分解爲：偏差，方差與噪聲之和。即誤差=偏差+方差+噪聲之和

偏差度量了學習算法的期望預測與真實結果的偏離程度,即刻畫了學習算法本身的擬合能力

方差度量了同樣大小的訓練集的變動所導致的學習性能的變化,即刻畫了數據擾動所造成的影響

噪聲則表達了在當前任務上任何學習算法所能達到的期望泛化誤差的下界

L1正則化：
加上L1正則項，目標函數需要同時滿足cost和正則項都要小，如上圖所示，w1和w2所滿足的區域和cost的曲線相交點才能滿足所需，而交點處於座標軸，則w1=0，即參數解會產生稀疏項

L2正則化：
同樣的，當cost值固定，如圖中紅色曲線，w1和w2的曲線與紅色曲線的相交處，就是使得總體目標函數最小的位置

經過L1和L2正則化，參數值比較小，從而使得模型不會過於複雜

二、pytorch中的L2正則項——weight decay

2.1 權值衰減概念

說明：

$\lambda$ 是超參， $0 < \lambda < 1$ ，用於調和loss和正則項的比例
1/2是用於方便求導

權值衰減：
從上圖可知，加上L2正則化項後，w的更新公式中，wi乘以了一個小於1的正數，因此權值會發生衰減，故L2正則化也叫做權值衰減

2.2 L2正則化測試

# -*- 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)  # 設置隨機種子
n_hidden = 200
max_iter = 2000
disp_interval = 200
lr_init = 0.01


# ============================ step 1/5 數據 ============================
def gen_data(num_data=10, x_range=(-1, 1)):

    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__()
        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)


net_normal = MLP(neural_num=n_hidden)
net_weight_decay = MLP(neural_num=n_hidden)

# ============================ step 3/5 優化器 ============================
optim_normal = torch.optim.SGD(net_normal.parameters(), lr=lr_init, momentum=0.9)
optim_wdecay = torch.optim.SGD(net_weight_decay.parameters(), lr=lr_init, momentum=0.9, weight_decay=1e-2)

# ============================ step 4/5 損失函數 ============================
loss_func = torch.nn.MSELoss()

# ============================ step 5/5 迭代訓練 ============================

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:

        # 可視化
        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)

        # 繪圖
        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()