學習筆記|Pytorch使用教程23(正則化之weight_decay)

學習筆記|Pytorch使用教程23

本學習筆記主要摘自“深度之眼”，做一個總結，方便查閱。
使用Pytorch版本爲1.2

• 正則化與偏差方差分解
• pytorch中的L2正則項weight decay

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

• 損失函數:衡量模型輸出與真實標籤的差異
  $Loss= f\left(y^{\wedge}, y\right)$
• 代價函數(Cost Function) :
  $Cost=\frac{1}{N} \sum_{t}^{N} f\left(y_{i}^{\wedge}, y_{i}\right)$
• 目標函數(Objective Function) :
  $Obj = Cost + Regularization Term$
• L1 Regularization Term：$\sum_{i}^{N}\left|w_{i}\right|$
• L2 Regularization Term：$\sum_{i}^{N} w_{i}^{2}$
  (座標是L1，右邊是L2)
  

二.pytorch中的L2正則項weight decay

L2 Regularization = weight decay (權值衰減)
目標函數(Objective Function) :
$Obj = Cost + Regularization Term$
$Obj=Loss+\frac{\lambda }{2}\ast \sum_{N}^{i}w_{i}^{2}$
權值衰減推導過程：
$\begin{aligned} w_{i+1}=w_{i}-\frac{\partial O b j}{\partial w_{i}} &=w_{i}-\frac{\partial Loss}{\partial w_{i}} \\ w_{i+1}=w_{i}-\frac{\partial O b j}{\partial w_{i}} &=w_{i}-\left(\frac{\partial Loss}{\partial w_{i}}+\lambda \star w_{i}\right) \\ &=w_{i}(1-\lambda)-\frac{\partial \log s}{\partial w_{i}} \end{aligned}$

測試代碼：

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()

輸出：


進入tensorboard：
沒有帶L2正則化的，整個權值變化不大。

帶L2正則化的，權值不斷縮減。

在代碼optim_wdecay.step()設置斷點，並進入(step into)

    def step(self, closure=None):
        """Performs a single optimization step.

        Arguments:
            closure (callable, optional): A closure that reevaluates the model
                and returns the loss.
        """
        loss = None
        if closure is not None:
            loss = closure()

        for group in self.param_groups:
            weight_decay = group['weight_decay']
            momentum = group['momentum']
            dampening = group['dampening']
            nesterov = group['nesterov']

            for p in group['params']:
                if p.grad is None:
                    continue
                d_p = p.grad.data
                if weight_decay != 0:
                    d_p.add_(weight_decay, p.data)
                if momentum != 0:
                    param_state = self.state[p]
                    if 'momentum_buffer' not in param_state:
                        buf = param_state['momentum_buffer'] = torch.clone(d_p).detach()
                    else:
                        buf = param_state['momentum_buffer']
                        buf.mul_(momentum).add_(1 - dampening, d_p)
                    if nesterov:
                        d_p = d_p.add(momentum, buf)
                    else:
                        d_p = buf

                p.data.add_(-group['lr'], d_p)

        return loss

在代碼的d_p.add_(weight_decay, p.data)進行權值衰減。其公式是：d_p = d_p + p.data * weight_decay
在代碼p.data.add_(-group['lr'], d_p)進行梯度更新。