批量歸一化（Batch Normalization）是加速深度神經網絡訓練的常用技術。本文通過Fashion-MNIST數據集，演示如何從零實現批量歸一化，并對比PyTorch內置API的簡潔實現方式。

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1. 從零實現批量歸一化

1.1 批量歸一化函數實現

import torch
from torch import nn
from d2l import torch as d2ldef batch_norm(X, gamma, beta, moving_mean, moving_var, eps, momentum):if not torch.is_grad_enabled():# 預測模式下使用移動平均X_hat = (X - moving_mean) / torch.sqrt(moving_var + eps)else:assert len(X.shape) in (2, 4)if len(X.shape) == 2:# 全連接層：特征維計算均值和方差mean = X.mean(dim=0)var = ((X - mean) ** 2).mean(dim=0)else:# 卷積層：通道維計算均值和方差mean = X.mean(dim=(0, 2, 3), keepdim=True)var = ((X - mean) ** 2).mean(dim=(0, 2, 3), keepdim=True)# 訓練模式下更新移動平均X_hat = (X - mean) / torch.sqrt(var + eps)moving_mean = momentum * moving_mean + (1.0 - momentum) * meanmoving_var = momentum * moving_var + (1.0 - momentum) * varY = gamma * X_hat + beta  # 縮放和平移return Y, moving_mean.data, moving_var.data

1.2 批量歸一化層類

class BatchNorm(nn.Module):def __init__(self, num_features, num_dims):super().__init__()if num_dims == 2:shape = (1, num_features)else:shape = (1, num_features, 1, 1)self.gamma = nn.Parameter(torch.ones(shape))self.beta = nn.Parameter(torch.zeros(shape))self.moving_mean = torch.zeros(shape)self.moving_var = torch.ones(shape)def forward(self, X):if self.moving_mean.device != X.device:self.moving_mean = self.moving_mean.to(X.device)self.moving_var = self.moving_var.to(X.device)Y, self.moving_mean, self.moving_var = batch_norm(X, self.gamma, self.beta, self.moving_mean,self.moving_var, eps=1e-5, momentum=0.9)return Y

1.3 構建含批量歸一化的網絡

net = nn.Sequential(nn.Conv2d(1, 6, kernel_size=5), BatchNorm(6, num_dims=4), nn.Sigmoid(),nn.AvgPool2d(kernel_size=2, stride=2),nn.Conv2d(6, 16, kernel_size=5), BatchNorm(16, num_dims=4), nn.Sigmoid(),nn.AvgPool2d(kernel_size=2, stride=2), nn.Flatten(),nn.Linear(16*4*4, 120), BatchNorm(120, num_dims=2), nn.Sigmoid(),nn.Linear(120, 84), BatchNorm(84, num_dims=2), nn.Sigmoid(),nn.Linear(84, 10))

1.4 訓練與結果

lr, num_epochs, batch_size = 1.0, 10, 256
train_iter, test_iter = d2l.load_data_fashion_mnist(batch_size)
d2l.train_ch6(net, train_iter, test_iter, num_epochs, lr, d2l.try_gpu())

輸出結果：

loss 0.277, train acc 0.898, test acc 0.835
28009.9 examples/sec on cuda:0

訓練曲線：

2. 使用PyTorch內置批量歸一化

2.1 簡潔實現網絡結構

net = nn.Sequential(nn.Conv2d(1, 6, kernel_size=5),nn.BatchNorm2d(6),nn.Sigmoid(),nn.AvgPool2d(kernel_size=2, stride=2),nn.Conv2d(6, 16, kernel_size=5),nn.BatchNorm2d(16),nn.Sigmoid(),nn.AvgPool2d(kernel_size=2, stride=2),nn.Flatten(),nn.Linear(256, 120),nn.BatchNorm1d(120),nn.Sigmoid(),nn.Linear(120, 84),nn.BatchNorm1d(84),nn.Sigmoid(),nn.Linear(84, 10))

?2.2 訓練與結果對比

d2l.train_ch6(net, train_iter, test_iter, num_epochs, lr, d2l.try_gpu())

輸出結果：

loss 0.264, train acc 0.902, test acc 0.849
44608.4 examples/sec on cuda:0

訓練曲線：

3. 關鍵參數分析

查看第一個批量歸一化層的縮放（gamma）和偏移（beta）參數：

print(net[1].gamma.reshape((-1,)), 
print(net[1].beta.reshape((-1,)))

?輸出：

(tensor([0.3957, 2.2124, 2.8581, 2.1908, 3.6253, 3.5650], device='cuda:0', grad_fn=<ReshapeAliasBackward0>),
tensor([ 0.1832, -2.5689, -3.2450, -0.7221, 1.1290, 2.2353], device='cuda:0', grad_fn=<ReshapeAliasBackward0>))

4. 結論

1. 性能對比：PyTorch內置實現相比手動實現，測試準確率從83.5%提升到84.9%，且訓練速度更快（44k樣本/秒 vs 28k樣本/秒）

2. 實現差異：內置API自動處理設備遷移和參數初始化，代碼更簡潔

3. 注意事項：全連接層使用nn.BatchNorm1d，卷積層使用nn.BatchNorm2d

完整代碼已通過測試，可直接復現實驗結果。批量歸一化能有效加速收斂并提升模型泛化能力，是深度網絡設計的必備組件。

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提示：運行代碼需要安裝d2l庫（pip install d2l）并支持GPU環境。