深度神经网络中的Batch Normalization介绍及实现

之前在经典网络DenseNet介绍_fengbingchun的博客-CSDN博客_densenet中介绍DenseNet时，网络中会有BN层，即Batch Normalization，在每个Dense Block中都会有BN参与运算，下面对BN进行介绍并给出C++和PyTorch实现。

Batch Normalization即批量归一化由Sergey loffe等人于2015年提出，论文名为：《Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift》，论文见：https://arxiv.org/pdf/1502.03167.pdf 。

Batch Normalization是一种算法方法，它使深度神经网络的训练更快、更稳定。它可在激活函数前也可在激活函数后进行。它依赖于batch size，当batch size较小时，性能退化严重。在训练和测试阶段，它的计算方式不同。

对于CNN，使用BN更好；对于RNN，使用LN(Layer Normalization)更好。

在训练过程中，由于每层输入的分布随着前一层的参数发生变化而发生变化，因此训练深度神经网络很复杂。由于需要较低的学习率和仔细的参数初始化，这会减慢训练速度，并且使得训练具有饱和非线性的模型变得非常困难。我们将这种现象称为内部协变量偏移(internal covariate shift)，并通过归一化层输入来解决该问题。

Batch Normalization用于训练小批量样本(mini-batch)。它允许我们使用更高的学习率，并且不必太小心初始化。它还充当正则化器，在某些情况下消除了Dropout的需要。

Batch Normalization实现算法如下，截图来自原始论文：

在一个mini-batch中，在每一个BN层中，对每个样本的同一通道，计算它们的均值和方差，再对数据进行归一化，归一化到平均值为0，标准差为1 的常态分布，最后使用两个可学习参数gamma和beta对归一化的数据进行缩放和移位。此外，在训练过程中还保存了每个mini-batch每一BN层的均值和方差，最后求所有mini-batch均值和方差的期望值，以此来作为推理过程中该BN层的均值和方差。

Batch Normalization优点：

(1).在不影响收敛性的情况下，可使用更大的学习率，使训练更快、更稳定；

(2).具有正则化效果，防止过拟合，可去除Dropout和局部响应归一化(Local Response Normalization, LRN)；

(3).由于训练数据打乱顺序，使得每个epoch中mini-batch都不一样，对不同mini-batch做归一化可以起到数据增强的效果；

(4).缓减梯度爆炸和梯度消失。

以下是C++实现：

batch_normalization.hpp:

#ifndef FBC_SRC_NN_BATCH_NORM_HPP_
#define FBC_SRC_NN_BATCH_NORM_HPP_#include <vector>
#include <memory>
#include <algorithm>namespace ANN {class BatchNorm {
public:BatchNorm(int number, int channels, int height, int width) : number_(number), channels_(channels), height_(height), width_(width){mean_.resize(channels_);std::fill(mean_.begin(), mean_.end(), 0.);variance_.resize(channels_);std::fill(variance_.begin(), variance_.end(), 0.);}int LoadData(const float* data, int length);std::unique_ptr<float []> Run();void SetGamma(float gamma) { gamma_ = gamma; }float GetGamma() const { return gamma_; }void SetBeta(float beta) { beta_ = beta; }float GetBeta() const { return beta_; }void SetMean(std::vector<float> mean) { mean_ = mean; }std::vector<float> GetMean() const { return mean_; }void SetVariance(std::vector<float> variance) { variance_ = variance; }std::vector<float> GetVariance() const { return variance_; }void SetEpsilon(float epsilon) { epsilon_ = epsilon; }private:int number_; // mini-batchint channels_;int height_;int width_;std::vector<float> mean_;std::vector<float> variance_;float gamma_  = 1.; // 缩放 float beta_ = 0.; // 平移float epsilon_ = 1e-5; // small positive value to avoid zero-divisionstd::vector<float> data_;
};} // namespace ANN#endif // FBC_SRC_NN_BATCH_NORM_HPP_

batch_normalization.cpp:

#include "batch_normalization.hpp"
#include <string.h>
#include <vector>
#include <cmath>
#include "common.hpp"namespace ANN {int BatchNorm::LoadData(const float* data, int length)
{CHECK(number_ * channels_ * height_ * width_ == length);data_.resize(length);memcpy(data_.data(), data, length * sizeof(float));return 0;
}std::unique_ptr<float[]> BatchNorm::Run()
{int spatial_size = height_ * width_;for (int n = 0; n < number_; ++n) {int offset = n * (channels_ * spatial_size);for (int c = 0; c < channels_; ++c) {const float* p = data_.data() + offset + (c * spatial_size);for (int k = 0; k < spatial_size; ++k) {mean_[c] += *p++;}}}std::transform(mean_.begin(), mean_.end(), mean_.begin(), [=](float_t x) { return x / (number_ * spatial_size); });for (int n = 0; n < number_; ++n) {int offset = n * (channels_ * spatial_size);for (int c = 0; c < channels_; ++c) {const float* p = data_.data() + offset + (c * spatial_size);for (int k = 0; k < spatial_size; ++k) {variance_[c] += std::pow(*p++ - mean_[c], 2.);}}}std::transform(variance_.begin(), variance_.end(), variance_.begin(), [=](float_t x) { return x / (std::max(1., number_*spatial_size*1.)); });std::vector<float> stddev(channels_);for (int c = 0; c < channels_; ++c) {stddev[c] = std::sqrt(variance_[c] + epsilon_);}std::unique_ptr<float[]> output(new float[number_ * channels_ * spatial_size]);for (int n = 0; n < number_; ++n) {const float* p1 = data_.data() + n * (channels_ * spatial_size);float* p2 = output.get() + n * (channels_ * spatial_size);for (int c = 0; c < channels_; ++c) {for (int k = 0; k < spatial_size; ++k) {*p2++ = (*p1++ - mean_[c]) / stddev[c];}}}return output;
}} // namespace ANN

funset.cpp:

int test_batch_normalization()
{const std::vector<float> data = { 11.1, -2.2, 23.3, 54.4, 58.5, -16.6,-97.7, -28.8, 49.9, -61.3, 52.6, -33.9,-2.45, -15.7, 72.4, 9.1, 47.2, 21.7};const int number = 3, channels = 1, height = 1, width = 6;ANN::BatchNorm bn(number, channels, height, width);bn.LoadData(data.data(), data.size());std::unique_ptr<float[]> output = bn.Run();fprintf(stdout, "result:\n");for (int n = 0; n < number; ++n) {const float* p = output.get() + n * (channels * height * width);for (int c = 0; c < channels; ++c) {for (int h = 0; h < height; ++h) {for (int w = 0; w < width; ++w) {fprintf(stdout, "%f, ", p[c * (height * width) + h * width + w]);}fprintf(stdout, "\n");}}}return 0;
}

执行结果如下：

以下是调用PyTorch接口实现：源码来自于https://zh.d2l.ai/chapter_convolutional-modern/batch-norm.html

import torch
from torch import nn
import numpy as np# reference: https://zh.d2l.ai/chapter_convolutional-modern/batch-norm.html
# BatchNorm reimplementation
def batch_norm(X, gamma, beta, moving_mean, moving_var, eps, momentum):# 通过is_grad_enabled来判断当前模式是训练模式还是预测模式if not torch.is_grad_enabled():# 如果是在预测模式下，直接使用传入的移动平均所得的均值和方差X_hat = (X - moving_mean) / torch.sqrt(moving_var + eps)else:assert len(X.shape) in (2, 4)if len(X.shape) == 2:# 使用全连接层的情况，计算特征维上的均值和方差mean = X.mean(dim=0)var = ((X - mean) ** 2).mean(dim=0)else:# 使用二维卷积层的情况，计算通道维上（axis=1）的均值和方差。# 这里我们需要保持X的形状以便后面可以做广播运算mean = X.mean(dim=(0, 2, 3), keepdim=True)var = ((X - mean) ** 2).mean(dim=(0, 2, 3), keepdim=True)# 训练模式下，用当前的均值和方差做标准化X_hat = (X - mean) / torch.sqrt(var + eps)# 更新移动平均的均值和方差moving_mean = momentum * moving_mean + (1.0 - momentum) * meanmoving_var = momentum * moving_var + (1.0 - momentum) * varY = gamma * X_hat + beta  # 缩放和移位return Y, moving_mean.data, moving_var.dataclass BatchNorm(nn.Module):# num_features：完全连接层的输出数量或卷积层的输出通道数。# num_dims：2表示完全连接层，4表示卷积层def __init__(self, num_features, num_dims):super().__init__()if num_dims == 2:shape = (1, num_features)else:shape = (1, num_features, 1, 1)# 参与求梯度和迭代的拉伸和偏移参数，分别初始化成1和0self.gamma = nn.Parameter(torch.ones(shape))self.beta = nn.Parameter(torch.zeros(shape))# 非模型参数的变量初始化为0和1self.moving_mean = torch.zeros(shape)self.moving_var = torch.ones(shape)def forward(self, X):# 如果X不在内存上，将moving_mean和moving_var复制到X所在显存上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)# 保存更新过的moving_mean和moving_varY, 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# N = 3, C = 1, H = 1, W = 6
data =  [[[[11.1, -2.2, 23.3, 54.4, 58.5, -16.6]]],[[[-97.7, -28.8, 49.9, -61.3, 52.6, -33.9]]],[[[-2.45, -15.7, 72.4, 9.1, 47.2, 21.7]]]]
input = torch.FloatTensor(data) # [N, C, H, W]
print("input shape:", input.shape)model = BatchNorm(1, 2)
output = model(input)
print("output:", output)print("test finish")

执行结果如下：可见，C++和PyTorch实现结果相同

以下是调用tiny-dnn接口的测试代码：

int test_dnn_batch_normalization()
{const std::vector<float> data = { 11.1, -2.2, 23.3, 54.4, 58.5, -16.6,-97.7, -28.8, 49.9, -61.3, 52.6, -33.9,-2.45, -15.7, 72.4, 9.1, 47.2, 21.7 };const int number = 3, channels = 1, height = 1, width = 6;const int spatial_size = height * width;tiny_dnn::tensor_t in_data(number), out_data(number);for (int n = 0; n < number; ++n) {in_data[n].resize(spatial_size * channels);out_data[n].resize(spatial_size * channels);int offset = n * (spatial_size * channels);memcpy(in_data[n].data(), data.data() + offset, sizeof(float)*spatial_size*channels);std::fill(out_data[n].begin(), out_data[n].end(), 0.);}std::vector<tiny_dnn::tensor_t*> in(1), out(1);in[0] = &in_data;out[0] = &out_data;tiny_dnn::batch_normalization_layer bn(spatial_size, channels);bn.forward_propagation(in, out);fprintf(stdout, "tiny_dnn result:\n");for (int n = 0; n < number; ++n) {for (int s = 0; s < spatial_size * channels; ++s)fprintf(stdout, "%f  ", out_data[n][s]);fprintf(stdout, "\n");}return 0;
}

执行结果如下：与上面的C++和PyTorch代码结果若有不同，原因是tiny-dnn源码中math_functions.h文件在求平均方差时除数为num_examples*spatial_dim-1.0f，而不是num_examples*spatial_dim

GitHub:

https://github.com/fengbingchun/NN_Test

https://github.com/fengbingchun/PyTorch_Test