深度神经网络中的Batch Normalization介绍及实现
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之前在https://blog.csdn.net/fengbingchun/article/details/114493591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>
// Blog:
namespace ANN {
class BatchNorm {
public:
BatchNorm(int number, int channels, int height, int width) : number_(number), channels_(channels), height_(height), width_(width) {}
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-batch
int channels_;
int height_;
int width_;
std::vector<float> mean_;
std::vector<float> variance_;
float gamma_ = 1.;
float beta_ = 0.;
float epsilon_ = 1e-5;
std::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()
{
mean_.resize(channels_ * height_ * width_);
memset(mean_.data(), 0, mean_.size() * sizeof(float));
for (int n = 0; n < number_; ++n) {
const float* p = data_.data() + 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) {
mean_[c * height_ * width_ + h * width_ + w] += p[c * height_ * width_ + h * width_ + w];
}
}
}
}
for (int len = 0; len < channels_ * height_ * width_; ++len) {
mean_[len] /= number_;
}
variance_.resize(channels_ * height_ * width_);
memset(variance_.data(), 0, variance_.size() * sizeof(float));
for (int n = 0; n < number_; ++n) {
const float* p = data_.data() + 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) {
variance_[c * height_ * width_ + h * width_ + w] += std::pow(p[c * height_ * width_ + h * width_ + w] - mean_[c * height_ * width_ + h * width_ + w], 2.);
}
}
}
}
for (int len = 0; len < channels_ * height_ * width_; ++len) {
variance_[len] /= number_;
}
std::unique_ptr<float[]> output(new float[number_ * channels_ * height_ * width_]);
for (int n = 0; n < number_; ++n) {
const float* p1 = data_.data() + n * (channels_ * height_ * width_);
float* p2 = 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) {
p2[c * height_ * width_ + h * width_ + w] = (p1[c * height_ * width_ + h * width_ + w] - mean_[c * height_ * width_ + h * width_ + w]) /
std::sqrt(variance_[c * height_ * width_ + h * width_ + w] + epsilon_);
}
}
}
}
return output;
}
} // namespace ANNfunset.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：test_batch_normalization.py
import torch
from torch import nn
import numpy as np
# reference: https://github.com/Johann-Huber/batchnorm_pytorch/blob/main/batch_normalization_in_pytorch.ipynb
# BatchNorm reimplementation
class myBatchNorm2d(nn.Module):
def __init__(self, input_size = None , epsilon = 1e-5, momentum = 0.99):
super(myBatchNorm2d, self).__init__()
assert input_size, print('Missing input_size parameter.')
# Batch mean & var must be defined during training
self.mu = torch.zeros(1, input_size)
self.var = torch.ones(1, input_size)
# For numerical stability
self.epsilon = epsilon
# Exponential moving average for mu & var update
self.it_call = 0 # training iterations
self.momentum = momentum # EMA smoothing
# Trainable parameters
self.beta = torch.nn.Parameter(torch.zeros(1, input_size))
self.gamma = torch.nn.Parameter(torch.ones(1, input_size))
# Batch size on which the normalization is computed
self.batch_size = 0
def forward(self, x):
# [batch_size, input_size]
self.it_call += 1
if self.training:
print("Info: training ...")
if( self.batch_size == 0 ):
# First iteration : save batch_size
self.batch_size = x.shape[0]
# Training : compute BN pass
#batch_mu = (x.sum(dim=0)/x.shape[0]).unsqueeze(0) # [1, input_size]
batch_mu = torch.mean(x, dim=0)
#batch_var = (x.var(dim=0)/x.shape[0]).unsqueeze(0)*2 # [1, input_size]
batch_var = torch.var(x, unbiased=False, dim=0)
#print("batch_mu:", batch_mu)
#print("batch_var:", batch_var)
x_normalized = (x-batch_mu)/torch.sqrt(batch_var + self.epsilon) # [batch_size, input_size]
x_bn = self.gamma * x_normalized + self.beta # [batch_size, input_size]
# Update mu & std
if(x.shape[0] == self.batch_size):
running_mu = batch_mu
running_var = batch_var
else:
running_mu = batch_mu*self.batch_size/x.shape[0]
running_var = batch_var*self.batch_size/x.shape[0]
self.mu = running_mu * (self.momentum/self.it_call) + \\
self.mu * (1 - (self.momentum/self.it_call))
self.var = running_var * (self.momentum/self.it_call) + \\
self.var * (1 - (self.momentum/self.it_call))
else:
print("Info: inference ...")
# Inference: compute BN pass using estimated mu & var
if (x.shape[0] == self.batch_size):
estimated_mu = self.mu
estimated_var = self.var
else :
estimated_mu = self.mu*x.shape[0]/self.batch_size
estimated_var = self.var*x.shape[0]/self.batch_size
x_normalized = (x-estimated_mu)/torch.sqrt(estimated_var + self.epsilon) # [batch_size, input_size]
x_bn = self.gamma * x_normalized + self.beta # [batch_size, input_size]
return x_bn # [batch_size, output_size=input_size]
# N = 3, C = 1, H = 1, W = 6
input_size = 1 # channel
bn = myBatchNorm2d(input_size)
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:", input)
output = bn.forward(input)
print("output:", output)
'''
print("######################")
a = np.array(data)
print(np.mean(a, axis=0))
print(np.var(a, axis=0))
'''执行结果如下：可见，C++和PyTorch实现结果相同
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