# Caffe BatchNormalization 推导

## 谈谈理论与公式推导

$y_i = \frac{x_i-\overline{x}}{\sqrt{\delta^2+\epsilon}},\quad\overline x = \frac{1}{m}\sum_{i=1}^{m}x_i,\quad \delta^2 = \frac{1}{m}\sum_{i=1}^{m}(x_i-\overline x)^2,\quad 求\frac{\partial L}{\partial x_i}$

$\frac{\partial L}{\partial x_i} = \sum_{j=1}^{m}{\frac{\partial L}{\partial y_j}*\frac{\partial y_j}{\partial x_i}}$

(1) $$\overline x$$$$x_i$$的导函数

$\frac{\partial \overline x}{\partial x_i} = \frac{1}{m}$

(2) $$\delta^2$$$$x_i$$的导函数

$\frac{\partial \delta^2}{\partial x_i} = \frac{1}{m}(\sum_{j=1}^{m}2*(x_j-\overline x)*(-\frac{1}{m}))+2(x_i-\overline x)$

$\frac{\partial y_j}{\partial x_i} = \frac{\partial{\frac{x_j -\overline x}{\sqrt{\delta^2+\epsilon}}}}{\partial x_i}$

$\frac{\partial y_j}{\partial x_i} = -\frac{1}{m}(\delta^2+\epsilon)^{-1/2}-\frac{1}{m}(\delta^2+\epsilon)^{-3/2}(x_i-\overline x)(x_j - \overline x)\quad\quad i \neq j$

$\frac{\partial y_j}{\partial x_i} = (1-\frac{1}{m})(\delta^2+\epsilon)^{-1/2}-\frac{1}{m}(\delta^2+\epsilon)^{-3/2}(x_i-\overline x)(x_j - \overline x)\quad\quad i = j$

$\frac{\partial L}{\partial x_i} = \frac{\partial L}{\partial y_i}*(\delta^2+\epsilon)^{-1/2} + \sum_{j=1}^{m}\frac{\partial L}{\partial y_j}*(-\frac{1}{m}(\delta^2+\epsilon)^{-1/2}-\frac{1}{m}(\delta^2+\epsilon)^{-3/2}(x_i-\overline x)(x_j-\overline x))$

$\frac{\partial L}{\partial x_i} = (\delta^2+\epsilon)^{-1/2}(\frac{\partial L}{\partial y_i}- \sum_{j=1}^{m}\frac{\partial L}{\partial y_j}\frac{1}{m}-\sum_{j=1}^{m}\frac{\partial L}{\partial y_j}\frac{1}{m}(\delta^2+\epsilon)^{-1}(x_i-\overline x)(x_j-\overline x)) \\ =(\delta^2+\epsilon)^{-1/2}(\frac{\partial L}{\partial y_i}- \sum_{j=1}^{m}\frac{\partial L}{\partial y_j}\frac{1}{m}-\sum_{j=1}^{m}\frac{\partial L}{\partial y_j}\frac{1}{m}y_jy_i \\ =(\delta^2+\epsilon)^{-1/2}(\frac{\partial L}{\partial y_i}- \frac{1}{m}\sum_{j=1}^{m}\frac{\partial L}{\partial y_j}-\frac{1}{m}y_i\sum_{j=1}^{m}\frac{\partial L}{\partial y_j}y_j)$

  // if Y = (X-mean(X))/(sqrt(var(X)+eps)), then
//
// dE(Y)/dX =
//   (dE/dY - mean(dE/dY) - mean(dE/dY \cdot Y) \cdot Y)
//     ./ sqrt(var(X) + eps)
//
// where \cdot and ./ are hadamard product and elementwise division,


## 谈谈具体的源码实现

HW的归一化，求出NC个均值与方差，然后N个均值与方差求出一个均值与方差的Vector，size为C，即相同通道的一个mini_batch的样本求出一个mean和variance

### 成员变量

BN层的成员变量比较多，由于在bn的实现中，需要记录mean_,variance_,归一化的值，同时根据训练和测试实现也有所差异。

  Blob<Dtype> mean_,variance_,temp_,x_norm; //temp_保存(x-mean_x)^2
bool use_global_stats_;//标注训练与测试阶段
Dtype moving_average_fraction_;
int channels_;
Dtype eps_; // 防止分母为0

// 中间变量，理解了BN的具体过程即可明了为什么需要这些
Blob<Dtype> batch_sum_multiplier_; // 长度为N*1，全为1，用以求和
Blob<Dtype> num_by_chans_; // 临时保存H*W的结果，length为N*C
Blob<Dtype> spatial_sum_multiplier_; // 统计HW的均值方差使用


### 成员函数

#### LayerSetUp,层次的建立，相应数据的读取

//LayerSetUp函数的具体实现
template <typename Dtype>
void LayerSetUp(const vector<Blob<Dtype>*>& bottom,
const vector<Blob<Dtype>*>& top){
// 参见proto中添加的BatchNormLayer
BathcNormParameter param = this->layer_param_.batch_norm_param();
moving_average_fraction_ = param.moving_average_fraction();//默认0.99

//这里有点多余，好处是防止在测试的时候忘写了use_global_stats时默认true
use_global_stats_ = this->phase_ == TEST;
if (param.has_use_global_stat()) {
use_global_stats_ = param.use_global_stats();
}

if (bottom[0]->num_axes() == 1) { //这里基本看不到为什么.....???
channels_  = 1;
}
else{ // 基本走下面的通道，因为输入是NCHW
channels_ = bottom[0]->shape(1);
}
eps_ = param.eps(); // 默认1e-5
if (this->blobs_.size() > 0) {  // 测试的时候有值了，保存了均值方差和系数
//保存mean,variance,
}
else{
// BN层的内部参数的初始化
this->blobs_.resize(3); // 均值滑动，方差滑动，滑动系数
vector<int>sz;
sz.push_back(channels_);
this->blobs_[0].reset(new Blob<Dtype>(sz)); // C
this->blobs_[1].reset(new Blob<Dtype>(sz)); // C
sz[0] = 1;
this->blobs_[2].reset(new Blob<Dtype>(sz)); // 1
for (size_t i = 0; i < 3; i++) {
caffe_set(this->blobs_[i]->count(),Dtype(0),
this->blobs_[i]->mutable_cpu_data());
}
}
}


#### Reshape,根据BN层在网络的位置，调整bottom和top的shape

Reshape层主要是完成中间变量的值，由于是按照通道求取均值和方差，而CaffeBlob是NCHW,因此先求取了HW,后根据BatchN求最后的输出C,因此有了中间的batch_sum_multiplier_和spatial_sum_multiplier_以及num_by_chans_其中num_by_chans_与前两者不想同，前两者为方便计算，初始为1，而num_by_chans_为中间过渡

template <typename Dtype>
void BatchNormLayer<Dtype>::Reshape(const vector<Blob<Dtype>*>& bottom,
const vector<Blob<Dtype>*>& top) {
if (bottom[0]->num_axes() >= 1) {
CHECK_EQ(bottom[0]->shape(1),channels_);
}
top[0]->ReshapeLike(*bottom[0]); // Reshape(bottom[0]->shape());
vector<int>sz;
sz.push_back(channels_);
mean_.Reshape(sz);
variance_.Reshape(sz);
temp_.ReshapeLike(*bottom[0]);
x_norm_.ReshapeLike(*bottom[0]);
sz[0] = bottom[0]->shape(0); //N
// 后续会初始化为1，为求Nbatch的均值和方差
batch_sum_multiplier_.Reshape(sz);
caffe_set(batch_sum_multiplier_.count(),Dtype(1),
batch_sum_multiplier_.mutable_cpu_data());

int spatial_dim = bottom[0]->count(2);//H*W
if (spatial_sum_multiplier_.num_axes() == 0 ||
spatial_sum_multiplier_.shape(0) != spatial_dim) {
sz[0] = spatial_dim;
spatial_sum_multiplier_.Reshape(sz); //初始化1，方便求和
caffe_set(spatial_sum_multiplier_.count(),Dtype(1)
spatial_sum_multiplier_.mutable_cpu_data());
}

// N*C,保存H*W后的结果,会在计算中结合data与spatial_dim求出
int numbychans = channels_*bottom[0]->shape(0);
if (num_by_chans_.num_axes() == 0 ||
num_by_chans_.shape(0) != numbychans) {
sz[0] = numbychans;
num_by_chans_.Reshape(sz);
}
}


#### Forward 前向计算

template <typename Dtype>
void BatchNormLayer<Dtype>::Forward_cpu(const vector<Blob<Dtype>*>& bottom,
const vector<Blob<Dtype>*>& top) {
// 想要完成前向计算，必须计算相应的均值与方差，此处的均值与方差均为向量的形式c

const Dtype* bottom_data = bottom[0]->cpu_data();
Dtype* top_data = top[0]->mutable_cpu_data();
int num = bottom[0]->shape(0);// N
int spatial_dim = bottom[0]->count(2); //H*W
if (bottom[0] != top[0]) {
caffe_copy(top[0]->count(),bottom_data,top_data);//先复制一下
}

if (use_global_stats_) { // 测试阶段,使用全局的均值
const Dtype scale_factory = this_->blobs_[2]->cpu_data()[0] == 0?
0:1/this->blobs_[2]->cpu_data()[0];
// 直接载入训练的数据 alpha*x = y
caffe_cpu_scale(mean_.count(),scale_factory,
this_blobs_[0]->cpu_data(),mean_.mutable_cpu_data());
caffe_cpu_scale(variance_.count(),scale_factory,
this_blobs_[1]->cpu_data(),variance_.mutable_cpu_data());
}
else{ //训练阶段  compute mean
//1.计算均值,先计算HW的，在包含N
// caffe_cpu_gemv 实现 y =  alpha*A*x+beta*y;
// 输出的是channels_*num,
//每次处理的列是spatial_dim，由于spatial_sum_multiplier_初始为1，即NCHW中的
// H*W各自相加，得到N*C*average，此处多除以了num，下一步可以不除以
caffe_cpu_gemv<Dtype>(CBlasNoTrans,channels_*num,spatial_dim,
1./(spatial_dim*num),bottom_data,spatial_sum_multiplier_.cpu_data()
,0.,num_by_chans_.mutable_cpu_data());

//2.计算均值，计算N各的平均值.
// 由于输出的是channels个均值，因此需要转置
// 上一步得到的N*C的均值，再按照num求均值，因为batch_sum全部为1,
caffe_cpu_gemv<Dtype>(CBlasTrans,num,channels_,1,
num_by_chans_.cpu_data(),batch_sum_multiplier_.cpu_data(),
0,mean_.mutable_cpu_data());
}
// 此处的均值已经保存在mean_中了
// 进行 x - mean_x 操作，需要注意按照通道，即先确定x属于哪个通道.
// 因此也是进行两种，先进行H*W的减少均值
// caffe_cpu_gemm 实现alpha * A*B + beta* C
// 输入是num*1 * 1* channels_,输出是num*channels_
caffe_cpu_gemm<Dtype>(CBlasNoTrans,CBlasNoTrans,num,channels_,1,1,
batch_sum_multiplier_.cpu_data(),mean_.cpu_data(),0,
num_by_chans_.mutable_cpu_data());

//同上，输入是num*channels_*1 * 1* spatial = NCHW
// top_data = top_data - mean;
caffe_cpu_gemm<Dtype>(CBlasNoTrans,CBlasNoTrans,num*channels_,
spatial_dim,1,-1,num_by_chans_.cpu_data(),
spatial_sum_multiplier_.cpu_data(),1, top_data());

// 解决完均值问题，接下来就是解决方差问题
if (use_global_stats_) { // 测试的方差上述已经读取了
// compute variance using var(X) = E((X-EX)^2)
// 此处的top已经为x-mean_x了
caffe_powx(top[0]->count(),top_data,Dtype(2),
temp_.mutable_cpu_data());//temp_保存(x-mean_x)^2

// 同均值一样，此处先计算spatial_dim的值
caffe_cpu_gemv<Dtype>(CblasNoTrans,num*channels_,spatial_dim,
1./(num*spatial_dim),temp_.cpu_data(),
spatial_sum_multiplier_.cpu_data(),0,
num_by_chans_.mutable_cpu_data();
)
caffe_cpu_gemv<Dtype>(CBlasTrans,num,channels_,1.,
num_by_chans_.cpu_data(),batch_sum_multiplier_.cpu_data(),
0,variance_.mutable_cpu_data());// E((X_EX)^2)

//均值和方差计算完成后，需要更新batch的滑动系数
this->blobs_[2]->mutable_cpu_data()[0] *= moving_average_fraction_;
this->blobs_[2]->mutable_cpu_data()[0] += 1;
caffe_cpu_axpby(mean_.count(),Dtype(1),mean_.cpu_data(),
moving_average_fraction_,this->blobs_[0]->mutable_cpu_data());

int m = bottom[0]->count()/channels_;
Dtype bias_correction_factor = m > 1? Dtype(m)/(m-1):1;
caffe_cpu_axpby(variance_.count(),bias_correction_factor,
variance_.cpu_data(),moving_average_fraction_,
this->blobs_[1]->mutable_cpu_data());
}

// 方差求个根号,加上eps为防止分母为0
caffe_powx(variance_.count(),variance_.cpu_data(),Dtype(0.5),
variance_.mutable_cpu_data());

// top_data = x-mean_x/sqrt(variance_),此处的top_data已经转化为x-mean_x了
// 同减均值，也要分C--N*C和  N*C --- N*C*H*W
// N*1 *  1*C == N*C
caffe_cpu_gemm<Dtype>(CBlasNoTrans,CBlasNoTrans,num,channels_,1,1,
batch_sum_multiplier_.cpu_data(),variance_.cpu_data(),0,
num_by_chans_.mutable_cpu_data());
// NC*1 * 1* spatial_dim = NCHW
caffe_cpu_gemm<Dtype>(CBlasNoTrans,CBlasNoTrans,num*channels_,spatial_dim,
1, 1.,num_by_chans_.cpu_data(),spatial_sum_multiplier_.cpu_data(), 0,
temp_.mutable_cpu_data());
// temp最终保存的是sqrt（方差+eps)
caffe_cpu_div(top[0].count(),top_data,temp_.cpu_data(),top_data);
}


#### Backward过程，根据梯度，反向计算

Backward过程会根据前面所推导的公式进行计算，具体的实现如下面所示.

template <typename Dtype>
void BatchNormLayer<Dtype>::Backward_cpu(const vector<Blob<Dtype>*>& top,
const vector<bool>& propagate_down,const vector<Blob<Dtype>*>& bottom) {
const Dtype* top_diff;
if (bottom[0] != top[0]) { // 判断是否同名
top_diff = top[0]->cpu_diff();
}
else{
caffe_copy(x_norm_.count(),top[0]->cpu_diff(),x_norm_.mutable_cpu_diff());
top_diff = x_norm_.cpu_diff();
}
Dtype* bottom_diff = bottom[0]->mutable_cpu_diff();
if (use_global_stats_) { // 测试阶段
caffe_div(temp_.count(),top_diff,temp_.cpu_data(),bottom_diff);
return ; // 测试阶段不需要计算梯度。
}
const Dtype* top_data = x_norm_.cpu_data();
int num = bottom[0]->shape(0); //n
int spatial_dim = bottom[0]->count(2); // H*W

// 根据推导的公式开始具体计算。
// dE(Y)/dX =
//   (top_diff- mean(top_diff) - mean(top_diff \cdot Y) \cdot Y)
//     ./ sqrt(var(X) + eps)

// sum(top_diff \cdot Y) ,y为x_norm_ NCHW,求取的均先求C通道的均值
caffe_mul(temp_.count(),top_data,top_diff,bottom_diff);
//NC*HW* HW*1 =  NC*1
caffe_cpu_gemv<Dtype>(CblasNoTrans,channels_*num,spatial_dim,1.,
bottom_diff,spatial_sum_multiplier_.cpu_data(),0,
num_by_chans_.mutable_cpu_data());
// (NC)^T*1 * N*1 =  C*1
caffe_cpu_gemv<Dtype>(CBlasTrans,num,channels_,1.,
num_by_chans_.cpu_data(),batch_sum_multiplier_.cpu_data(),
0,mean_.mutable_cpu_data());

// N*1  * 1* C = N* C
caffe_cpu_gemm<Dtype>(CblasNoTrans,CblasNoTrans,num,channels_,1,1,
batch_sum_multiplier_.cpu_data(),mean_.cpu_data(),0,
num_by_chans_.mutable_cpu_data());
// N*C *1  * 1* HW =  NC* HW
caffe_cpu_gemm<Dtype>(CblasNoTrans,CblasNoTrans,num*channels_,spatial_dim,
1,1.,num_by_chans_.cpu_data(),spatial_sum_multiplier_.cpu_data(),0,
bottom_diff);
//相当与 sum (DE/DY .\cdot Y)

// sum(dE/dY \cdot Y) \cdot Y
caffe_mul(temp_.count(), top_data, bottom_diff, bottom_diff);

// 完成了右边一个部分，还有前面的 sum(DE/DY)和DE/DY
// 再完成sum(DE/DY)
caffe_cpu_gemv<Dtype>(CblasNoTrans,channels_*num,spatial_dim,1,
top_diff,spatial_sum_multiplier_.cpu_data(),0.,
num_by_chans_.mutable_cpu_data());
caffe_cpu_gemv<Dtype>(CBlasTrans,num,channels_,1.,
num_by_chans_.cpu_data(),batch_sum_multiplier_.cpu_data(),0,
mean_.mutable_cpu_data());
caffe_cpu_gemm<Dtype>(CblasNoTrans,CblasNoTrans,num,channels_,1,
1,batch_sum_multiplier_.cpu_data(),mean_.cpu_data(),0,
num_by_chans_.mutable_cpu_data());
// 现在完成了sum(DE/DY)+y*sum(DE/DY.\cdot y)
caffe_cpu_gemm<Dtype>(CblasNoTrans,CblasNoTrans,num*channels_,spatial_dim,
1,1.,num_by_chans_.cpu_data(),spatial_sum_multiplier_.cpu_data(),1,
bottom_diff);

//top_diff - 1/m * (sum(DE/DY)+y*sum(DE/DY.\cdot y))
caffe_cpu_axpby(bottom[0]->count(),Dtype(1),top_diff,
Dtype(-1/(num*spatial_dim)),bottom_diff);

// 前面还有常数项 variance_+eps
caffe_div(temp_.count(),bottom_diff,temp_.cpu_data(),bottom_diff);
}


backward的过程也是先求出通道的值，然后广播到整个feature_map,来回两次，然后调用axpby完成 top_diff - 1/m* (sum(top_diff)+ysum(top_diffy)))这里的y针对通道进行。

posted @ 2017-11-09 20:41  圆滚滚的小峰峰  阅读(3099)  评论(0编辑  收藏  举报