BatchNormTrainingBackprop

BatchNormTrainingBackprop  // Compute mean and variance backprop from the input.

Description

Computes the input, gamma and beta backprop increments.

Inputs

Name	Element Type	Shape
input	real	\((\bullet, C, \ldots)\)
gamma	same as input	\((C)\)
beta	same as input	\((C)\)
mean	same as input	\((C)\)
variance	same as input	\((C)\)
normalized_delta	same as input	same as input

Attributes

Name	Type	Notes
epsilon	double	Small bias added to variance to avoid division by 0.

Outputs

Name	Element Type	Shape
input_delta	same as input	Same as input
gamma_delta	same as gamma	\((C)\)
beta_delta	same as beta	\((C)\)

Mathematical Definition

It is easiest to simplify by looking at a single channel and flattening the remaining axes into a vector; so gamma and beta are scalars, and input is an \(N\)-element vector.

The step by step forward training computation is

\[\begin{split}\mathtt{mean} &= \frac{\sum{\mathtt{input}_i}}{N}\\ \mathtt{centered}_i &= \mathtt{input}_i - \mathtt{mean}\\ \mathtt{square}_i &= \mathtt{centered}_i^2\\ \mathtt{variance} &= \frac{\sum \mathtt{square}_i}{N}\\ \mathtt{invsqrt} &= \frac{1}{\sqrt{\mathtt{variance}+\epsilon}}\\ \mathtt{gmul} &= \texttt{gamma}\cdot \mathtt{invsqrt}\\ \mathtt{normed}_i &= \mathtt{centered}_i\mathtt{gmul}+\texttt{beta}\end{split}\]

Using the notation \(\overline{\texttt{name}}\) for \(\texttt{name_delta}\) and \(\overline{x} \leftarrow y\) to mean the backprop value for \(\texttt{x_delta}\) is a sum that includes \(y\).

We work backwards

\[\begin{split}\overline{\texttt{beta}}&\leftarrow \overline{\texttt{normed}}\\ \overline{\texttt{gmul}}&\leftarrow \sum \overline{\texttt{normed}}_i\\ \overline{\texttt{centered}}_i&\leftarrow\overline{\texttt{normed}}_i\texttt{gmul}\\ \overline{\texttt{gamma}}&\leftarrow \overline{\texttt{gmul}}\cdot\texttt{invsqrt}\\ \overline{\texttt{invsqrt}}&\leftarrow\texttt{gamma}\cdot\overline{\texttt{gmul}}\\ \overline{\texttt{variance}}&\leftarrow -\frac{\overline{\texttt{invsqrt}}\cdot\texttt{invsqrt}}{2\cdot(\texttt{variance}+\epsilon)}\\ \overline{\texttt{square}}_i&\leftarrow\frac{\overline{\texttt{variance}}}{N}\\ \overline{\texttt{centered}}_i&\leftarrow 2\cdot\texttt{centered}_i\cdot\overline{\texttt{square}}_i\\ \overline{\texttt{input}}_i&\leftarrow\overline{\texttt{centered}}_i\\ \overline{\texttt{mean}}&\leftarrow\sum\overline{\texttt{centered}}_i\\ \overline{\texttt{input}}_i&\leftarrow\frac{\overline{\texttt{mean}}}{N}\end{split}\]

C++ Interface

class BatchNormTrainingBackprop : public ngraph::op::Op

Public Functions

void validate_and_infer_types()

Throws if the node is invalid.