Group Normalization¶

General¶

The group normalization primitive performs a forward or backward group normalization operation on tensors with numbers of dimensions equal to 3 or more.

Forward¶

The group normalization operation is defined by the following formulas. We show formulas only for 2D spatial data which are straightforward to generalize to cases of higher and lower dimensions. Variable names follow the standard Naming Conventions.

\[\dst(n, g \cdot C_G + c_g, h, w) = \gamma(g \cdot C_G + c_g) \cdot \frac{\src(n, g \cdot C_G + c_g, h, w) - \mu(n, g)} {\sqrt{\sigma^2(n, g) + \varepsilon}} + \beta(g \cdot C_G + c_g),\]

where

\(C_G = \frac{C}{G}\),
\(c_g \in [0, C_G).\),
\(\gamma(c), \beta(c)\) are optional scale and shift for a channel (see dnnl_use_scale and dnnl_use_shift flags),
\(\mu(n, g), \sigma^2(n, g)\) are mean and variance for a group of channels in a batch (see dnnl_use_global_stats flag), and
\(\varepsilon\) is a constant to improve numerical stability.

Mean and variance are computed at runtime or provided by a user. When mean and variance are computed at runtime, the following formulas are used:

\(\mu(n, g) = \frac{1}{(C/G)HW} \sum\limits_{c_ghw} \src(n, g \cdot C_G + c_g, h, w)_{}\),
\(\sigma^2(n, g) = \frac{1}{(C/G)HW} \sum\limits_{c_ghw} {}_{} (\src(n, g \cdot C_G + c_g, h, w) - \mu(n, g))^2\).

The \(\gamma(c)\) and \(\beta(c)\) tensors are considered learnable.

Note

The group normalization primitive computes population mean and variance and not the sample or unbiased versions that are typically used to compute running mean and variance.
Using the mean and variance computed by the group normalization primitive, running mean and variance \(\hat\mu\) and \(\hat\sigma^2\) can be computed as

\[\begin{split}\hat\mu := \alpha \cdot \hat\mu + (1 - \alpha) \cdot \mu, \\ \hat\sigma^2 := \alpha \cdot \hat\sigma^2 + (1 - \alpha) \cdot \sigma^2.\end{split}\]

Difference Between Forward Training and Forward Inference¶

If mean and variance are computed at runtime (i.e., dnnl_use_global_stats is not set), they become outputs for the propagation kind dnnl_forward_training (because they would be required during the backward propagation) and are not exposed for the propagation kind dnnl_forward_inference.

Backward¶

The backward propagation computes \(\diffsrc(n, c, h, w)\), \(\diffgamma(c)^*\), and \(\diffbeta(c)^*\) based on \(\diffdst(n, c, h, w)\), \(\src(n, c, h, w)\), \(\mu(n, g)\), \(\sigma^2(n, g)\), \(\gamma(c) ^*\), and \(\beta(c) ^*\).

The tensors marked with an asterisk are used only when the primitive is configured to use \(\gamma(c)\) and \(\beta(c)\) (i.e., dnnl_use_scale or dnnl_use_shift are set).

Execution Arguments¶

Depending on the flags and propagation kind, the group normalization primitive requires different inputs and outputs. For clarity, a summary is shown below.

Flags	dnnl_forward_inference	dnnl_forward_training	dnnl_backward	dnnl_backward_data
dnnl_normalization_flags_none	Inputs : \(\src\) Outputs : \(\dst\)	Inputs : \(\src\) Outputs : \(\dst\) , \(\mu\) , \(\sigma^2\)	Inputs : \(\diffdst\) , \(\src\) , \(\mu\) , \(\sigma^2\) Outputs : \(\diffsrc\)	Same as for dnnl_backward
dnnl_use_global_stats	Inputs : \(\src\) , \(\mu\) , \(\sigma^2\) Outputs : \(\dst\)	Inputs : \(\src\) , \(\mu\) , \(\sigma^2\) Outputs : \(\dst\)	Inputs : \(\diffdst\) , \(\src\) , \(\mu\) , \(\sigma^2\) Outputs : \(\diffsrc\)	Same as for dnnl_backward
dnnl_use_scale	Inputs : \(\src\) , \(\gamma\) Outputs : \(\dst\)	Inputs : \(\src\) , \(\gamma\) Outputs : \(\dst\) , \(\mu\) , \(\sigma^2\)	Inputs : \(\diffdst\) , \(\src\) , \(\mu\) , \(\sigma^2\) , \(\gamma\) Outputs : \(\diffsrc\) , \(\diffgamma\)	Not supported
dnnl_use_shift	Inputs : \(\src\) , \(\beta\) Outputs : \(\dst\)	Inputs : \(\src\) , \(\beta\) Outputs : \(\dst\) , \(\mu\) , \(\sigma^2\)	Inputs : \(\diffdst\) , \(\src\) , \(\mu\) , \(\sigma^2\) , \(\beta\) Outputs : \(\diffsrc\) , \(\diffbeta\)	Not supported
dnnl_use_global_stats \| dnnl_use_scale \| dnnl_use_shift	Inputs : \(\src\) , \(\mu\) , \(\sigma^2\) , \(\gamma\) , \(\beta\) Outputs : \(\dst\)	Inputs : \(\src\) , \(\mu\) , \(\sigma^2\) , \(\gamma\) , \(\beta\) Outputs : \(\dst\)	Inputs : \(\diffdst\) , \(\src\) , \(\mu\) , \(\sigma^2\) , \(\gamma\) , \(\beta\) Outputs : \(\diffsrc\) , \(\diffgamma\) , \(\diffbeta\)	Not supported

When executed, the inputs and outputs should be mapped to an execution argument index as specified by the following table.

Primitive Input/Output	Execution Argument Index
\(\src\)	DNNL_ARG_SRC
\(\gamma\)	DNNL_ARG_SCALE
\(\beta\)	DNNL_ARG_SHIFT
mean ( \(\mu\) )	DNNL_ARG_MEAN
variance ( \(\sigma^2\) )	DNNL_ARG_VARIANCE
\(\dst\)	DNNL_ARG_DST
\(\diffdst\)	DNNL_ARG_DIFF_DST
\(\diffsrc\)	DNNL_ARG_DIFF_SRC
\(\diffgamma\)	DNNL_ARG_DIFF_SCALE
\(\diffbeta\)	DNNL_ARG_DIFF_SHIFT
\(\text{binary post-op}\)	DNNL_ARG_ATTR_MULTIPLE_POST_OP(binary_post_op_position) \| DNNL_ARG_SRC_1

Implementation Details¶

General Notes¶

The different flavors of the primitive are partially controlled by the flags parameter that is passed to the primitive descriptor creation function (e.g., dnnl::group_normalization_forward::primitive_desc()). Multiple flags can be set using the bitwise OR operator (|).
For forward propagation, the mean and variance might be either computed at runtime (in which case they are outputs of the primitive) or provided by a user (in which case they are inputs). In the latter case, a user must set the dnnl_use_global_stats flag. For the backward propagation, the mean and variance are always input parameters.
Both forward and backward propagation support in-place operations, meaning that \(\src\) can be used as input and output for forward propagation, and \(\diffdst\) can be used as input and output for backward propagation. In case of an in-place operation, the original data will be overwritten. Note, however, that backward propagation requires the original \(\src\), hence the corresponding forward propagation should not be performed in-place.

Data Type Support¶

The operation supports the following combinations of data types:

Propagation	Source / Destination	Mean / Variance / ScaleShift
forward / backward	f32, bf16, f16	f32
forward	s8	f32

Warning

There might be hardware- or implementation-specific restrictions. Check the Implementation Limitations section below.

Data Representation¶

Mean and Variance¶

The mean (\(\mu\)) and variance (\(\sigma^2\)) are separate 2D tensors of size \(N \times G\).

The format of the corresponding memory object must be dnnl_nc (dnnl_ab).

Scale and Shift¶

If dnnl_use_scale or dnnl_use_shift are used, the scale (\(\gamma\)) and shift (\(\beta\)) are separate 1D tensors of shape \(C\).

The format of the corresponding memory object must be dnnl_x (dnnl_a).

Source, Destination, and Their Gradients¶

The group normalization primitive expects data to be \(N \times C \times SP_n \times \cdots \times SP_0\) tensor.

The group normalization primitive is optimized for the following memory formats:

Spatial	Logical tensor	Implementations optimized for memory formats
1D	NCW	dnnl_ncw ( dnnl_abc ), dnnl_nwc ( dnnl_acb )
2D	NCHW	dnnl_nchw ( dnnl_abcd ), dnnl_nhwc ( dnnl_acdb )
3D	NCDHW	dnnl_ncdhw ( dnnl_abcde ), dnnl_ndhwc ( dnnl_acdeb )

Post-Ops and Attributes¶

Attributes enable you to modify the behavior of the group normalization primitive. The following attributes are supported by the group normalization primitive:

Propagation	Type	Operation	Description	Restrictions
forward	attribute	Scales	Scales the corresponding tensor by the given scale factor(s).	Supported only for int8 group normalization and one scale per tensor is supported.
forward	Post-op	Binary	Applies a Binary operation to the result	General binary post-op restrictions
forward	Post-op	Eltwise	Applies an Eltwise operation to the result.

Implementation Limitations¶

Refer to Data Types for limitations related to data types support.

Performance Tips¶

Mixing different formats for inputs and outputs is functionally supported but leads to highly suboptimal performance.
Use in-place operations whenever possible (see caveats in General Notes).

Examples¶

Group Normalization Primitive Example

This C++ API example demonstrates how to create and execute a Group Normalization primitive in forward training propagation mode.

Key optimizations included in this example:

In-place primitive execution;
Source memory format for an optimized primitive implementation;

Eltwise Layer Normalization