Batch Normalization¶

General¶

The batch normalization primitive performs a forward or backward batch normalization operation on tensors with number of dimensions equal to 2 or more.

Forward¶

The batch 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.

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

where

\(\gamma(c), \beta(c)\) are optional scale and shift for a channel (see dnnl_use_scale and dnnl_use_shift flags),
\(\mu(c), \sigma^2(c)\) are mean and variance for a channel (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:

\(\mu(c) = \frac{1}{NHW} \sum\limits_{nhw} \src(n, c, h, w)_{}\),
\(\sigma^2(c) = \frac{1}{NHW} \sum\limits_{nhw} {}_{} (\src(n, c, h, w) - \mu(c))^2\).

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

In training mode, the primitive also optionally supports:

fusion with ReLU activation with zero negative slope applied to the result (see dnnl_fuse_norm_relu flag).
fusion with binary addition and ReLU activation with zero negative slope applied to the result (see dnnl_fuse_norm_add_relu flag).

Note

The batch 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 batch 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.
If batch normalization is created with ReLU fusion (i.e., dnnl_fuse_norm_relu or dnnl_fuse_norm_add_relu are set), for the propagation kind dnnl_forward_training the primitive would produce a workspace memory as one extra output. This memory is required to compute the backward propagation. When the primitive is executed with propagation kind dnnl_forward_inference, the workspace is not produced. Behavior would be the same as creating a batch normalization primitive with ReLU as a post-op (see section below).

Backward¶

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

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 batch 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 : \(\diffdst\) , \(\src\) , \(\mu\) , \(\sigma^2\) 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 : \(\diffdst\) , \(\src\) , \(\mu\) , \(\sigma^2\) , \(\gamma\) Outputs : \(\diffsrc\) , \(\diffgamma\) , \(\diffbeta\)	Not supported
`flags` \| dnnl_fuse_norm_relu	Inputs : same as with `flags` Outputs : same as with `flags`	Inputs : same as with `flags` Outputs : same as with `flags` , Workspace	Inputs : same as with `flags` , Workspace Outputs : same as with `flags`	Same as for dnnl_backward if `flags` do not contain dnnl_use_scale or dnnl_use_shift ; not supported otherwise
`flags` \| dnnl_fuse_norm_add_relu	Inputs : same as with `flags` and \(\src_1\) for fused binary addition Outputs : same as with `flags`	Inputs : same as with `flags` and \(\src_1\) for fused binary addition Outputs : same as with `flags` , Workspace	Inputs : same as with `flags` , Workspace Outputs : same as with `flags` and \(\diffsrc_1\) for fused binary addition	Same as for dnnl_backward if `flags` do not contain dnnl_use_scale or dnnl_use_shift ; not supported otherwise

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
\(\src_1\)	DNNL_ARG_SRC_1
\(\gamma\)	DNNL_ARG_SCALE
\(\beta\)	DNNL_ARG_SHIFT
mean ( \(\mu\) )	DNNL_ARG_MEAN
variance ( \(\sigma^2\) )	DNNL_ARG_VARIANCE
\(\dst\)	DNNL_ARG_DST
workspace	DNNL_ARG_WORKSPACE
\(\diffdst\)	DNNL_ARG_DIFF_DST
\(\diffsrc\)	DNNL_ARG_DIFF_SRC
\(\diffsrc_1\)	DNNL_ARG_DIFF_SRC_1
\(\diffgamma\)	DNNL_ARG_DIFF_SCALE
\(\diffbeta\)	DNNL_ARG_DIFF_SHIFT

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::batch_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 original \(\src\), hence the corresponding forward propagation should not be performed in-place.
As mentioned above, the batch normalization primitive can be fused with binary addition and ReLU activation (dnnl_fuse_norm_add_relu). In this case:
- on the forward propagation the primitive has one additional input, \(\src_1\), that should have memory descriptor equal to primitive dst_desc memory descriptor.
- on the backward propagation the primitive has one additional output, \(\diffsrc_1\), that should have memory descriptor equal to primitive diff_dst_desc memory descriptor.
As mentioned above, the batch normalization primitive can be fused with ReLU activation (dnnl_fuse_norm_relu) or binary addition and ReLU activation (dnnl_fuse_norm_add_relu) even in the training mode. In this case, on the forward propagation the primitive has one additional output, workspace, that should be passed during the backward propagation.

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 1D tensors of size \(C\).

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

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_a.

Source, Destination, and Their Gradients¶

Like other CNN primitives, the batch normalization primitive expects data to be \(N \times C \times SP_n \times \cdots \times SP_0\) tensor.

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

Spatial	Logical tensor	Implementations optimized for memory formats
0D	NC	dnnl_nc ( dnnl_ab )
1D	NCW	dnnl_ncw ( dnnl_abc ), dnnl_nwc ( dnnl_acb ), optimized^
2D	NCHW	dnnl_nchw ( dnnl_abcd ), dnnl_nhwc ( dnnl_acdb ), optimized^
3D	NCDHW	dnnl_ncdhw ( dnnl_abcde ), dnnl_ndhwc ( dnnl_acdeb ), optimized^

Here optimized^ means the format that comes out of any preceding compute-intensive primitive.

Post-Ops and Attributes¶

Post-ops and attributes enable you to modify the behavior of the batch normalization primitive by chaining certain operations after the batch normalization operation. The following post-ops are supported by batch normalization primitives:

Propagation	Type	Operation	Description
forward	post-op	eltwise	Applies an Eltwise operation to the result (currently only dnnl_eltwise_relu algorithm is supported)

Note

As mentioned in Primitive Attributes, the post-ops should be used for inference only. For instance, using ReLU as a post-op would not produce the additional output workspace that is required to compute backward propagation correctly. Hence, in case of training one should use the dnnl_fuse_norm_relu or dnnl_fuse_norm_add_relu directly.

Implementation Limitations¶

Refer to Data Types for limitations related to data types support.
For the data types that have forward propagation support only, mean and variance must be provided by a user (i.e., dnnl_use_global_stats is set).
CPU implementations do not support the fusion with binary addition and ReLU activation (dnnl_fuse_norm_add_relu).

Performance Tips¶

For backward propagation, use the same memory format for \(\src\), \(\diffdst\), and \(\diffsrc\) (the format of the \(\diffdst\) and \(\diffsrc\) are always the same because of the API). Different formats are functionally supported but lead to highly suboptimal performance.
Use in-place operations whenever possible (see caveats in General Notes).
GPU implementations support an experimental algorithm with single pass statistics calculations. Please review experimental features for more details.

Examples¶

Batch Normalization Primitive Example

This C++ API example demonstrates how to create and execute a Batch 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;
Fused post-ops via operation descriptor flags;

RNN Binary