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.
where
\(\gamma(c), \beta(c)\) are optional scale and shift for a channel (see dnnl_use_scaleshift, 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).
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 is 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 postop (see section below).
Backward¶
The backward propagation computes \(\diffsrc(n, c, h, w)\), \(\diffgamma(c)^*\), and \(\diffbeta(c)^*\) based on \(\diffdst(n, c, h, w)\), \(\src(n, c, h, w)\), \(\mu(c)\), \(\sigma^2(c)\), \(\gamma(c) ^*\), 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_scaleshift, 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.
Inputs : \(\src\) Outputs : \(\dst\) 
Inputs : \(\src\) Outputs : \(\dst\) , \(\mu\) , \(\sigma^2\) 
Inputs : \(\diffdst\) , \(\src\) , \(\mu\) , \(\sigma^2\) Outputs : \(\diffsrc\) 
Same as for dnnl_backward 

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 

Inputs : \(\src\) , \(\gamma\) , \(\beta\) Outputs : \(\dst\) 
Inputs : \(\src\) , \(\gamma\) , \(\beta\) Outputs : \(\dst\) , \(\mu\) , \(\sigma^2\) 
Inputs : \(\diffdst\) , \(\src\) , \(\mu\) , \(\sigma^2\) , \(\gamma\) , \(\beta\) Outputs : \(\diffsrc\) , \(\diffgamma\) , \(\diffbeta\) 
Not supported 

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 

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 

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 

Inputs : same as with Outputs : same as with 
Inputs : same as with Outputs : same as with , Workspace 
Inputs : same as with , Workspace Outputs : same as with 
Same as for dnnl_backward if do not contain dnnl_use_scaleshift ; 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 
\(\gamma, \beta\) 
DNNL_ARG_SCALE_SHIFT 
\(\gamma\) 
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 
\(\diffgamma, \diffbeta\) 
DNNL_ARG_DIFF_SCALE_SHIFT 
\(\diffgamma\) 
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 operation descriptor initialization function (e.g., dnnl::batch_normalization_forward::desc::desc()). Multiple flags can be set using the bitwise OR operator (
). Flag dnnl_use_scaleshift can not be mixed with dnnl_use_scale or dnnl_use_shift.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.
The memory format and data type for
src
anddst
are assumed to be the same, and in the API they are typically referred to asdata
(e.g., seedata_desc
in dnnl::batch_normalization_forward::desc::desc()). The same is true fordiff_src
anddiff_dst
. The corresponding memory descriptors are referred to asdiff_data_desc
.Both forward and backward propagation support inplace 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 inplace operation, the original data will be overwritten. Note, however, that backward propagation requires original \(\src\), hence the corresponding forward propagation should not be performed inplace.
As mentioned above, the batch normalization primitive can be fused with ReLU activation 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 
f32 
forward 
f16 
f32 
forward 
s8 
f32 
Warning
There might be hardware or implementationspecific 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_scaleshift is used, the scale (\(\gamma\)) and shift (\(\beta\)) are combined in a single 2D tensor of shape \(2 \times C\).
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_nc (dnnl_ab).
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 

1D 
NCW 

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 computeintensive primitive.
Postops and Attributes¶
Postops and attributes enable you to modify the behavior of the batch normalization primitive by chaining certain operations after the batch normalization operation. The following postops are supported by batch normalization primitives:
Propagation 
Type 
Operation 
Description 

forward 
postop 
eltwise 
Applies an Eltwise operation to the result (currently only dnnl_eltwise_relu algorithm is supported) 
Note
As mentioned in Primitive Attributes, the postops should be used for inference only. For instance, using ReLU as a postop 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 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).
Performance Tips¶
For backward propagation, use the same memory format for
src
,diff_dst
, anddiff_src
(the format of thediff_dst
anddiff_src
are always the same because of the API). Different formats are functionally supported but lead to highly suboptimal performance.Use inplace operations whenever possible (see caveats in General Notes).
Examples¶
Engine 
Name 
Comments 


CPU/GPU 
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: 