Batch Normalization¶
The batch normalization primitive performs a forward or backward batch normalization operation on tensors with number of dimensions equal to 2 or more. Variable names follow the standard Conventions.
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.
The different flavors of the primitive are controlled by the flags
parameter that is passed to the operation descriptor initialization function
like dnnl::batch_normalization_forward::desc::desc()
. Multiple flags can
be combined using the bitwise OR operator (
).
Forward¶
where
\(\gamma(c)\) and \(\beta(c)\) are optional scale and shift for a channel (controlled using the
use_scaleshift
flag),\(\mu(c)\) and \(\sigma^2(c)\) are mean and variance for a channel (controlled using the
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 the training mode, the primitive also optionally supports fusion with ReLU
activation with zero negative slope applied to the result (see
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 \(\hat\mu_i\) and \(\hat\sigma^2_i\) where \(i\) is iteration number, can be computed as:
Difference Between Forward Training and Forward Inference¶
If mean and variance are computed at runtime (i.e.,
use_global_stats
is not set), they become outputs for the propagation kindforward_training
(because they would be required during the backward propagation) and are not exposed for the propagation kindforward_inference
.If batch normalization is created with ReLU fusion (i.e.,
fuse_norm_relu
is set), for the propagation kindforward_training
the primitive would produce aworkspace
memory as one extra output. This memory is required to compute the backward propagation. When the primitive is executed with propagation kindforward_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.,
use_scaleshift
is 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.
In: \(\src\); Out: \(\dst\) 
In: \(\src\); Out: \(\dst\), \(\mu\), \(\sigma^2\) 
In: \(\diffdst\), \(\src\), \(\mu\), \(\sigma^2\); Out: \(\diffsrc\) 
Same as for 

In: \(\src\), \(\mu\), \(\sigma^2\); Out: \(\dst\) 
In: \(\src\), \(\mu\), \(\sigma^2\); Out: \(\dst\) 
In: \(\diffdst\), \(\src\), \(\mu\), \(\sigma^2\); Out: \(\diffsrc\) 
Same as for 


In: \(\src\), \(\gamma\), \(\beta\); Out: \(\dst\) 
In: \(\src\), \(\gamma\), \(\beta\); Out: \(\dst\), \(\mu\), \(\sigma^2\) 
In: \(\diffdst\), \(\src\), \(\mu\), \(\sigma^2\), \(\gamma\), \(\beta\); Out: \(\diffsrc\), \(\diffgamma\), \(\diffbeta\) 
Not supported 

In: \(\src\), \(\mu\), \(\sigma^2\), \(\gamma\), \(\beta\); Out: \(\dst\) 
In: \(\src\), \(\mu\), \(\sigma^2\), \(\gamma\), \(\beta\); Out: \(\dst\) 
In: \(\diffdst\), \(\src\), \(\mu\), \(\sigma^2\), \(\gamma\), \(\beta\); Out: \(\diffsrc\), \(\diffgamma\), \(\diffbeta\) 
Not supported 

In: same as with 
In: same as with 
In: same as with 
Same as for 
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 

\(\src\) 

\(\gamma, \beta\) 

mean (\(\mu\)) 

variance (\(\sigma\)) 

\(\dst\) 

workspace 

\(\diffdst\) 

\(\diffsrc\) 

\(\diffgamma, \diffbeta\) 
Operation Details¶
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
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
indnnl::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 Types Support¶
The operation supports the following combinations of data types.
Note
Here we abbreviate data types names for readability. For example, dnnl::memory::data_type::f32
is
abbreviated to f32
.
Propagation 
Source / Destination 
Mean / Variance / ScaleShift 

forward / backward 

forward 

forward 
Data Representation¶
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 

3D 
NCDHW 
Here optimized means the format chosen by the preceding computeintensive primitive.
Statistics Tensors¶
The mean (\(\mu\)) and variance (\(\sigma^2\)) are separate 1D tensors of size \(C\).
The format of the corresponding memory object must be x
(a
).
If used, the scale (\(\gamma\)) and shift (\(\beta\)) are combined in a single 2D tensor of shape \(2 \times C\).
The format of the corresponding memory object must be nc
(ab
).
Postops and Attributes¶
Propagation 
Type 
Operation 
Description 

forward 
postop 
eltwise 
Applies an eltwise operation to the output. 
Note
Using ReLU as a postop does not produce additional output in the
workspace
that is required to compute backward propagation correctly.
Hence, one should use the fuse_norm_relu
flag for training.
API¶

struct
dnnl
::
batch_normalization_forward
: public dnnl::primitive¶ Batch normalization forward propagation primitive.
Public Functions

batch_normalization_forward
()¶ Default constructor. Produces an empty object.

batch_normalization_forward
(const primitive_desc &pd)¶ Constructs a batch normalization forward propagation primitive.
 Parameters
pd
: Primitive descriptor for a batch normalization forward propagation primitive.

struct
desc
¶ Descriptor for a batch normalization forward propagation primitive.
Public Functions

desc
(prop_kind aprop_kind, const memory::desc &data_desc, float epsilon, normalization_flags flags)¶ Constructs a batch normalization descriptor for forward propagation.
 Note
Inplace operation is supported: the dst can refer to the same memory as the src.
 Parameters
aprop_kind
: Propagation kind. Possible values are dnnl::prop_kind::forward_training and dnnl::prop_kind::forward_inference.data_desc
: Source and destination memory descriptors.epsilon
: Batch normalization epsilon parameter.flags
: Batch normalization flags (dnnl::normalization_flags).


struct
primitive_desc
: public dnnl::primitive_desc¶ Primitive descriptor for a batch normalization forward propagation primitive.
Public Functions

primitive_desc
()¶ Default constructor. Produces an empty object.

primitive_desc
(const desc &adesc, const engine &aengine, bool allow_empty = false)¶ Constructs a primitive descriptor for a batch normalization forward propagation primitive.
 Parameters
adesc
: Descriptor for a batch normalization forward propagation primitive.aengine
: Engine to use.allow_empty
: A flag signifying whether construction is allowed to fail without throwing an exception. In this case an empty object will be produced. This flag is optional and defaults to false.

primitive_desc
(const desc &adesc, const primitive_attr &attr, const engine &aengine, bool allow_empty = false)¶ Constructs a primitive descriptor for a batch normalization forward propagation primitive.
 Parameters
adesc
: Descriptor for a batch normalization forward propagation primitive.attr
: Primitive attributes to use.aengine
: Engine to use.allow_empty
: A flag signifying whether construction is allowed to fail without throwing an exception. In this case an empty object will be produced. This flag is optional and defaults to false.

memory::desc
src_desc
() const¶ Returns a source memory descriptor.
 Return
Source memory descriptor.
 Return
A zero memory descriptor if the primitive does not have a source parameter.

memory::desc
dst_desc
() const¶ Returns a destination memory descriptor.
 Return
Destination memory descriptor.
 Return
A zero memory descriptor if the primitive does not have a destination parameter.

memory::desc
weights_desc
() const¶ Returns a weights memory descriptor.
 Return
Weights memory descriptor.
 Return
A zero memory descriptor if the primitive does not have a weights parameter.

memory::desc
workspace_desc
() const¶ Returns the workspace memory descriptor.
 Return
Workspace memory descriptor.
 Return
A zero memory descriptor if the primitive does not require workspace parameter.



struct
dnnl
::
batch_normalization_backward
: public dnnl::primitive¶ Batch normalization backward propagation primitive.
Public Functions

batch_normalization_backward
()¶ Default constructor. Produces an empty object.

batch_normalization_backward
(const primitive_desc &pd)¶ Constructs a batch normalization backward propagation primitive.
 Parameters
pd
: Primitive descriptor for a batch normalization backward propagation primitive.

struct
desc
¶ Descriptor for a batch normalization backward propagation primitive.
Public Functions

desc
(prop_kind aprop_kind, const memory::desc &diff_data_desc, const memory::desc &data_desc, float epsilon, normalization_flags flags)¶ Constructs a batch normalization descriptor for backward propagation.
 Parameters
aprop_kind
: Propagation kind. Possible values are dnnl::prop_kind::backward_data and dnnl::prop_kind::backward (diffs for all parameters are computed in this case).diff_data_desc
: Diff source and diff destination memory descriptor.data_desc
: Source memory descriptor.epsilon
: Batch normalization epsilon parameter.flags
: Batch normalization flags (dnnl::normalization_flags).


struct
primitive_desc
: public dnnl::primitive_desc¶ Primitive descriptor for a batch normalization backward propagation primitive.
Public Functions

primitive_desc
()¶ Default constructor. Produces an empty object.

primitive_desc
(const desc &adesc, const engine &aengine, const batch_normalization_forward::primitive_desc &hint_fwd_pd, bool allow_empty = false)¶ Constructs a primitive descriptor for a batch normalization backward propagation primitive.
 Parameters
adesc
: Descriptor for a batch normalization backward propagation primitive.aengine
: Engine to use.hint_fwd_pd
: Primitive descriptor for a batch normalization forward propagation primitive. It is used as a hint for deciding which memory format to use.allow_empty
: A flag signifying whether construction is allowed to fail without throwing an exception. In this case an empty object will be produced. This flag is optional and defaults to false.

primitive_desc
(const desc &adesc, const primitive_attr &attr, const engine &aengine, const batch_normalization_forward::primitive_desc &hint_fwd_pd, bool allow_empty = false)¶ Constructs a primitive descriptor for a batch normalization backward propagation primitive.
 Parameters
adesc
: Descriptor for a batch normalization backward propagation primitive.attr
: Primitive attributes to use.aengine
: Engine to use.hint_fwd_pd
: Primitive descriptor for a batch normalization forward propagation primitive. It is used as a hint for deciding which memory format to use.allow_empty
: A flag signifying whether construction is allowed to fail without throwing an exception. In this case an empty object will be produced. This flag is optional and defaults to false.

memory::desc
src_desc
() const¶ Returns a source memory descriptor.
 Return
Source memory descriptor.
 Return
A zero memory descriptor if the primitive does not have a source parameter.

memory::desc
weights_desc
() const¶ Returns a weights memory descriptor.
 Return
Weights memory descriptor.
 Return
A zero memory descriptor if the primitive does not have a weights parameter.

memory::desc
dst_desc
() const¶ Returns a destination memory descriptor.
 Return
Destination memory descriptor.
 Return
A zero memory descriptor if the primitive does not have a destination parameter.

memory::desc
diff_src_desc
() const¶ Returns a diff source memory descriptor.
 Return
Diff source memory descriptor.
 Return
A zero memory descriptor if the primitive does not have a diff source memory with.

memory::desc
diff_dst_desc
() const¶ Returns a diff destination memory descriptor.
 Return
Diff destination memory descriptor.
 Return
A zero memory descriptor if the primitive does not have a diff destination parameter.

memory::desc
diff_weights_desc
() const¶ Returns a diff weights memory descriptor.
 Return
Diff weights memory descriptor.
 Return
A zero memory descriptor if the primitive does not have a diff weights parameter.

memory::desc
mean_desc
() const¶ Returns memory descriptor for mean.
 Return
Memory descriptor for mean.

