Layer Normalization

API reference: C, C++

The layer normalization primitive performs a forward or backward layer normalization operation on 2-5D data tensor.

The layer normalization operation performs normalization over the last logical axis of data tensor and is defined by the following formulas. We show formulas only for 3D data which are straightforward to generalize to cases of higher dimensions. Variable names follow the standard Naming Conventions.

Forward

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

where

• \(\gamma(c), \beta(c)\) are optional scale and shift for a channel (see dnnl_use_scaleshift flag),
• \(\mu(t, n), \sigma^2(t, n)\) are computed at run-time or provided by a user mean and variance (see dnnl_use_global_stats flag), and
• \(\varepsilon\) is a constant to improve numerical stability.

When mean and variance are computed at a run-time the following formulas are used:

• \(\mu(t, n) = \frac{1}{C} \sum\limits_{c} src(t, n, c)_{}\),
• \(\sigma^2(t, n) = \frac{1}{C} \sum\limits_{c} {}_{} (src(t, n, c) - \mu(t, n))^2\).

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

Difference Between Forward Training and Forward Inference

• If mean and variance are computed at run-time (i.e., dnnl_use_global_stats is not set), they become outputs for the propagation kind dnnl_forward_training (since they would be required during the backward propagation). Data layout for mean and variance must be specified when initializing layer normalization descriptor by passing memory descriptor for statistics (e.g., by passing stat_desc in dnnl::layer_normalization_forward::desc::desc()). Mean and variance are not exposed for the propagation kind dnnl_forward_inference.

Backward

The backward propagation computes \(diff\_src(t, n, c)\), \(diff\_\gamma(c)^*\), and \(diff\_\beta(c)^*\) based on \(diff\_dst(t, n, c)\), \(src(t, n, c)\), \(\mu(t, n)\), \(\sigma^2(t, n)\), \(\gamma(c) ^*\), 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 is set).

Execution Arguments

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

Propagation	Flag	Input	Out
forward inference		src	dst
forward inference	S	src, scaleshift	dst
forward inference	G	src, mean, variance, scaleshift	dst
forward training		src	dst, mean, variance
forward training	S	src, scaleshift	dst, mean, variance
forward training	G	src, mean, variance, scaleshift	dst
backward data		src, mean, variance, diff_dst	diff_src
backward data & weights	S	src, mean, variance, diff_dst, scaleshift	diff_src, diff_scaleshift

Implementation Details

General Notes

1. 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::layer_normalization_forward::desc::desc()). Multiple flags can be set using the bitwise OR operator (|).
2. For forward propagation, the mean and variance might be either computed at run-time (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.
3. The memory format and data type for src and dst are assumed to be the same, and in the API are typically referred as data (e.g., see data_desc in dnnl::layer_normalization_forward::desc::desc()). The same holds for diff_src and diff_dst. The corresponding memory descriptors are referred to as diff_data_desc.
4. Both forward and backward propagation support in-place operations, meaning that src can be used as input and output for forward propagation, and diff_dst can be used as input and output for backward propagation. In case of in-place operation, the original data will be overwritten.

Data Type Support

The operation supports the following combinations of data types:

Propagation	Source / Destination	Mea
forward / backward	f32	f32

Data Representation

Mean and Variance

The mean ( \(\mu\)) and variance ( \(\sigma^2\)) are separate tensors with number of dimensions equal to ( \(data\_ndims - 1\)) and size \((data\_dim[0], data\_dim[1], ..., data\_dim[ndims - 2]).\)

Corresponding memory object can have arbitrary memory format. Unless mean and variance are computed at runtime and not exposed (i.e. propagation kind is dnnl_forward_inference and dnnl_use_global_stats is not set), user should provide memory descriptor for statistics when initializing layer normalization descriptor. For best performance it is adviced to use memory format that follows the data memory format, i.e. data format is dnnl_tnc, best performance can be expected for statistics with dnnl_tn format and suboptimal for statistics with dnnl_nc format.

Scale and Shift

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 dnnl_nc (dnnl_ab).

Source, Destination, and Their Gradients

Layer normalization primitive works with arbitrary data tensor, however it was designed for RNN data tensors(i.e. dnnl_nc, dnnl_tnc, dnnl_ldnc). Unlike CNN data tensors, RNN data tensors have a single feature dimension. Layer normalization performs normalization over the last logical dimension (feature dimension for RNN tensors) across non-feature dimensions.

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

Logical tensor	Imp
NC	dnnl_nc (dnnl_ab)
TNC	dnnl_tnc (dnnl_abc), dnnl_ntc (dnnl_bac)
LDNC	dnnl_ldnc (dnnl_abcd)

Performance Tips

1. For data tensors (src, dst, diff_src, diff_dst) use memory formats for which last logical axis is the last in the physical memory layout.
2. For mean/variance use memory format that follows the data memory format, i.e. data format is dnnl_tnc, best performance can be expected for statistics with dnnl_tn and suboptimal for statistics with dnnl_nc format.
3. For backward propagation, use the same memory format for src, diff_dst, and diff_src (the format of the diff_dst and diff_src are always the same because of the API). Different formats are functionally supported but lead to highly suboptimal performance.
4. Use in-place operations whenever possible.