The layer normalization primitive performs a forward or backward layer normalization operation on a 2-5D data tensor.
The layer normalization operation performs normalization over the last logical axis of the 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.
\[ \dst(t, n, c) = \gamma(c) \cdot \frac{\src(t, n, c) - \mu(t, n)} {\sqrt{\sigma^2(t, n) + \varepsilon}} + \beta(c), \]
where
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:
The \(\gamma(c)\) and \(\beta(c)\) tensors are considered learnable.
The backward propagation computes \(\diffsrc(t, n, c)\), \(\diffgamma(c)^*\), and \(\diffbeta(c)^*\) based on \(\diffdst(t, n, c)\), \(src(t, n, c)\), \(\mu(t, n)\), \(\sigma^2(t, n)\), \(\gamma(c) ^*\), 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).
Depending on the flags and propagation kind, the layer normalization primitive requires different inputs and outputs. For clarity, a summary is shown below.
| 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_scaleshift | 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 |
| dnnl_use_global_stats | dnnl_use_scaleshift | 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 |
| \(\gamma, \beta\) | DNNL_ARG_SCALE_SHIFT |
| mean ( \(\mu\)) | DNNL_ARG_MEAN |
| variance ( \(\sigma\)) | DNNL_ARG_VARIANCE |
| \(\dst\) | DNNL_ARG_DST |
| \(\diffdst\) | DNNL_ARG_DIFF_DST |
| \(\diffsrc\) | DNNL_ARG_DIFF_SRC |
| \(\diffgamma\), \(\diffbeta\) | DNNL_ARG_DIFF_SCALE_SHIFT |
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 (|).src and dst are assumed to be the same, and in the API they are typically referred to as data (e.g., see data_desc in dnnl::layer_normalization_forward::desc::desc()). The same is true for diff_src and diff_dst. The corresponding memory descriptors are referred to as diff_data_desc.The operation supports the following combinations of data types:
| Propagation | Source / Destination | Mea |
|---|---|---|
| forward / backward | f32 | f32 |
| forward | f16 | f32 |
The mean ( \(\mu\)) and variance ( \(\sigma^2\)) are separate tensors with number of dimensions equal to ( \(data\_ndims - 1\)) and size \((data\_dim[0], data\_dim[1], ..., data\_dim[ndims - 2])\).
The corresponding memory object can have an 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), the user should provide a memory descriptor for statistics when initializing the layer normalization descriptor. For best performance, it is advised to use the memory format that follows the data memory format; i.e., if the data format is dnnl_tnc, the best performance can be expected for statistics with the dnnl_tn format and suboptimal for statistics with the dnnl_nt format.
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).
The layer normalization primitive works with an 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) |
src, dst, diff_src, diff_dst), use memory formats for which the last logical axis is the last in the physical memory layout.mean/variance, use the memory format that follows the data memory format; i.e., if the data format is dnnl_tnc, the best performance can be expected for statistics with dnnl_tn and suboptimal for statistics with the dnnl_nt format.src, diff_dst, and diff_src (the format of diff_dst and diff_src are always the same because of the API). Different formats are functionally supported but lead to highly suboptimal performance.| Engine | Name | Com |
|---|---|---|
| CPU/GPU | Layer Normalization Primitive Example | This C++ API example demonstrates how to create and execute a Layer normalization primitive in forward propagation mode. Key optimizations included in this example:
|