Deep Neural Network Library (DNNL)
1.2.0

Performance library for Deep Learning

Layer Normalization

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.

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

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.

- 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.

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).

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 | Output |
---|---|---|---|

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 |

- 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 (`|`

). - 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.
- 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`

. - 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.

The operation supports the following combinations of data types:

Propagation | Source / Destination | Mean / Variance / ScaleShift |
---|---|---|

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 \((data\_dim[0], data\_dim[1], ..., data\_dim[ndims - 2]).\)

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 advised 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.

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).

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 | Implementations optimized for memory formats |
---|---|

NC | dnnl_nc (dnnl_ab) |

TNC | dnnl_tnc (dnnl_abc), dnnl_ntc (dnnl_bac) |

LDNC | dnnl_ldnc (dnnl_abcd) |

- 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. - 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. - 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. - Use in-place operations whenever possible.