.. SPDX-FileCopyrightText: 2019-2020 Intel Corporation .. .. SPDX-License-Identifier: CC-BY-4.0 .. default-domain:: cpp .. include:: /elements/oneDNN/source/replacements.inc.rst .. _layer_normalization-label: ################### Layer normalization ################### 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 :ref:conventions-label. ******* Forward ******* .. math:: \dst(t, n, c) = \gamma(c) \cdot \frac{\src(t, n, c) - \mu(t, n)} {\sqrt{\sigma^2(t, n) + \varepsilon}} + \beta(c), where - :math:\gamma(c), \beta(c) are optional scale and shift for a channel (see the |use_scaleshift| flag), - :math:\mu(t, n), \sigma^2(t, n) are mean and variance (see |use_global_stats| flag), and - :math:\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: - :math:\mu(t, n) = \frac{1}{C} \sum\limits_{c} \src(t, n, c)_{}, - :math:\sigma^2(t, n) = \frac{1}{C} \sum\limits_{c} {}_{} (\src(t, n, c) - \mu(t, n))^2. The :math:\gamma(c) and :math:\beta(c) tensors are considered learnable. 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 kind |forward_training| (because they would be required during the backward propagation). Data layout for mean and variance must be specified during initialization of the layer normalization descriptor by passing the memory descriptor for statistics (e.g., by passing stat_desc in |layer_normalization_forward::desc::desc|). Mean and variance are not exposed for the propagation kind |forward_inference|. ******** Backward ******** The backward propagation computes :math:\diffsrc(t, n, c), :math:\diffgamma(c)^*, and :math:\diffbeta(c)^* based on :math:\diffdst(t, n, c), :math:src(t, n, c), :math:\mu(t, n), :math:\sigma^2(t, n), :math:\gamma(c) ^*, and :math:\beta(c) ^*. The tensors marked with an asterisk are used only when the primitive is configured to use :math:\gamma(c), and :math:\beta(c) (i.e., |use_scaleshift| is set). ******************* Execution Arguments ******************* Depending on the flags and propagation kind, the layer normalization primitive requires different inputs and outputs. For clarity, a summary is shown below. +----------------------------------------+------------------------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------------+-------------------------------------------------------------------------------------------------------------------------------------------------------------------+------------------------+ | | |forward_inference| | |forward_training| | |backward| | |backward_data| | +========================================+======================================================================================================+======================================================================================================+===================================================================================================================================================================+========================+ | |normalization_flags_none| | *In*: :math:\src *Out*: :math:\dst | *In*: :math:\src *Out*: :math:\dst, :math:\mu, :math:\sigma^2 | *In*: :math:\diffdst, :math:\src, :math:\mu, :math:\sigma^2 *Out*: :math:\diffsrc | Same as for |backward| | +----------------------------------------+------------------------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------------+-------------------------------------------------------------------------------------------------------------------------------------------------------------------+------------------------+ | |use_global_stats| | *In*: :math:\src, :math:\mu, :math:\sigma^2 *Out*: :math:\dst | *In*: :math:\src, :math:\mu, :math:\sigma^2 *Out*: :math:\dst | *In*: :math:\diffdst, :math:\src, :math:\mu, :math:\sigma^2 *Out*: :math:\diffsrc | Same as for |backward| | +----------------------------------------+------------------------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------------+-------------------------------------------------------------------------------------------------------------------------------------------------------------------+------------------------+ | |use_scaleshift| | *In*: :math:\src, :math:\gamma, :math:\beta *Out*: :math:\dst | *In*: :math:\src, :math:\gamma, :math:\beta *Out*: :math:\dst, :math:\mu, :math:\sigma^2 | *In*: :math:\diffdst, :math:\src, :math:\mu, :math:\sigma^2, :math:\gamma, :math:\beta *Out*: :math:\diffsrc, :math:\diffgamma, :math:\diffbeta | Not supported | +----------------------------------------+------------------------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------------+-------------------------------------------------------------------------------------------------------------------------------------------------------------------+------------------------+ | |use_global_stats| \| |use_scaleshift| | *In*: :math:\src, :math:\mu, :math:\sigma^2, :math:\gamma, :math:\beta *Out*: :math:\dst | *In*: :math:\src, :math:\mu, :math:\sigma^2, :math:\gamma, :math:\beta *Out*: :math:\dst | *In*: :math:\diffdst, :math:\src, :math:\mu, :math:\sigma^2, :math:\gamma, :math:\beta *Out*: :math:\diffsrc, :math:\diffgamma, :math:\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 =============================================== ========================= :math:\src |DNNL_ARG_SRC| :math:\gamma, \beta |DNNL_ARG_SCALE_SHIFT| mean (:math:\mu) |DNNL_ARG_MEAN| variance (:math:\sigma) |DNNL_ARG_VARIANCE| :math:\dst |DNNL_ARG_DST| :math:\diffdst |DNNL_ARG_DIFF_DST| :math:\diffsrc |DNNL_ARG_DIFF_SRC| :math:\diffgamma, :math:\diffbeta |DNNL_ARG_DIFF_SCALE_SHIFT| =============================================== ========================= ***************** Operation Details ***************** 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., |layer_normalization_forward::desc::desc|). Multiple flags can be combined using the bitwise OR operator (|). 2. 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. 3. The memory format and data type for 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. 4. Both forward and backward propagation support in-place operations, meaning that :math:\src can be used as input and output for forward propagation, and :math:\diffdst can be used as input and output for backward propagation. In case of an in-place operation, the original data will be overwritten. Note, however, that backward propagation requires original :math:\src, hence the corresponding forward propagation should not be performed in-place. ****************** Data Types Support ****************** The layer normalization supports the following combinations of data types. .. note:: Here we abbreviate data types names for readability. For example, |_f32| is abbreviated to |f32|. ================== ==================== ============================ Propagation Source / Destination Mean / Variance / ScaleShift ================== ==================== ============================ forward / backward |f32| |f32| forward |f16| |f32| ================== ==================== ============================ ******************* Data Representation ******************* Mean and Variance ================= The mean (:math:\mu) and variance (:math:\sigma^2) are separate tensors with number of dimensions equal to (:math:data\_ndims - 1) and size :math:(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 |forward_inference| and |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 |tnc|, the best performance can be expected for statistics with the |tn| format and suboptimal for statistics with the |nt| format. Scale and Shift =============== If used, the scale (:math:\gamma) and shift (:math:\beta) are combined in a single 2D tensor of shape :math:2 \times C. The format of the corresponding memory object must be |nc| (|ab|). Source, Destination, and Their Gradients ======================================== The layer normalization primitive works with an arbitrary data tensor; however, it was designed for RNN data tensors (i.e., |nc|, |tnc|, |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 |nc| (|ab|) TNC |tnc| (|abc|), |ntc| (|bac|) LDNC |ldnc| (|abcd|) ============== ============================================ *** API *** .. doxygenstruct:: dnnl::layer_normalization_forward :project: oneDNN :members: .. doxygenstruct:: dnnl::layer_normalization_backward :project: oneDNN :members: .. vim: ts=3 sw=3 et spell spelllang=en