Softmax {#dev_guide_softmax}
============================

>
> [API Reference](@ref dnnl_api_softmax)
>

## General

The softmax primitive performs forward or backward softmax or logsoftmax
operation along a particular axis on data with arbitrary dimensions. All other
axes are treated as independent (batch).

### Forward

In general form, the operation is defined by the following formulas (the
variable names follow the standard @ref dev_guide_conventions).

Softmax:

\f[
    \dst(\overline{ou}, c, \overline{in}) =
        \frac
        {e^{\src(\overline{ou}, c, \overline{in}) - \nu(\overline{ou}, \overline{in})}}
        {
            \sum\limits_{ic}
                e^{\src(\overline{ou}, ic, \overline{in}) - \nu(\overline{ou}, \overline{in})}
        }
\f]

Logsoftmax:

\f[
    \dst(\overline{ou}, c, \overline{in}) =
        \ln\left({\frac
        {
            e^{\src(\overline{ou}, c, \overline{in}) - \nu(\overline{ou}, \overline{in})}
        }
        {
            \sum\limits_{ic}
                e^{\src(\overline{ou}, ic, \overline{in}) - \nu(\overline{ou}, \overline{in})}
        }}\right) =
        \left(\src(\overline{ou}, c, \overline{in}) - \nu(\overline{ou}, \overline{in})\right)
            - \ln\left(
                    \sum\limits_{ic}
                    e^{\src(\overline{ou}, ic, \overline{in}) - \nu(\overline{ou}, \overline{in})}
                 \right)
\f]

Above

- \f$c\f$ is the axis over which the operation is computed on,
- \f$\overline{ou}\f$ is the outermost index (to the left of the axis),
- \f$\overline{in}\f$ is the innermost index (to the right of the axis), and
- \f$\nu\f$ is used to produce numerically stable results and defined as:

\f[
    \nu(\overline{ou}, \overline{in}) =
        \max\limits_{ic}
        \src(\overline{ou}, ic, \overline{in})
\f]

#### Difference Between Forward Training and Forward Inference

There is no difference between the #dnnl_forward_training
and #dnnl_forward_inference propagation kinds.

### Backward

The backward propagation computes \f$\diffsrc(ou, c, in)\f$, based on
\f$\diffdst(ou, c, in)\f$ and \f$\dst(ou, c, in)\f$.

## Execution Arguments
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                                                              |
| \dst                        | DNNL_ARG_DST                                                              |
| \diffsrc                    | DNNL_ARG_DIFF_SRC                                                         |
| \diffdst                    | DNNL_ARG_DIFF_DST                                                         |
| \f$src scale\f$             | DNNL_ARG_ATTR_SCALES \| DNNL_ARG_SRC                                      |
| \f$dst scale\f$             | DNNL_ARG_ATTR_SCALES \| DNNL_ARG_DST                                      |
| \f$\text{binary post-op}\f$ | DNNL_ARG_ATTR_MULTIPLE_POST_OP(binary_post_op_position) \| DNNL_ARG_SRC_1,|
|                             | DNNL_ARG_ATTR_MULTIPLE_POST_OP(binary_post_op_position) \| DNNL_ARG_SRC_2 |

## Implementation Details

### General Notes

1. Both forward and backward propagation support in-place 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 in-place operation, the original data will be overwritten. This support is
   limited to cases when data types of \src and \dst or \diffsrc and \diffdst
   are identical.

### Post-ops and Attributes

Attributes enable you to modify the behavior of the softmax primitive.
The following attributes are supported by the softmax primitive:

| Propagation | Type      | Operation                                                             | Description                                                   | Restrictions                                                           |
|:------------|:----------|:----------------------------------------------------------------------|:--------------------------------------------------------------|:-----------------------------------------------------------------------|
| forward     | attribute | [Scales](@ref dnnl::primitive_attr::set_scales_mask)                  | Scales the corresponding tensor by the given scale factor(s). | Supported only for int8 softmax and one scale per tensor is supported. |
| forward     | post-op   | [Binary](@ref dnnl::post_ops::append_binary)                          | Applies a @ref dnnl_api_binary operation to the result        | General binary post-op restrictions                                    |
| forward     | Post-op   | [Eltwise](@ref dnnl::post_ops::append_eltwise)                        | Applies an @ref dnnl_api_eltwise operation to the result.     |                                                                        |
| forward     | attribute | [Accumulation mode](@ref dnnl::primitive_attr::set_accumulation_mode) | Defines the implementation's accumulation arithmetic.         | Only the values `strict`, `relaxed`, and `any` are supported.          |

#### Accumulation Mode

You can optimize performance of the forward operation when the source and
destination floating-point data types of the operation are equal and different
from `f32`. When the destination data type is different from `f32`, additional
memory will be used to accumulate data and store it in the destination memory
buffer for a requested data type. Using the additional memory can be opted-out
with an accumulation mode setting set to
[relaxed](@ref dnnl::accumulation_mode::relaxed) or
[any](@ref dnnl::accumulation_mode::any), which will use the precision of
destination data type to accumulate intermediate results directly into the
destination memory buffer. This performance optimization, however, results in
in a minor decrease in accuracy. Depending on the actual data, the difference
between `strict` and `relaxed` accumulation can reach several units in the last
piece (ulps).

### Data Type Support

The softmax primitive supports the following combinations of data types:

| Propagation | Source                      | Destination                 |
|:------------|:----------------------------|:----------------------------|
| forward     | f32, f64, bf16, f16, u8, s8 | f32, f64, bf16, f16, u8, s8 |
| backward    | f32, f64, bf16, f16         | f32, f64, bf16, f16         |

### Data Representation

#### Source, Destination, and Their Gradients

The softmax primitive works with arbitrary data tensors. There is no special
meaning associated with any logical dimensions. However, the softmax axis is
typically referred to as channels (hence in formulas \f$c\f$ is used).


## Implementation Limitations

1. Refer to @ref dev_guide_data_types for limitations related to data types
   support.

2. **GPU**
   - Only tensors of 6 or fewer dimensions are supported.

## Performance Tips

1. Use in-place operations whenever possible.

## Examples

See @ref dev_guide_examples page for a complete list. Softmax examples are listed in the
[Activation Functions](@ref examples_activation) section.
