Performance library for Deep Learning
1.96.0
Eltwise

API Reference

## General

### Forward

The eltwise primitive applies an operation to every element of the tensor (the variable names follow the standard Naming Conventions):

$\dst(\overline{s}) = Operation(\src(\overline{s})),$

where $$\overline{s} = (s_n, .., s_0)$$.

The following operations are supported:

Operation oneDNN algorithm kind Forward formula Backward formula (from src) Backward formula (from dst)
abs dnnl_eltwise_abs $$d = \begin{cases} s & \text{if}\ s > 0 \\ -s & \text{if}\ s \leq 0 \end{cases}$$ $$ds = \begin{cases} dd & \text{if}\ s > 0 \\ -dd & \text{if}\ s < 0 \\ 0 & \text{if}\ s = 0 \end{cases}$$
bounded_relu dnnl_eltwise_bounded_relu $$d = \begin{cases} \alpha & \text{if}\ s > \alpha \geq 0 \\ s & \text{if}\ 0 < s \leq \alpha \\ 0 & \text{if}\ s \leq 0 \end{cases}$$ $$ds = \begin{cases} dd & \text{if}\ 0 < s \leq \alpha, \\ 0 & \text{otherwise}\ \end{cases}$$
clip dnnl_eltwise_clip $$d = \begin{cases} \beta & \text{if}\ s > \beta \geq \alpha \\ s & \text{if}\ \alpha < s \leq \beta \\ \alpha & \text{if}\ s \leq \alpha \end{cases}$$ $$ds = \begin{cases} dd & \text{if}\ \alpha < s \leq \beta \\ 0 & \text{otherwise}\ \end{cases}$$
elu dnnl_eltwise_elu
dnnl_eltwise_elu_use_dst_for_bwd
$$d = \begin{cases} s & \text{if}\ s > 0 \\ \alpha (e^s - 1) & \text{if}\ s \leq 0 \end{cases}$$ $$ds = \begin{cases} dd & \text{if}\ s > 0 \\ dd \cdot \alpha e^s & \text{if}\ s \leq 0 \end{cases}$$ $$ds = \begin{cases} dd & \text{if}\ d > 0 \\ dd \cdot (d + \alpha) & \text{if}\ d \leq 0 \end{cases}. See\ (2).$$
exp dnnl_eltwise_exp
dnnl_eltwise_exp_use_dst_for_bwd
$$d = e^s$$ $$ds = dd \cdot e^s$$ $$ds = dd \cdot d$$
gelu_erf dnnl_eltwise_gelu_erf $$d = 0.5 s (1 + \mathop{erf}[\frac{s}{\sqrt{2}}])$$ $$ds = dd \cdot \left(0.5 + 0.5 \, \mathop{erf}\left({\frac{s}{\sqrt{2}}}\right) + \frac{s}{\sqrt{2\pi}}e^{-0.5s^{2}}\right)$$
gelu_tanh dnnl_eltwise_gelu_tanh $$d = 0.5 s (1 + \tanh[\sqrt{\frac{2}{\pi}} (s + 0.044715 s^3)])$$ $$See\ (1).$$
linear dnnl_eltwise_linear $$d = \alpha s + \beta$$ $$ds = \alpha \cdot dd$$
log dnnl_eltwise_log $$d = \log_{e}{s}$$ $$ds = \frac{dd}{s}$$
logistic dnnl_eltwise_logistic
dnnl_eltwise_logistic_use_dst_for_bwd
$$d = \frac{1}{1+e^{-s}}$$ $$ds = \frac{dd}{1+e^{-s}} \cdot (1 - \frac{1}{1+e^{-s}})$$ $$ds = dd \cdot d \cdot (1 - d)$$
pow dnnl_eltwise_pow $$d = \alpha s^{\beta}$$ $$ds = dd \cdot \alpha \beta s^{\beta - 1}$$
relu dnnl_eltwise_relu
dnnl_eltwise_relu_use_dst_for_bwd
$$d = \begin{cases} s & \text{if}\ s > 0 \\ \alpha s & \text{if}\ s \leq 0 \end{cases}$$ $$ds = \begin{cases} dd & \text{if}\ s > 0 \\ \alpha \cdot dd & \text{if}\ s \leq 0 \end{cases}$$ $$ds = \begin{cases} dd & \text{if}\ d > 0 \\ \alpha \cdot dd & \text{if}\ d \leq 0 \end{cases}. See\ (2).$$
round dnnl_eltwise_round $$d = round(s)$$
soft_relu dnnl_eltwise_soft_relu $$d = \log_{e}(1+e^s)$$ $$ds = \frac{dd}{1 + e^{-s}}$$
sqrt dnnl_eltwise_sqrt
dnnl_eltwise_sqrt_use_dst_for_bwd
$$d = \sqrt{s}$$ $$ds = \frac{dd}{2\sqrt{s}}$$ $$ds = \frac{dd}{2d}$$
square dnnl_eltwise_square $$d = s^2$$ $$ds = dd \cdot 2 s$$
swish dnnl_eltwise_swish $$d = \frac{s}{1+e^{-\alpha s}}$$ $$ds = \frac{dd}{1 + e^{-\alpha s}}(1 + \alpha s (1 - \frac{1}{1 + e^{-\alpha s}}))$$
tanh dnnl_eltwise_tanh
dnnl_eltwise_tanh_use_dst_for_bwd
$$d = \tanh{s}$$ $$ds = dd \cdot (1 - \tanh^2{s})$$ $$ds = dd \cdot (1 - d^2)$$

$$(1)\ ds = dd \cdot 0.5 (1 + tanh[\sqrt{\frac{2}{\pi}} (s + 0.044715 s^3)]) \cdot (1 + \sqrt{\frac{2}{\pi}} (s + 0.134145 s^3) \cdot (1 - tanh[\sqrt{\frac{2}{\pi}} (s + 0.044715 s^3)]) )$$

$$(2)\ \text{Operation is supported only for } \alpha \geq 0.$$

#### 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 $$\diffsrc(\overline{s})$$, based on $$\diffdst(\overline{s})$$ and $$\src(\overline{s})$$. However, some operations support a computation using $$\dst(\overline{s})$$ memory produced during forward propagation. Refer to the table above for a list of operations supporting destination as input memory and the corresponding formulas.

#### Exceptions

The eltwise primitive with algorithm round does not support backward propagation.

## 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
$$binary post-op$$ DNNL_ARG_ATTR_MULTIPLE_POST_OP(binary_post_op_position) | DNNL_ARG_SRC_1

## Implementation Details

### General Notes

1. All eltwise primitives have a common initialization function (e.g., dnnl::eltwise_forward::desc::desc()) which takes both parameters $$\alpha$$, and $$\beta$$. These parameters are ignored if they are unused.
2. 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::eltwise_forward::desc::desc()). The same holds for $$\diffsrc$$ and $$\diffdst$$. The corresponding memory descriptors are referred to as diff_data_desc.
3. 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 an in-place operation, the original data will be overwritten. Note, however, that some algorithms for backward propagation require original $$\src$$, hence the corresponding forward propagation should not be performed in-place for those algorithms. Algorithms that use $$\dst$$ for backward propagation can be safely done in-place.
4. For some operations it might be beneficial to compute backward propagation based on $$\dst(\overline{s})$$, rather than on $$\src(\overline{s})$$, for improved performance.
Note
For operations supporting destination memory as input, $$\dst$$ can be used instead of $$\src$$ when backward propagation is computed. This enables several performance optimizations (see the tips below).

### Data Type Support

The eltwise primitive supports the following combinations of data types:

Propagation Source / Destination Int
forward / backward f32, bf16 f32
forward f16 f16
forward s32 / s8 / u8 f32
Warning
There might be hardware and/or implementation specific restrictions. Check Implementation Limitations section below.

Here the intermediate data type means that the values coming in are first converted to the intermediate data type, then the operation is applied, and finally the result is converted to the output data type.

### Data Representation

The eltwise primitive works with arbitrary data tensors. There is no special meaning associated with any logical dimensions.

### Post-ops and Attributes

Propagation Type Operation Description Restrictions
Forward Post-op Binary Applies a Binary operation to the result General binary post-op restrictions

## Implementation Limitations

1. Refer to Data Types for limitations related to data types support.

## Performance Tips

1. For backward propagation, use the same memory format for $$\src$$, $$\diffdst$$, and $$\diffsrc$$ (the format of the $$\diffdst$$ and $$\diffsrc$$ are always the same because of the API). Different formats are functionally supported but lead to highly suboptimal performance.
2. Use in-place operations whenever possible (see caveats in General Notes).
3. As mentioned above for all operations supporting destination memory as input, one can use the $$\dst$$ tensor instead of $$\src$$. This enables the following potential optimizations for training:
• Such operations can be safely done in-place.
• Moreover, such operations can be fused as a post-op with the previous operation if that operation does not require its $$\dst$$ to compute the backward propagation (e.g., if the convolution operation satisfies these conditions).

## Examples

Engine Name Com
CPU/GPU Element-Wise Primitive Example This C++ API example demonstrates how to create and execute an Element-wise primitive in forward training propagation mode.