# Convolution¶

API Reference

## General¶

The convolution primitive computes forward, backward, or weight update for a batched convolution operation on 1D, 2D, or 3D spatial data with bias.

The convolution operation is defined by the following formulas. We show formulas only for 2D spatial data which are straightforward to generalize to cases of higher and lower dimensions. Variable names follow the standard Naming Conventions.

Let $$\src$$, $$\weights$$ and $$\dst$$ be $$N \times IC \times IH \times IW$$, $$OC \times IC \times KH \times KW$$, and $$N \times OC \times OH \times OW$$ tensors respectively. Let $$\bias$$ be a 1D tensor with $$OC$$ elements.

Furthermore, let the remaining convolution parameters be:

Parameter

Depth

Height

Width

Comment

$$PD_L$$

$$PH_L$$

$$PW_L$$

In the API we use padding_l to indicate the corresponding vector of paddings ( _l in the name stands for left )

$$PD_R$$

$$PH_R$$

$$PW_R$$

In the API we use padding_r to indicate the corresponding vector of paddings ( _r in the name stands for right )

Stride

$$SD$$

$$SH$$

$$SW$$

Convolution without strides is defined by setting the stride parameters to 1

Dilation

$$DD$$

$$DH$$

$$DW$$

Non-dilated convolution is defined by setting the dilation parameters to 0

The following formulas show how oneDNN computes convolutions. They are broken down into several types to simplify the exposition, but in reality the convolution types can be combined.

To further simplify the formulas, we assume that $$\src(n, ic, ih, iw) = 0$$ if $$ih < 0$$, or $$ih \geq IH$$, or $$iw < 0$$, or $$iw \geq IW$$.

### Forward¶

#### Regular Convolution¶

$\begin{split}\dst(n, oc, oh, ow) = \bias(oc) \\ + \sum_{ic=0}^{IC-1}\sum_{kh=0}^{KH-1}\sum_{kw=0}^{KW-1} \src(n, ic, oh \cdot SH + kh - PH_L, ow \cdot SW + kw - PW_L) \cdot \weights(oc, ic, kh, kw).\end{split}$

Here:

• $$OH = \left\lfloor{\frac{IH - KH + PH_L + PH_R}{SH}} \right\rfloor + 1,$$

• $$OW = \left\lfloor{\frac{IW - KW + PW_L + PW_R}{SW}} \right\rfloor + 1.$$

#### Convolution with Groups¶

In the API, oneDNN adds a separate groups dimension to memory objects representing $$\weights$$ tensors and represents weights as $$G \times OC_G \times IC_G \times KH \times KW$$ 5D tensors for 2D convolutions with groups.

$\begin{split}\dst(n, g \cdot OC_G + oc_g, oh, ow) = \bias(g \cdot OC_G + oc_g) \\ + \sum_{ic_g=0}^{IC_G-1}\sum_{kh=0}^{KH-1}\sum_{kw=0}^{KW-1} \src(n, g \cdot IC_G + ic_g, oh \cdot SH + kh - PH_L, ow \cdot SW + kw - PW_L) \cdot \weights(g, oc_g, ic_g, kh, kw),\end{split}$

where

• $$IC_G = \frac{IC}{G}$$,

• $$OC_G = \frac{OC}{G}$$, and

• $$oc_g \in [0, OC_G).$$

The case when $$OC_G = IC_G = 1$$ is also known as a depthwise convolution.

#### Convolution with Dilation¶

$\begin{split}\dst(n, oc, oh, ow) = \bias(oc) \\ + \sum_{ic=0}^{IC-1}\sum_{kh=0}^{KH-1}\sum_{kw=0}^{KW-1} \src(n, ic, oh \cdot SH + kh \cdot (DH + 1) - PH_L, ow \cdot SW + kw \cdot (DW + 1) - PW_L) \cdot \weights(oc, ic, kh, kw).\end{split}$

Here:

• $$OH = \left\lfloor{\frac{IH - DKH + PH_L + PH_R}{SH}} \right\rfloor + 1,$$ where $$DKH = 1 + (KH - 1) \cdot (DH + 1)$$, and

• $$OW = \left\lfloor{\frac{IW - DKW + PW_L + PW_R}{SW}} \right\rfloor + 1,$$ where $$DKW = 1 + (KW - 1) \cdot (DW + 1)$$.

#### Deconvolution (Transposed Convolution)¶

Deconvolutions (also called fractionally strided convolutions or transposed convolutions) work by swapping the forward and backward passes of a convolution. One way to put it is to note that the weights define a convolution, but whether it is a direct convolution or a transposed convolution is determined by how the forward and backward passes are computed.

#### 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 $$\diffsrc$$ based on $$\diffdst$$ and $$\weights$$.

The weights update computes $$\diffweights$$ and $$\diffbias$$ based on $$\diffdst$$ and $$\src$$.

Note

The optimized memory formats $$\src$$ and $$\weights$$ might be different on forward propagation, backward propagation, and weights update.

## 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

$$\weights$$

DNNL_ARG_WEIGHTS

$$\bias$$

DNNL_ARG_BIAS

$$\dst$$

DNNL_ARG_DST

$$\diffsrc$$

DNNL_ARG_DIFF_SRC

$$\diffweights$$

DNNL_ARG_DIFF_WEIGHTS

$$\diffbias$$

DNNL_ARG_DIFF_BIAS

$$\diffdst$$

DNNL_ARG_DIFF_DST

$$depthwise$$

DNNL_ARG_ATTR_POST_OP_DW

$$\text{binary post-op}$$

DNNL_ARG_ATTR_MULTIPLE_POST_OP(binary_post_op_position) | DNNL_ARG_SRC_1

## Implementation Details¶

N/A.

### Data Types¶

Convolution primitive supports the following combination of data types for source, destination, and weights memory objects:

Propagation

Source

Weights

Destination

Bias

forward

f32

f32

f32, s8

f32

forward

f16

f16

f16, f32, u8, s8

f16, f32

forward

u8, s8

s8

u8, s8, s32, f32, f16, bf16

u8, s8, s32, f32, f16, bf16

forward

bf16

bf16

f32, bf16

f32, bf16

backward

f32, bf16

bf16

bf16

backward

f32

f32

f32

f32

weights update

bf16

f32, bf16

bf16, s8, u8

f32, bf16

Warning

There might be hardware and/or implementation specific restrictions. Check Implementation Limitations section below.

### Data Representation¶

Like other CNN primitives, the convolution primitive expects the following tensors:

Spatial

Source / Destination

Weights

1D

$$N \times C \times W$$

$$[G \times ] OC \times IC \times KW$$

2D

$$N \times C \times H \times W$$

$$[G \times ] OC \times IC \times KH \times KW$$

3D

$$N \times C \times D \times H \times W$$

$$[G \times ] OC \times IC \times KD \times KH \times KW$$

Physical format of data and weights memory objects is critical for convolution primitive performance. In the oneDNN programming model, convolution is one of the few primitives that support the placeholder memory format tag dnnl::memory::format_tag::any (shortened to any from now on) and can define data and weight memory objects format based on the primitive parameters. When using any it is necessary to first create a convolution primitive descriptor and then query it for the actual data and weight memory objects formats.

While convolution primitives can be created with memory formats specified explicitly, the performance is likely to be suboptimal.

The table below shows the combinations for which plain memory formats the convolution primitive is optimized for.

Spatial

Convolution Type

Data / Weights logical tensor

Implementation optimized for memory formats

1D, 2D, 3D

any

optimized

1D

f32, bf16

NCW / OIW, GOIW

1D

1D

int8

NCW / OIW

2D

f32, bf16

NCHW / OIHW, GOIHW

2D

2D

int8

NCHW / OIHW, GOIHW

3D

f32, bf16

NCDHW / OIDHW, GOIDHW

3D

3D

int8

NCDHW / OIDHW

### Post-ops and Attributes¶

Post-ops and attributes enable you to modify the behavior of the convolution primitive by applying the output scale to the result of the primitive and by chaining certain operations after the primitive. The following attributes and post-ops are supported:

Propagation

Type

Operation

Description

Restrictions

forward

attribute

Output scale

Scales the result of convolution by given scale factor(s)

int8 convolutions only

forward

attribute

Zero points

Sets zero point(s) for the corresponding tensors

int8 convolutions only

forward

post-op

Eltwise

Applies an Eltwise operation to the result

forward

post-op

Sum

Adds the operation result to the destination tensor instead of overwriting it

forward

post-op

Binary

Applies a Binary operation to the result

General binary post-op restrictions

To facilitate dynamic quantization, the primitive supports run-time output scales. That means a user could configure attributes with output scales set to the DNNL_RUNTIME_F32_VAL wildcard value instead of the actual scales, if the scales are not known at the primitive descriptor creation stage. In this case, the user must provide the scales as an additional input memory object with argument DNNL_ARG_ATTR_OUTPUT_SCALES during the execution stage.

Similarly to run-time output scales, the primitive supports run-time zero points. The wildcard value for zero points is DNNL_RUNTIME_S32_VAL. The following masks are supported by the primitive:

• 0, which applies one zero point value to an entire tensor, and

• 2, which applies a zero point value per each element in a IC or OC dimension for DNNL_ARG_SRC or DNNL_ARG_DST arguments respectively.

During the execution stage, the corresponding memory object must be passed as an argument with its index set to (DNNL_ARG_ATTR_ZERO_POINTS | DNNL_ARG_${MEMORY_INDEX}). Possible ${MEMORY_INDEX} values are DNNL_ARG_SRC and DNNL_ARG_DST.

• For instance, a source tensor zero points memory argument would be passed with index (DNNL_ARG_ATTR_ZERO_POINTS | DNNL_ARG_SRC).

Note

The library does not prevent using post-ops in training, but note that not all post-ops are feasible for training usage. For instance, using ReLU with non-zero negative slope parameter as a post-op would not produce an additional output workspace that is required to compute backward propagation correctly. Hence, in this particular case one should use separate convolution and eltwise primitives for training.

The library supports any number and order of post operations, but only the following sequences deploy optimized code:

Type of convolutions

Post-ops sequence supported

f32 and bf16 convolution

eltwise, sum, sum -> eltwise

int8 convolution

eltwise, sum, sum -> eltwise, eltwise -> sum

The attributes and post-ops take effect in the following sequence:

• Source zero point attribute,

• Output scale attribute,

• Post-ops, in the order they were attached,

• Destination zero point attribute.

The operations during attributes and post-ops applying are done in single precision floating point data type. The conversion to the actual destination data type happens just before the actual storing.

#### Example 1¶

Consider the following pseudo-code:

primitive_attr attr;
attr.set_post_ops({
{ sum={scale=beta} },
{ eltwise={scale=gamma, type=tanh, alpha=ignore, beta=ignored } }
});

convolution_forward(src, weights, dst, attr);

The would lead to the following:

$\dst(\overline{x}) = \gamma \cdot \tanh \left( \alpha \cdot conv(\src, \weights) + \beta \cdot \dst(\overline{x}) \right)$

#### Example 2¶

The following pseudo-code:

primitive_attr attr;
attr.set_post_ops({
{ eltwise={scale=gamma, type=relu, alpha=eta, beta=ignored } },
{ sum={scale=beta} }
});

convolution_forward(src, weights, dst, attr);

That would lead to the following:

$\dst(\overline{x}) = \beta \cdot \dst(\overline{x}) + \gamma \cdot ReLU \left( \alpha \cdot conv(\src, \weights), \eta \right)$

#### Example 3¶

The following pseudo-code:

primitive_attr attr;
attr.set_post_ops({
{ eltwise={scale=gamma, type=relu, alpha=eta, beta=ignored } }
});

convolution_forward(src, weights, dst, attr);

That would lead to the following:

$\dst(\overline{x}) = \gamma \cdot ReLU \left( \alpha \cdot conv(\src - shift_{src}, \weights), \eta \right) + shift_{dst}$

## Algorithms¶

oneDNN implements convolution primitives using several different algorithms:

• Direct. The convolution operation is computed directly using SIMD instructions. This is the algorithm used for the most shapes and supports int8, f32 and bf16 data types.

• Winograd. This algorithm reduces computational complexity of convolution at the expense of accuracy loss and additional memory operations. The implementation is based on the Fast Algorithms for Convolutional Neural Networks by A. Lavin and S. Gray. The Winograd algorithm often results in the best performance, but it is applicable only to particular shapes. Moreover, Winograd only supports int8 and f32 data types.

• Implicit GEMM. The convolution operation is reinterpreted in terms of matrix-matrix multiplication by rearranging the source data into a scratchpad memory. This is a fallback algorithm that is dispatched automatically when other implementations are not available. GEMM convolution supports the int8, f32, and bf16 data types.

### Direct Algorithm¶

oneDNN supports the direct convolution algorithm on all supported platforms for the following conditions:

• Data and weights memory formats are defined by the convolution primitive (user passes any).

• The number of channels per group is a multiple of SIMD width for grouped convolutions.

• For each spatial direction padding does not exceed one half of the corresponding dimension of the weights tensor.

• Weights tensor width does not exceed 14.

In case any of these constraints are not met, the implementation will silently fall back to an explicit GEMM algorithm.

oneDNN supports the Winograd convolution algorithm on systems with Intel(R) Advanced Vector Extensions 512 (Intel(R) AVX-512) support and Intel Deep Learning Boost (Intel DL Boost) under the following conditions:

• Data and weights memory formats are defined by the convolution primitive (user passes any as the data format).

• The spatial domain is two-dimensional.

• The weights shape is 3x3, there are no groups, dilation or strides ($$KH = KW = 3$$, $$SH = SW = 1$$, and $$DH = DW = 0$$).

• The data type is either int8 or f32.

In case any of these constraints is not met, the implementation will silently fall back to the direct algorithm.

The Winograd convolution algorithm implementation additionally chooses tile size based on the problem shape and propagation kind :

• For forward_inference oneDNN supports $$F(2 \times 2, 3 \times 3)$$ or $$F(4 \times 4, 3 \times 3)$$

• oneDNN supports only $$F(4 \times 4, 3 \times 3)$$ Winograd for all the training propagation kinds.

The following side effects should be weighed against the (potential) performance boost achieved from using the Winograd algorithm:

• Memory consumption. Winograd implementation in oneDNN requires additional scratchpad memory to store intermediate results. As more convolutions using Winograd are added to the topology, the amount of memory required can grow significantly. This growth can be controlled if the scratchpad memory can be reused across multiple primitives. See Primitive Attributes: Scratchpad for more details.

• Accuracy. In some cases Winograd convolution produce results that are significantly less accurate than results from the direct convolution.

Create a Winograd convolution by simply creating a convolution descriptor (step 6 in simple network example specifying the Winograd algorithm. The rest of the steps are exactly the same.

auto conv1_desc = convolution_forward::desc(
conv1_src_md, conv1_weights_md, conv1_bias_md, conv1_dst_md,
conv1_strides, conv1_padding_l, conv1_padding_r);

### Automatic Algorithm Selection¶

oneDNN supports dnnl::algorithm::convolution_auto algorithm that instructs the library to automatically select the best algorithm based on the heuristics that take into account tensor shapes and the number of logical processors available. (For automatic selection to work as intended, use the same thread affinity settings when creating the convolution as when executing the convolution.)

## Implementation Limitations¶

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

2. CPU

• Winograd are implemented only for processors with Intel AVX-512 and Intel DL Boost instruction sets

• Run-time output scales are not supported

• Integer $$\dst$$ is not supported for floating point $$\src$$ and $$\weights$$

• f16 $$\dst$$ is not supported for integer $$\src$$ and $$\weights$$

• backward convolution with bias is not supported

## Performance Tips¶

• Use dnnl::memory::format_tag::any for source, weights, and destinations memory format tags when create a convolution primitive to allow the library to choose the most appropriate memory format.

## Examples¶

### convolution_example_cpp - CPU/GPU¶

This C++ API example demonstrates how to create and execute a Convolution primitive in forward propagation mode in two configurations - with and without groups.

Key optimizations included in this example:

• Creation of optimized memory format from the primitive descriptor;

• Primitive attributes with fused post-ops.