Matrix Multiplication {#dev_guide_matmul} ========================================= > > [API Reference](@ref dnnl_api_matmul) > ## General The matrix multiplication (MatMul) primitive computes the product of two 2D tensors with optional bias addition (the variable names follow the standard @ref dev_guide_conventions): \f[ \dst(m, n) = \sum_{k=0}^{K - 1} \left( \src(m, k) \cdot \weights(k, n) \right) + \bias(m, n) \f] The MatMul primitive also supports batching multiple independent matrix multiplication operations, in which case the tensors can be up to 12D: \f[ \dst(bs_0, bs_1, \ldots, m, n) = \sum_{k=0}^{K - 1} \left( \src(bs_0, bs_1, \ldots, m, k) \cdot \weights(bs_0, bs_1, \ldots, k, n) \right) + \bias(bs_0, bs_1, \ldots, m, n) \f] MatMul also supports implicit broadcast semantics i.e., \src can be broadcasted into \weights if the corresponding dimension in \src is 1 (and vice versa). However, all tensors (including \bias, if it exists) must have the same number of dimensions. The shape of \dst only depends on \src and \weights tensors. The \bias cannot change the dimensions of \dst by broadcasting. In other words, for every dimension, the following constraint must hold true: `dimension(bias) == dimension(dst) || dimension(bias) == 1`. ## 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 | | \f$\text{dropout output mask}\f$ | DNNL_ARG_ATTR_DROPOUT_MASK | | \f$\text{dropout probability}\f$ | DNNL_ARG_ATTR_DROPOUT_PROBABILITY | | \f$\text{dropout rng seed}\f$ | DNNL_ARG_ATTR_DROPOUT_SEED | | \f$\text{binary post-op}\f$ | DNNL_ARG_ATTR_MULTIPLE_POST_OP(binary_post_op_position) \| DNNL_ARG_SRC_1 | | \f$\text{prelu post-op}\f$ | DNNL_ARG_ATTR_MULTIPLE_POST_OP(prelu_post_op_position) \| DNNL_ARG_WEIGHTS | ## Implementation Details ### General Notes 1. The MatMul primitive supports input and output tensors with run-time specified shapes and memory formats. The run-time specified dimensions or strides are specified using the #DNNL_RUNTIME_DIM_VAL wildcard value during the primitive initialization and creation stage. At the execution stage, the user must pass fully specified memory objects so that the primitive is able to perform the computations. Note that the less information about shapes or format is available at the creation stage, the less performant execution will be. In particular, if the shape is not known at the creation stage, you cannot use the special format tag #dnnl::memory::format_tag::any to enable an implementation to choose the most appropriate memory format for the corresponding input or output shapes. On the other hand, run-time specified shapes enable users to create a primitive once and use it in different situations. 2. Inconsistency with dimensions being "primitive-creation-time-defined" vs "runtime-defined" is invalid. For example, \src and \weights with dimensions set to `{3, 4, 4}` and `{DNNL_RUNTIME_DIM_VAL, 4, 4}` respectively is invalid. 3. The broadcasting shape consistency check is not done for the dimensions with #DNNL_RUNTIME_DIM_VAL. Make sure the dimensions for the tensors are valid. 4. Multiple batch dimensions and broadcasting of batch dimensions of `src` and `weights` are supported for both CPU and GPU engines. Check the tutorials below to see #DNNL_RUNTIME_DIM_VAL support in use. ### Data Types The MatMul primitive supports the following combinations of data types for source, destination, weights, and bias tensors: | Source | Weights | Destination | Bias | |:-----------------|:---------------------|:---------------------------------|:----------------------------| | f64 | f64 | f64 | f64, f32, f16, bf16, s8, u8 | | f32 | f32 | f32 | f32, bf16, f16, u8, s8 | | f16 | f16, u8, s8, u4, s4 | f16, u8, s8 | f32 | | f16 | f16, u8, s8 | f32 | f32, f16 | | bf16 | bf16, u8, s8, u4, s4 | f32, bf16 | f32, bf16 | | f32, bf16, f16 | u8, s8 | f32, bf16, f16 | f32, bf16, f16 | | f8_e5m2, f8_e4m3 | f8_e5m2, f8_e4m3 | f32, f16, bf16, f8_e5m2, f8_e4m3 | f32, bf16, f16 | | u8, s8 | s8 | u8, s8, s32, f32, f16, bf16 | u8, s8, s32, f32, f16, bf16 | ### Data Representation The MatMul primitive expects the following tensors: | Dims | Source | Weights | Destination | Bias | |:-----|:--------------------------------|:--------------------------------|:--------------------------------|:----------------------------------------------------------| | 2D | M \f$\times\f$ K | K \f$\times\f$ N | M \f$\times\f$ N | None or \f$(M \text{ or } 1) \times (N \text{ or } 1)\f$ | | ND | S \f$\times\f$ M \f$\times\f$ K | W \f$\times\f$ K \f$\times\f$ N | D \f$\times\f$ M \f$\times\f$ N | None or B | where for the sake of notational convenience, we have \f[ S = \prod_{i = 0}^{ND - 3} \mathrm{src\_dims}[i], \; W = \prod_{i = 0}^{ND - 3} \mathrm{weights\_dims}[i] \\ D = \prod_{i = 0}^{ND - 3} \mathrm{\dst\_dims}[i], \; B = \prod_{i = 0}^{ND - 1} \left( \mathrm{\dst\_dims}[i] \mbox{ or } 1 \right) \f] The MatMul primitive is generally optimized for the case in which memory objects use plain memory formats. Additionally, the \src and \weights must have at least one of the axes `m` or `k` and `n` or `k` contiguous (i.e., stride=1) respectively. However, it is recommended to use the placeholder memory format #dnnl::memory::format_tag::any if an input tensor is reused across multiple executions. In this case, the primitive will set the most appropriate memory format for the corresponding input tensor. The memory format of the destination tensor should always be plain with `n` axis contiguous. For example, #dnnl::memory::format_tag::ab for the 2D case and #dnnl::memory::format_tag::abc or #dnnl::memory::format_tag::bac for the 3D one. ### Attributes and Post-ops Attributes and post-ops enable modifying the behavior of the MatMul primitive. The following attributes and post-ops are supported: | Type | Operation | Description | Restrictions | |:----------|:---------------------------------------------------------------|:------------------------------------------------------------------------------|:------------------------------------| | Attribute | [Scales](@ref dnnl::primitive_attr::set_scales_mask) | Scales the result by given scale factor(s) | | | Attribute | [Zero-points](@ref dnnl::primitive_attr::set_zero_points_mask) | Sets zero point(s) for the corresponding tensors | Int8 computations only | | Attribute | [Dropout](@ref dnnl::primitive_attr::set_dropout) | Applies pseudo-random dropout to destination buffer, also fills mask buffer | | | Post-op | [Eltwise](@ref dnnl::post_ops::append_eltwise) | Applies an @ref dnnl_api_eltwise operation to the result | | | Post-op | [Sum](@ref dnnl::post_ops::append_sum) | Adds the operation result to the destination tensor instead of overwriting it | | | Post-op | [Binary](@ref dnnl::post_ops::append_binary) | Applies a @ref dnnl_api_binary operation to the result | General binary post-op restrictions | | Post-op | [Prelu](@ref dnnl::post_ops::append_prelu) | Applies an @ref dnnl_api_prelu operation to the result | | The following masks are supported by the primitive: - 0, which applies one scale / zero point value to an entire tensor, and - 2, which applies a scale value per column along the `n`dimension for `DNNL_ARG_WEIGHTS`. When scales and/or zero-points masks are specified, the user must provide the corresponding scales and/or zero-points as additional input memory objects with argument `DNNL_ARG_ATTR_SCALES | DNNL_ARG_${MEMORY_INDEX}` or `DNNL_ARG_ATTR_ZERO_POINTS | DNNL_ARG_${MEMORY_INDEX}` during the execution stage. For instance, a source tensor zero points memory argument would be passed with index (`DNNL_ARG_ATTR_ZERO_POINTS | DNNL_ARG_SRC`). When Dropout is specified, at the execution stage the user must provide 2 input memory objects with `DNNL_ARG_ATTR_DROPOUT_PROBABILITY` (1x1x...x1 f32 value from 0.f to 1.f) and `DNNL_ARG_DROPOUT_SEED` (1x1x...x1 s32 value from INT_MIN to INT_MAX), and 1 output memory object with `DNNL_ARG_ATTR_DROPOUT_MASK` (u8 memory buffer that shares its shape with the destination buffer). @note Please check tutorials below to see run-time attributes in use. ## Implementation Limitations 1. Check @ref dev_guide_data_types. 2. **GPU** - Supports up to 6 dimensions. - Source zero point mask of `0` is only supported. - Sum post-op doesn't support data type other than destination data type. - Bias of bf16 data type is supported for configuration with bf16 source data type and weights bf16 data type, and up to three dimensional matrices. - Only reference support is available for f8_e4m3. Optimized implementation for f8_e5m2 is available only on Intel(R) Data Center GPU Max Series. - Configuration with int8 source data type, s8 weight data type and bf16 destination data type don't support: * Destination zero point. * Runtime dimensions. * Three and higher dimensional matrices. - The layout of dropout mask has to be exactly the same as that of dst. 3. **CPU** - Configuration with int8 source data type, s8 weight data type and f16 destination data type isn't supported. - Configuration with floating point source data type, integer weights data type and floating point destination data type is not optimized. - Only reference support for fp8 data types (f8_e5m2, f8_e4m3) is is available on CPU. - The layout of dropout mask has to be exactly the same as that of dst. ## Performance Tips - Use #dnnl::memory::format_tag::any for either of the input tensors if and only if the shape of the corresponding tensor is fully known at creation time and it is possible to cache reordered tensors across multiple primitive executions. For instance, a good candidate for reuse are the weights tensors during inference: their shapes and data types are known in advance; thus they can be reordered during the first inference pass and can be reused during the subsequent passes. However, if any of the input tensors cannot be reused, it is best to force the primitive to use the same format as that used by the tensors. ## Examples The following examples are available: ### Matrix Multiplication Primitive Examples [MatMul Primitive Example](@ref matmul_example_cpp) @copydetails matmul_example_cpp_short [MatMul Tutorial: Comparison with SGEMM](@ref cpu_sgemm_and_matmul_cpp) (CPU only) @copydetails cpu_sgemm_and_matmul_cpp_short [MatMul Tutorial: INT8 Inference](@ref inference_int8_matmul_cpp) @copydetails inference_int8_matmul_cpp_short [MatMul Tutorial: Quantization](@ref cpu_matmul_quantization_cpp) (CPU only) @copydetails cpu_matmul_quantization_cpp_short [MatMul Tutorial: Weights decompression](@ref weights_decompression_matmul_cpp) (CPU only) @copydetails weights_decompression_matmul_cpp_short