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ALT

README > CUTLASS GEMM API

CUTLASS GEMM API

CUTLASS presents a uniform programming model for matrix multiply-accumulate operations at each level of the hierarchy. This document focuses on device-level, threadblock-level GEMMs, warp-level GEMMs, thread-level GEMMs, and instruction-level GEMMs.

CUTLASS GEMM Model

CUTLASS implements the basic GEMM triple loop nest with a tiled structure mirroring the execution model hierarchy.

The following pseudocode describes the model for a GEMM kernel targeting a warp-synchronous matrix multiply instruction like mma.sync. The entire operation is referred to as "Gemm," as it is assumed that an epilogue operation performs the general matrix update similar to BLAS.

                                                                            // cutlass::gemm::device::Gemm
                                                                            //
for (int cta_n = 0; cta_n < GemmN; cta_n += CtaTileN) {                     // for each CTA       } CTA-level concurrency
  for (int cta_m = 0; cta_m < GemmM; cta_m += CtaTileM) {                   //    for each CTA    }
                                                                            //    
                                                                            // cutlass::gemm::threadblock::Mma
                                                                            //
    for (int cta_k = 0; cta_k < GemmK; cta_k += CtaTileK) {                 //       "GEMM mainloop" - no unrolling - one iteration of this loop is one "stage"
                                                                            //
      for (int warp_n = 0; warp_n < CtaTileN; warp_n += WarpTileN) {        // for each warp      } warp-level concurrency
        for (int warp_m = 0; warp_m < CtaTileM; warp_m += WarpTileM) {      //    for each warp   }
                                                                            //
          for (int warp_k = 0; warp_k < CtaTileK; warp_k += MmaK) {         //       fully unroll across CtaTileK - one iteration of this loop is one "k Group" == "MmaK"
                                                                            //
            for (int mma_k = 0; mma_k < WarpTileK; mma_k += MmaK) {         // cutlass::gemm::warp::Mma
              for (int mma_n = 0; mma_n < WarpTileN; mma_n += MmaN) {       //
                for (int mma_m = 0; mma_m < WarpTileM; mma_m += MmaM) {     //
                                                                            //
                  mma_instruction(d, a, b, c);                              // cutlass::arch::mma - warp-wide matrix multiply instruction

                }   // for mma_m
              }   // for mma_n
            }   // for mma_k

          }   // for warp_k
        }   // for warp_m
      }   // for warp_n

    }   // for cta_k
  }   // for cta_m
}   // for cta_n

The outer-most loops correspond to CTA-level hardware concurrency and are not explicitly written as loops in the code. These are implied by CUDA grid launch semantics.

The comment cutlass::gemm::threadblock::Mma refers to the threadblock-scoped matrix multiply-accumulate concept. This is the computation performed by one threadblock to compute a matrix product in registers. The "GEMM main loop" is listed.

The comment cutlass::gemm::warp::Mma refers to the computation performed by each warp. This is a nested loop executing a sequence of accumulated outer products.

The inner-most operation corresponds directly to hardware support. In this example, the nested structure terminates with warp-synchronous matrix multiply instructions targeting Tensor Cores. Alternatively, GEMMs targeting single-thread instructions may have an additional series of nested loops corresponding to thread-level concurrency.

CUTLASS GEMM Components

This loop nest is expressed in CUTLASS via the following components which are specialized for data type, layout, and math instruction.

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These components are described in the following sections.

Device-wide GEMM API

The device-level GEMM API is intended to streamline instantiation and execution of the standard GEMM computation across the GPU. This operator is intended to be used in host-side .cu code and has semantics similar to cuBLAS.

The device-wide GEMM API is embodied by the following operators:

Example: launch a mixed-precision GEMM targeting Volta Tensor Cores.

  using Gemm = cutlass::gemm::device::Gemm<
    cutlass::half_t,                           // ElementA
    cutlass::layout::ColumnMajor,              // LayoutA
    cutlass::half_t,                           // ElementB
    cutlass::layout::ColumnMajor,              // LayoutB
    cutlass::half_t,                           // ElementOutput
    cutlass::layout::ColumnMajor,              // LayoutOutput
    float,                                     // ElementAccumulator
    cutlass::arch::OpClassTensorOp,            // tag indicating Tensor Cores
    cutlass::arch::Sm70                        // tag indicating target GPU compute architecture
  >;

  Gemm gemm_op;
  cutlass::Status status;
 
  //
  // Launch GEMM on the device
  //
 
  status = gemm_op({
    {m, n, k},
    {ptrA, lda},
    {ptrB, ldb},
    {ptrC, ldc},
    {ptrD, ldd},
    {alpha, beta}
  });

  if (status != cutlass::Status::kSuccess) {
    return -1;
  }

Threadblock-level GEMM API

GEMMs at this scope are expected to efficiently load tiles of data from global memory into internal storage and then compute matrix products with warp-level GEMM operators.

The threadblock-scoped matrix multiply operation is embodied by cutlass::gemm::threadblock::MmaPipelined. This is a class inspired by std::transform_reduce() which computes the accumulated matrix product of a range of tiles defined by tile iterators.

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In the case of GEMM, the tile iterators are cutlass::transform::threadblock::PredicatedTileIterator to traverse a sequence of tiles in global memory with appropriate predication to avoid out-of-bounds memory accesses.

Concept. Threadblock-level matrix multiply accumulate operators are function objects satisfying the following concept.

struct Mma {
  /// Shape of warp-level matrix operation (concept: GemmShape)
  struct Shape;

  /// Data type of multiplicand A (concept: numeric type)
  struct ElementA;

  /// Layout of multiplicand A (concept: Layout)
  struct LayoutA;

  /// Data type of multiplicand B (concept: numeric type)
  struct ElementB;

  /// Layout of multiplicand B (concept: Layout)
  struct LayoutB;

  /// Data type of accumulator matrix C (concept: numeric type)
  struct ElementC;

  /// Layout of accumulator matrix C (concept: Layout)
  struct LayoutC;

  /// Iterator of A operand in shared memory - satisfies: ReadableRandomAccessTileIteratorConcept
  struct IteratorA;

  /// Fragment object loaded from IteratorA (concept: Array<ElementA, ..>)
  struct FragmentA;

  /// Iterator of B operand in shared memory - satisfies: ReadableRandomAccessTileIteratorConcept
  struct IteratorB;

  /// Fragment object loaded from IteratorB (concept: Array<ElementB, ..>)
  struct FragmentB;

  /// Iterator of C operand in shared memory - 
  ///    satisfies: ReadableRandomAccessTileIteratorConcept | WriteableRandomAccessTileIteratorConcept
  struct IteratorC;

  /// Fragment object loaded from IteratorC (concept: Array<ElementC, ..>)
  struct FragmentC;

  /// Warp-level matrix multiply operator (concept: satisfies gemm::warp::Mma)
  struct Operator;

  //
  // Method
  //

  /// Computes a matrix product accumulated in D
  CUTLASS_DEVICE
  void operator()(
    FragmentC &D, 
    IteratorA iter_A, 
    IteratorB iter_B, 
    FragmentC const &C);
};

Warp-level Matrix Multiply API

Warp-level GEMM operators load tiles from shared memory into registers and then compute matrix multiplies using either Tensor Cores or CUDA Cores. The result is accumulated in a register tile. Iterators are defined for each operand A, B, and C.

The warp-level GEMM API is a generalization of CUDA's WMMA API to achieve the following objectives:

  • native matrix multiply sizes of Tensor Cores
  • permuted shared memory layouts to ensure conflict-free accesses
  • pointer initilization outside of the mainloop
  • efficient traversal

Defining a warp-level matrix multiply in CUTLASS is similar to WMMA as shown below.

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The usage model is also similar. The following example computes a warp-level GEMM operation, accumulating a series of matrix products in a register-backed array. The input to a warp-level GEMM operation in CUTLASS must be data in shared memory loaded by iterators or on register-backed fragments.

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#include "cutlass/gemm/warp/default_mma_tensor_op.h"

using LayoutA = cutlass::layout::ColumnMajorTensorOpMultiplicandCongruous<
    cutlass::sizeof_bits<Element>::value, 64>;

using LayoutB = cutlass::layout::RowMajorTensorOpMultiplicandCongruous<
    cutlass::sizeof_bits<Element>::value, 64>;

using WarpMma = typename cutlass::gemm::warp::DefaultMmaTensorOp<
    cutlass::gemm::GemmShape<64, 64, 8>,                            // Overall warp-level GEMM operation
    cutlass::gemm::GemmShape<16, 8, 8>,                             // Target instruction
    cutlass::half_t, LayoutA,                                       // operand A type and layout
    cutlass::half_t, LayoutB,                                       // operand B type and layout
    float,                                                          // accumulator type
    cutlass::layout::RowMajor>::Type;                               // accumulator layout

//
// Define a GEMM operation loading data from shared memory
//
int const kGemmK = 32;

__shared__ ElementA smem_buffer_A[WarpMma::Shape::kM * kGemmK];
__shared__ ElementB smem_buffer_B[WarpMma::Shape::kN * kGemmK];

//
// Construct iterators into SMEM tiles
//

// leading dimensions inferred from matrix problem size
int lda = WarpMma::Shape::kM;
int ldb = WarpMma::Shape::kN;

// iterators into shared memory
WarpMma::IteratorA warp_iterator_A({smem_buffer_A, lda});
WarpMma::IteratorB warp_iterator_B({smem_buffer_B, ldb});

// Fragments in registers storing the operands
FragmentA frag_A;
FragmentB frag_B;
FragmentC accum;

WarpMma mma;

accum.clear();

//
// Accumulated outer product
//

#pragma unroll 1
for (int k = 0; k < kGemmK; k += WarpMma::Shape::kK) {

  
  iter_A.load(frag_A);  // Load fragments from A and B matrices
  iter_B.load(frag_B);

  ++iter_A; ++iter_B;   // Advance along GEMM K to next tile in A
                        //   and B matrices

                        // Compute matrix product
  mma(accum, frag_A, frag_B, accum);
}

Concept. Warp-level Mma operations are function objects satisfying the following concept.

struct Mma {
  /// Shape of warp-level matrix operation (concept: GemmShape)
  struct Shape;

  /// Data type of multiplicand A (concept: numeric type)
  struct ElementA;

  /// Layout of multiplicand A (concept: Layout)
  struct LayoutA;

  /// Data type of multiplicand B (concept: numeric type)
  struct ElementB;

  /// Layout of multiplicand B (concept: Layout)
  struct LayoutB;

  /// Data type of accumulator matrix C (concept: numeric type)
  struct ElementC;

  /// Layout of accumulator matrix C (concept: Layout)
  struct LayoutC;

  /// Iterator of A operand in shared memory - satisfies: ReadableRandomAccessTileIteratorConcept
  struct IteratorA;

  /// Fragment object loaded from IteratorA (concept: Array<ElementA, ..>)
  struct FragmentA;

  /// Iterator of B operand in shared memory - satisfies: ReadableRandomAccessTileIteratorConcept
  struct IteratorB;

  /// Fragment object loaded from IteratorB (concept: Array<ElementB, ..>)
  struct FragmentB;

  /// Iterator of C operand in shared memory - 
  ///     satisfies: ReadableRandomAccessTileIteratorConcept | WriteableRandomAccessTileIteratorConcept
  struct IteratorC;

  /// Fragment object loaded from IteratorC (concept: Array<ElementC, ..>)
  struct FragmentC;

  /// Indicates class of matrix operator (arch::OpClassSimt or arch::OpClassTensorOp)
  struct OperatorClass;

  //
  // Methods
  //

  /// Computes a matrix multiply-accumulate
  CUTLASS_DEVICE
  void operator()(
    FragmentC &D, 
    IteratorA A, 
    IteratorB B, 
    FragmentC const &C);
};

Tensor Core Operators. Warp-level matrix multiply operators targeting Tensor Cores may be defined with the following template arguments. The Policy type specifies implementation-level details which may be used to affect performance or internal implementation of the warp-level operator.

namespace cutlass {
namespace gemm {
namespace warp {

/// Structure to compute the matrix product targeting CUDA cores and SIMT math instructions.
template <
  /// Size of the Gemm problem - concept: gemm::GemmShape<>
  typename Shape_,
  /// Data type of A elements
  typename ElementA_,
  /// Layout of A matrix (concept: MatrixLayout)
  typename LayoutA_,
  /// Data type of B elements
  typename ElementB_,
  /// Layout of B matrix (concept: MatrixLayout)
  typename LayoutB_,
  /// Element type of C matrix
  typename ElementC_,
  /// Layout of C matrix (concept: MatrixLayout)
  typename LayoutC_,
  /// Shape of the warp in units of thread (concept: MmaSimtPolicy)
  typename Policy_,
  /// Used for partial specialization
  typename Enable = bool
>
class MmaTensorOp {}

} // namespace warp
} // namespace gemm
} // namespace cutlass

SIMT Math Instructions. Warp-level matrix multiply operators targeting CUDA Cores may be defined with the following template arguments. The Policy type specifies implementation-level details which may be used to affect performance or internal implementation of the warp-level operator.

/// Structure to compute the matrix product targeting CUDA cores and SIMT math instructions.
template <
  /// Size of the Gemm problem - concept: gemm::GemmShape<>
  typename Shape_,
  /// Data type of A elements
  typename ElementA_,
  /// Layout of A matrix (concept: MatrixLayout)
  typename LayoutA_,
  /// Data type of B elements
  typename ElementB_,
  /// Layout of B matrix (concept: MatrixLayout)
  typename LayoutB_,
  /// Element type of C matrix
  typename ElementC_,
  /// Layout of C matrix (concept: MatrixLayout)
  typename LayoutC_,
  /// Shape of the warp in units of thread (concept: MmaSimtPolicy)
  typename Policy_,
  /// Used for partial specialization
  typename Enable = bool
>
class MmaSimt;

Thread-level GEMM API

Thread-level GEMM operations perform matrix multiply-accumulate on data held in registers. These target CUDA Cores exclusively.

Concept. Thread-level matrix multiply operations are function objects satisfying the following concept.

struct Mma {

  /// Shape of warp-level matrix operation (concept: GemmShape)
  struct Shape;

  /// Data type of multiplicand A (concept: numeric type)
  struct ElementA;

  /// Layout of multiplicand A (concept: Layout)
  struct LayoutA;

  /// Fragment object loaded from IteratorA (concept: Array<ElementA, ..>)
  struct FragmentA;

  /// Data type of multiplicand B (concept: numeric type)
  struct ElementB;

  /// Layout of multiplicand B (concept: Layout)
  struct LayoutB;

  /// Fragment object loaded from IteratorA (concept: Array<ElementB, ..>)
  struct FragmentB;

  /// Data type of accumulator matrix C (concept: numeric type)
  struct ElementC;

  /// Layout of accumulator matrix C (concept: Layout)
  struct LayoutC;

  /// Fragment object loaded from IteratorA (concept: Array<ElementC, ..>)
  struct FragmentC;

  //
  // Methods
  //

  /// Computes a matrix multiply-accumulate
  CUTLASS_DEVICE
  void operator()(
    FragmentC &D, 
    FragmentA const &A, 
    FragmentB const &B, 
    FragmentC const &C);
};

The CUTLASS thread-level GEMM template accepts the following template arguments.

namespace cutlass {
namespace gemm {
namespace thread {

/// Structure to compute the matrix product
template <
  /// Size of the Gemm problem - concept: gemm::GemmShape<>
  typename Shape,
  /// Data type of A elements
  typename ElementA,
  /// Layout of A matrix (concept: MatrixLayout)
  typename LayoutA,
  /// Data type of B elements
  typename ElementB,
  /// Layout of B matrix (concept: MatrixLayout)
  typename LayoutB,
  /// Element type of C matrix
  typename ElementC,
  /// Layout of C matrix (concept: MatrixLayout)
  typename LayoutC,
  /// Concept: arch::OpMultiplyAdd or arch::Mma<>
  typename Operator = arch::OpMultiplyAdd,
  /// Used for partial specialization
  typename Enable = bool
>
struct Mma;

} // namespace thread
} // namespace gemm
} // namespace cutlass

Efficient Epilogue

CUTLASS GEMM operators perform mma followed by epilogue operation similar to cuBLAS. CUTLASS implements an efficient row-major epilogue. Thus, to achieve column-major GEMM, operands A & B are transposed and swapped.

To enable efficient row-major epilogue for both row-major and column-major output layout, CUTLASS' device-level GEMM operators cutlass::device::Gemm and cutlass::device::GemmUniversal provide two template definitions:

Efficient row-major epilogue for:

  • (i) GEMM operator on row-major source/output uses template (a). It runs row-major GEMM and an efficient row-major epilogue.
  • (ii) GEMM operator on column-major source/output uses template (b). It transposes and swaps operands A and B to enable efficient epilogue. A x B = C => Transpose(B) x Transpose(A) = Transpose(C). For column-major source (C) matrix, Transpose(C) is row-major, and efficient epilogue works on row-major.

Note that cuBLAS typically expects a column-major source (C) and output matrix (D). Thus, CUTLASS library only instantiates and generates GEMM operatos with column-major layout. However, CUTLASS by itself can run both row-major and column-major output layouts for all combinations of input layouts. Thus, CUTLASS supports the following layout combinations for input and output layouts:

  • {N,T} x {N,T} => {N,T} - NN, TN, TN, TT GEMM for both row-major and column-major output

Instruction-level operations

CUTLASS defines a template-based interface to Tensor Core operations to avoid resorting to inline PTX.

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