Grouped GEMM + dGLU (SM100)

This is an experimental API and subject to change.

Overview

Unified Grouped GEMM + dGLU fusion: A block-scaled grouped GEMM fused with a dGLU backward epilogue (dSwiGLU or dGeGLU) on NVIDIA Blackwell GPUs (SM100+), designed for MoE (Mixture of Experts) workloads. Implemented with CUTLASS/CUTE.

This is a unified API that supports both weight layout modes:

• Dense mode: All expert weights packed into a single contiguous (N, K, L) tensor

• Discrete mode: Per-expert weight pointers (no weight stacking required)

And both backward activation functions:

• dSwiGLU: act_func="dswiglu" (default)

• dGeGLU: act_func="dgeglu"

Groups are contiguous in the M dimension and described by padded_offsets (cumulative aligned end offsets).

This kernel performs:

1. Block-scaled grouped GEMM: Low-precision GEMM (FP4, FP8) with per-block scale factors across multiple expert groups

2. dGLU backward epilogue: Fused backward computation using the forward C tensor (input/gate interleaved)

3. Optional quantized output: Produces row and column scale factors for downstream quantization

Shapes

Equations

• Inputs

  • A: contiguous activation tensor across all groups, shape (valid_m, K, 1)

  • B (dense): weight tensor across all groups, shape (N, K, L)

  • B (discrete): per-expert weight pointers, b_ptrs shape (num_experts,) of int64

  • C: forward intermediate tensor with interleaved input/gate blocks, shape (valid_m, 2N, 1)

  • SFA: scale factor tensor for A, shape (32, 4, ceil(valid_m/128), 4, ceil(ceil(K/sf_vec_size)/4), 1)

  • SFB (dense): scale factor tensor for B, shape (32, 4, ceil(N/128), 4, ceil(ceil(K/sf_vec_size)/4), L)

  • SFB (discrete): per-expert SFB pointers, sfb_ptrs shape (num_experts,) of int64

  • padded_offsets: cumulative sum of aligned group M sizes, shape (L,). valid_m = padded_offsets[-1]

  • alpha: per-group scaling factors for GEMM, shape (L,)

  • beta: per-group scaling factors for C, shape (L,)

  • prob: per-row gating probabilities, shape (valid_m, 1, 1)

  • norm_const: normalization constant for FP8 quantization, shape (1,)

• Outputs

  • D_row: row-quantized dGLU output, shape (valid_m, 2N, 1)

  • D_col: column-quantized dGLU output, shape (valid_m, 2N, 1)

  • dprob: gradient of prob, shape (valid_m, 1, 1). Must be zero-initialized.

  • dbias (optional): per-expert bias gradient tensor, shape (L, 2N, 1)

  • SFD_row: row scale factors (when d_dtype is FP8), shape (32, 4, ceil(valid_m/128), 4, ceil(ceil((2N)/sf_vec_size)/4), 1)

  • SFD_col: column scale factors (when d_dtype is FP8), shape (32, 4, ceil((2N)/128), 4, ceil(ceil(valid_m/sf_vec_size)/4), 1)

  • amax: per-group amax (when d_dtype is bf16/float16), shape (L, 2, 1)

Step 1: Block-scaled grouped GEMM (per group g with rows m in [padded_offsets[g-1], padded_offsets[g])):

\( \text{ref}[m, n] = \alpha_g^2 \sum_{k} \text{dequantize}(A[m, k], \text{SFA}) \cdot \text{dequantize}(B[n, k, g], \text{SFB}) \)

Step 2: dGLU backward epilogue (performed with 32-column interleaving along 2N):

• Scale C by beta_g per group and deinterleave into input/gate halves by 32-wide blocks.

For dSwiGLU (act_func="dswiglu"):

• swish = gate * sigmoid(gate)

• dprob += sum(swish * input * ref) over 32-column chunks

• ab = ref * prob * swish

• dswiglu = ref * prob * input * sigmoid(gate) * (1 + gate * (1 - sigmoid(gate)))

• Interleave [ab, dswiglu] back into D_row/D_col with 32-column blocks.

For dGeGLU (act_func="dgeglu"): Uses sigmoid(1.702 * gate) scaling in the backward computation.

Step 3: Optional output quantization (when SFD outputs are generated):

\( \text{SFD_row}[m, n] = \text{norm_const} \cdot \max_{k \in \text{block}} |D[m, k]| \cdot \text{rcp_max} \)

\( D_{\text{quantized}}[m, n] = D[m, n] \cdot \frac{\text{norm_const}}{\text{SFD_row}[m, n]} \)

Diagram

 A (valid_m×K×1)    B (N×K×L) or b_ptrs    padded_offsets
 SFA                SFB or sfb_ptrs              |
   |                 |                           |
   |    +------------+                           |
   |    |                                        |
   v    v                                        v
  Dequantize → Grouped GEMM (per group ranges) → ref
                    |
                    | × alpha[group_idx]
                    v
               ref (valid_m×N×1)
                    |
 C (valid_m×2N×1) --× beta[group_idx]--> deinterleave 32-col blocks
                    |                    |
                    |                swish, sigmoid
                    |                    |
                    +--> dprob (sum over blocks)
                    |
                    +--> ab, dswiglu → interleave → D (valid_m×2N×1)
                                      |
                         +-----------+-----------+
                         |                       |
                         v                       v
                    Row Quantize            Col Quantize
                         |                       |
                         v                       v
                    D_row, SFD_row         D_col, SFD_col

---

API Usage

High-level Wrapper

Dense mode:

from cudnn import grouped_gemm_dglu_wrapper_sm100
from cuda.bindings import driver as cuda

stream = cuda.CUstream(torch.cuda.current_stream().cuda_stream)

outputs = grouped_gemm_dglu_wrapper_sm100(
    a_tensor=a,
    c_tensor=c,
    sfa_tensor=sfa,
    padded_offsets=padded_offsets,
    alpha_tensor=alpha,
    beta_tensor=beta,
    prob_tensor=prob,
    dprob_tensor=dprob,
    generate_dbias=True,
    # Dense mode weights:
    b_tensor=b,
    sfb_tensor=sfb,
    # Common:
    norm_const_tensor=norm_const,
    acc_dtype=torch.float32,
    d_dtype=torch.bfloat16,
    cd_major="n",
    mma_tiler_mn=(256, 256),
    cluster_shape_mn=(2, 1),
    sf_vec_size=32,
    act_func="dswiglu",
    epilogue_op=None,
    current_stream=stream,
)

# dictionary access:
d_row = outputs["d_row_tensor"]
d_col = outputs["d_col_tensor"]
dprob = outputs["dprob_tensor"]
dbias = outputs["dbias_tensor"]
amax = outputs["amax_tensor"]
sfd_row = outputs["sfd_row_tensor"]
sfd_col = outputs["sfd_col_tensor"]

# or tuple unpacking:
d_row, d_col, dprob, dbias, amax, sfd_row, sfd_col = outputs

Discrete mode:

outputs = grouped_gemm_dglu_wrapper_sm100(
    a_tensor=a,
    c_tensor=c,
    sfa_tensor=sfa,
    padded_offsets=padded_offsets,
    alpha_tensor=alpha,
    beta_tensor=beta,
    prob_tensor=prob,
    dprob_tensor=dprob,
    # Discrete mode weights:
    b_ptrs=b_ptrs,
    sfb_ptrs=sfb_ptrs,
    n=n_dim,
    b_dtype=torch.uint8,
    b_major="k",
    # Common:
    act_func="dgeglu",
    current_stream=stream,
)

Class API

Dense mode:

from cudnn import GroupedGemmDgluSm100
from cuda.bindings import driver as cuda

stream = cuda.CUstream(torch.cuda.current_stream().cuda_stream)

api = GroupedGemmDgluSm100(
    sample_a=a,
    sample_c=c,
    sample_d_row=d_row,
    sample_d_col=d_col,
    sample_sfa=sfa,
    sample_padded_offsets=padded_offsets,
    sample_alpha=alpha,
    sample_beta=beta,
    sample_prob=prob,
    sample_dprob=dprob,
    sample_dbias=dbias,
    # Dense mode:
    sample_b=b,
    sample_sfb=sfb,
    # Optional quantization outputs
    sample_sfd_row=sfd_row,
    sample_sfd_col=sfd_col,
    sample_amax=amax,
    sample_norm_const=norm_const,
    # Configuration
    acc_dtype=torch.float32,
    mma_tiler_mn=(256, 256),
    act_func="dswiglu",
    epilogue_op=None,
)
assert api.check_support()
api.compile()
api.execute(
    a_tensor=a, c_tensor=c, d_row_tensor=d_row, d_col_tensor=d_col,
    sfa_tensor=sfa, padded_offsets=padded_offsets, alpha_tensor=alpha,
    beta_tensor=beta, prob_tensor=prob, dprob_tensor=dprob, dbias_tensor=dbias,
    b_tensor=b, sfb_tensor=sfb,
    sfd_row_tensor=sfd_row, sfd_col_tensor=sfd_col,
    amax_tensor=amax, norm_const_tensor=norm_const,
    current_stream=stream,
)

In the class API, dbias generation is specialized at compile time: if sample_dbias is omitted, dbias_tensor must also be omitted at execute().

Discrete mode:

api = GroupedGemmDgluSm100(
    sample_a=a,
    sample_c=c,
    sample_d_row=d_row,
    sample_d_col=d_col,
    sample_sfa=sfa,
    sample_padded_offsets=padded_offsets,
    sample_alpha=alpha,
    sample_beta=beta,
    sample_prob=prob,
    sample_dprob=dprob,
    # Discrete mode:
    num_experts=num_experts,
    b_shape=(n, k),
    b_dtype=torch.uint8,
    # Configuration
    act_func="dgeglu",
    b_major="k",
    epilogue_op="relu",
)
assert api.check_support()
api.compile()
api.execute(
    a_tensor=a, c_tensor=c, d_row_tensor=d_row, d_col_tensor=d_col,
    sfa_tensor=sfa, padded_offsets=padded_offsets, alpha_tensor=alpha,
    beta_tensor=beta, prob_tensor=prob, dprob_tensor=dprob,
    b_ptrs=b_ptrs, sfb_ptrs=sfb_ptrs,
    current_stream=stream,
)

---

Parameters

Weight Mode

The weight mode is auto-detected from constructor arguments:

• Dense: Provide sample_b and sample_sfb (contiguous weight tensors)

• Discrete: Provide num_experts, b_shape, and b_dtype (per-expert pointer mode)

Providing both or neither raises ValueError.

Input/Output Tensors

• Input tensor A: a_tensor / sample_a

  • Shape: (valid_m, K, 1)

  • Stride: (K, 1, valid_m·K) – must be K-major

  • Dtype (ab_dtype): {float4_e2m1fn_x2, uint8, float8_e4m3fn, float8_e5m2}

• Input tensor B (dense mode): b_tensor / sample_b

  • Shape: (N, K, L) where L = num_groups

  • Stride: K-major or N-major. Must be K-major for FP4.

  • Dtype: Must match A

• Input B pointers (discrete mode): b_ptrs

  • Shape: (num_experts,) – 1-D int64 device tensor of per-expert B data pointers

• Input tensor C: c_tensor / sample_c

  • Shape: (valid_m, 2N, 1) – forward activation with interleaved input/gate

  • Stride: (2N, 1, valid_m·2N) – must be N-major

  • Dtype: {float32, float16, bfloat16, float8_e4m3fn, float8_e5m2}

• Output tensor D_row: d_row_tensor / sample_d_row

  • Shape: (valid_m, 2N, 1)

  • Stride: (2N, 1, valid_m·2N) – must be N-major

  • Dtype (d_dtype): {bfloat16, float32} for FP4; {float8_e4m3fn, float8_e5m2} for FP8

• Output tensor D_col: d_col_tensor / sample_d_col

  • Shape: (valid_m, 2N, 1) – must match D_row dtype and stride

• Input tensor prob: prob_tensor / sample_prob

  • Shape: (valid_m, 1, 1), dtype: float32

• Output tensor dprob: dprob_tensor / sample_dprob

  • Shape: (valid_m, 1, 1), dtype: float32

  • Must be zero-initialized

• Scaling tensors: alpha shape (L,), beta shape (L,), amax shape (L, 2, 1), norm_const shape (1,)

Common Parameters

• acc_dtype: Must be torch.float32

• mma_tiler_mn: Kernel tile size. Default: (256, 256)

  • TILE_M ∈ {128, 256}

  • TILE_N = 256

• cluster_shape_mn: Thread Block cluster shape. Default: (2, 1) when TILE_M=256, (1, 1) otherwise

• sf_vec_size: Scale factor vector size. {16, 32}. Default: 16

• vector_f32: Enable packed f32 operations. Default: False

• m_aligned: Must be 256. Default: 256

• discrete_col_sfd: Generate discrete col-major scale factors. Default: False

• act_func: Backward activation function. "dswiglu" (default) or "dgeglu"

• b_major (discrete only): B tensor major dimension. "k" (default) or "n". Must be "k" for FP4.

• epilogue_op: Optional post-processing. None (default), "identity", "relu", or "srelu"

Wrapper-specific Parameters

• d_dtype: Output D tensor data type. Default: torch.bfloat16

• cd_major: Must be "n". Default: "n"

• n (discrete only): B weight N dimension

• b_dtype (discrete only): B weight data type

Wrapper Return Values

Returns a TupleDict (dictionary + tuple unpacking):

• d_row_tensor: Row-quantized dGLU output

• d_col_tensor: Column-quantized dGLU output

• dprob_tensor: Gradient of prob

• amax_tensor: Per-group amax (when d_dtype is bf16/fp16)

• sfd_row_tensor: Row scale factors (when SFD enabled)

• sfd_col_tensor: Column scale factors (when SFD enabled)

---

Support Surface and Constraints

Layouts and Strides

• A must be K-major

• B must be K-major (dense) or K/N-major (discrete). Must be K-major for FP4.

• C, D_row, D_col must be N-major

• All tensors must be 16-byte aligned along the contiguous dimension

Data Types

Format	ab_dtype	sf_dtype	sf_vec_size	d_dtype
MXFP8	float8_e4m3fn or float8_e5m2	{float8_e8m0fnu, float8_e4m3fn}	32	{float8_e4m3fn, float8_e5m2}
NVF4	float4_e2m1fn_x2 or uint8	{float8_e4m3fn, float8_e8m0fnu}	{16, 32}	{bfloat16, float32}

Additional Type Constraints

• A and B must have the same dtype

• Scale factor tensors must have the same dtype

• D_row and D_col must have the same dtype

• dbias must be bfloat16

• sf_dtype=float8_e4m3fn is incompatible with sf_vec_size=32

• FP8 c_dtype with vector_f32=True is not supported

• FP4 ab_dtype only supports d_dtype in {bfloat16, float32}

• For non-dbias paths, FP4 ab_dtype with sf_vec_size=16 and d_dtype=float32 is not supported

• FP8 ab_dtype only supports d_dtype in {float8_e4m3fn, float8_e5m2}

Shapes and Divisibility

• N must be divisible by 32 (32-column blocks for input/gate interleaving)

• Expert count must be <= 1024

• Each group’s M dimension is aligned to m_aligned (256)

• In the class API, dbias is compiled in only when sample_dbias is provided; passing a runtime dbias_tensor without sample_dbias raises ValueError

Environment

• Requires CUDA with SM100+ compute capability (Blackwell GPUs)

---

Usage Examples

For usage examples, see test cases in test/python/fe_api/test_grouped_gemm_dglu.py (dense mode, unified API) and test/python/fe_api/test_discrete_grouped_gemm_dswiglu.py (discrete mode).