fla/ops/gla/chunk_fuse.py

# -*- coding: utf-8 -*-

# Copyright (c) 2023, Songlin Yang
# Gated Linear Attention Transformers with Hardware-Efficient Training: https://arxiv.org/abs/2312.06635
# on-the-fly computation without materializing hidden statets into HBMs

from typing import Tuple

import torch
import torch.nn.functional as F
import triton
import triton.language as tl
from einops import rearrange
from packaging import version
from torch.cuda.amp import custom_bwd, custom_fwd

from fla.ops.gla.chunk_util import (bwd_decay_global_cumsum, fwd_decay_cumsum,
                                    prepare_qg_kg)
from fla.utils import contiguous

inv_ln2 = 1.44269504

@triton.jit
def fused_chunk_gla_fwd_kernel(
    # B: batch_size, H: n_heads, T: seq_len, D: d_head
    q,  # query [B, H, L, D_head_K]
    k,  # key [B, H, L, D_head_K]
    v,  # value [B, H, L, D_head_V]
    g,  # cumulative sum of log decay [B, H, L, D_head_K]
    o,  # output [B, H, L, D_head_V]

    initial_state,  # initial state of the chunk [B, H, D_head_K, D_head_V]
    final_state,  # final state of the chunk [B, H, D_head_K, D_head_V]

    s_qk_h,  # stride size: L * D_head_K
    s_qk_t,  # stride size: D_head_K
    s_qk_d,  # stride size: 1

    s_vo_h,  # stride size: L * D_head_V
    s_vo_t,  # stride size: D_head_V
    s_vo_d,  # stride size: 1

    B,  # batch size
    H,  # n_heads
    T,  # seq_len
    scale,  # D_head_K ** -0.5
    BT: tl.constexpr,  # BLOCK SIZE along the sequence dimension, a.k.a. chunk size
    BK: tl.constexpr,  # BLOCK SIZE along the K dimension
    BV: tl.constexpr,  # BLOCK SIZE along the V dimension
    DK: tl.constexpr,  # D_head_K
    DV: tl.constexpr,  # D_head_V
    USE_INITIAL_STATE: tl.constexpr,
    STORE_FINAL_STATE: tl.constexpr,
    CHECK: tl.constexpr
):
    # indices
    i_v, i_k, i_bh = tl.program_id(0), tl.program_id(1), tl.program_id(2)

    b_h = tl.zeros([BK, BV], dtype=tl.float32)

    # make block pointers
    p_q = tl.make_block_ptr(q + i_bh * s_qk_h, (T, DK), (s_qk_t, s_qk_d), (0, i_k * BK), (BT, BK), (1, 0))
    p_db = g + i_bh * s_qk_h + (BT - 1) * s_qk_t + i_k * BK + tl.arange(0, BK)
    p_k = tl.make_block_ptr(k + i_bh * s_qk_h, (DK, T), (s_qk_d, s_qk_t), (i_k * BK, 0), (BK, BT), (0, 1))
    p_v = tl.make_block_ptr(v + i_bh * s_vo_h, (T, DV), (s_vo_t, s_vo_d), (0, i_v * BV), (BT, BV), (1, 0))
    p_o = tl.make_block_ptr(o + (i_bh + i_k * B * H) * s_vo_h, (T, DV), (s_vo_t, s_vo_d), (0, i_v * BV), (BT, BV), (1, 0))

    if USE_INITIAL_STATE:
        p_h = tl.make_block_ptr(initial_state + i_bh * DK * DV, (DK, DV), (DV, 1), (i_k * BK, i_v * BV), (BK, BV), (1, 0))
        b_h += tl.load(p_h, boundary_check=(0, 1)).to(tl.float32)
    
    mask = (i_k * BK + tl.arange(0, BK)) < DK

    for i in range(0, tl.cdiv(T, BT)):
        # [BK, BT]
        b_k = tl.load(p_k, boundary_check=(0, 1))
        # [BT, BV]
        b_o = tl.zeros([BT, BV], dtype=tl.float32)
        b_v = tl.load(p_v, boundary_check=(0, 1))
        # [BT, BK]
        b_q = tl.load(p_q, boundary_check=(0, 1))
        d_b = tl.load(p_db, mask=mask, other=0).to(tl.float32)
        if CHECK and i == 0:
            b_o = tl.dot(b_q.to(b_v.dtype), b_h.to(b_v.dtype), allow_tf32=False)
            b_h = b_h * tl.math.exp2(d_b)[:, None] + tl.dot(b_k.to(b_v.dtype), b_v, allow_tf32=False)
        else:
            b_o = tl.dot(b_q.to(b_v.dtype), b_h.to(b_v.dtype), allow_tf32=False)
            b_h = b_h * tl.math.exp2(d_b)[:, None] + tl.dot(b_k.to(b_v.dtype), b_v, allow_tf32=False)

        tl.store(p_o, b_o.to(p_o.dtype.element_ty), boundary_check=(0, 1))
        p_q = tl.advance(p_q, (BT, 0))
        p_k = tl.advance(p_k, (0, BT))
        p_v = tl.advance(p_v, (BT, 0))
        p_o = tl.advance(p_o, (BT, 0))
        p_db += BT * DK

    if STORE_FINAL_STATE:
        p_final = tl.make_block_ptr(final_state + i_bh * DK * DV, (DK, DV), (DV, 1), (i_k * BK, i_v * BV), (BK, BV), (1, 0))
        tl.store(p_final, b_h.to(p_final.dtype.element_ty), boundary_check=(0, 1))


# Similar to Algorithm1 of https://arxiv.org/abs/2006.16236
@triton.jit
def fused_chunk_gla_bwd_kernel(
    q, k, v, g,
    do,  # gradient of output [B, H, L, D_head_V]
    dq,  # gradient of query [NV, B, H, L, D_head_K]
    dk,  # gradient of key [NV, B, H, L, D_head_K]
    dv,  # gradient of value [NK, B, H, L, D_head_V]

    initial_state,  # initial state of the chunk [B, H, D_head_K, D_head_V]

    s_qk_h,  # stride size: L * D_head_K
    s_qk_t,  # stride size: D_head_K
    s_qk_d,  # stride size: 1

    s_vo_h,  # stride size: L * D_head_V
    s_vo_t,  # stride size: D_head_V
    s_vo_d,  # stride size: 1

    B,  # batch_size
    H,  # n_heads
    T,  # seq_len
    scale,  # D_head_K ** -0.5
    # clamp_min,  # minimum log value of the gate for numerical stability. default: -5
    BT: tl.constexpr,  # BLOCK SIZE along the sequence dimension, a.k.a. chunk size
    BK: tl.constexpr,  # BLOCK SIZE along the K dimension
    BV: tl.constexpr,  # BLOCK SIZE along the V dimension
    DK: tl.constexpr,  # D_head_K
    DV: tl.constexpr,  # D_head_V
    USE_INITIAL_STATE: tl.constexpr,
    CHECK: tl.constexpr
):
    i_v, i_k, i_bh = tl.program_id(0), tl.program_id(1), tl.program_id(2)
    # [BV, BK]
    b_h = tl.zeros([BV, BK], dtype=tl.float32)

    if USE_INITIAL_STATE:
        p_h = tl.make_block_ptr(initial_state + i_bh * DK * DV, (DV, DK), (1, DV), (i_v * BV, i_k * BK), (BV, BK), (0, 1))
        b_h += tl.load(p_h, boundary_check=(0, 1)).to(tl.float32)
    
    mask = (i_k * BK + tl.arange(0, BK)) < DK    
    for i in range(0, tl.cdiv(T, BT)):
        p_k = tl.make_block_ptr(k + i_bh * s_qk_h, (T, DK), (s_qk_t, s_qk_d), (i * BT, i_k * BK), (BT, BK), (1, 0))
        p_db = g + i_bh * s_qk_h + ((i+1) * BT - 1) * s_qk_t + i_k * BK + tl.arange(0, BK)
        p_v = tl.make_block_ptr(v + i_bh * s_vo_h, (DV, T), (s_vo_d, s_vo_t), (i_v * BV, i * BT), (BV, BT), (0, 1))
        p_do = tl.make_block_ptr(do + i_bh * s_vo_h, (T, DV), (s_vo_t, s_vo_d), (i * BT, i_v * BV), (BT, BV), (1, 0))
        p_dq = tl.make_block_ptr(dq + (i_bh+i_v*B*H)*s_qk_h, (T, DK), (s_qk_t, s_qk_d), (i * BT, i_k * BK), (BT, BK), (1, 0))
        b_dq = tl.zeros([BT, BK], dtype=tl.float32)
        # [BT, DK]
        b_k = tl.load(p_k, boundary_check=(0, 1))
        # b_g = tl.load(p_g, boundary_check=(0, 1)) * inv_ln2
        d_b = tl.load(p_db, mask=mask, other=0).to(tl.float32)

        # [DV, BT]
        b_v = tl.load(p_v, boundary_check=(0, 1))
        # [BT, DV]
        b_do = tl.load(p_do, boundary_check=(0, 1))
        # [DV, DK]
        if CHECK and i == 0:
            b_dq += tl.dot(b_do, b_h.to(b_do.dtype), allow_tf32=False)
            b_h = b_h * tl.math.exp2(d_b)[None, :] + tl.dot(b_v, b_k.to(b_v.dtype), allow_tf32=False)
        else:
            b_dq += tl.dot(b_do, b_h.to(b_do.dtype), allow_tf32=False)
            b_h = b_h * tl.math.exp2(d_b)[None, :] + tl.dot(b_v, b_k.to(b_v.dtype), allow_tf32=False)
        b_dq *= scale
        tl.store(p_dq, b_dq.to(p_dq.dtype.element_ty), boundary_check=(0, 1))

    # sync threads
    b_h = None
    tl.debug_barrier()
    # [BK, BV]
    b_dh = tl.zeros([BK, BV], dtype=tl.float32)

    # cum = tl.zeros([BK], dtype=tl.float32)
    for i in range(1, tl.cdiv(T, BT) + 1):
        p_q = tl.make_block_ptr(q + i_bh * s_qk_h, (DK, T), (s_qk_d, s_qk_t), (i_k * BK, T - i * BT), (BK, BT), (0, 1))
        p_k = tl.make_block_ptr(k + i_bh * s_qk_h, (T, DK), (s_qk_t, s_qk_d), (T - i * BT, i_k * BK), (BT, BK), (1, 0))
        p_db = g + i_bh * s_qk_h + (T - (i-1) * BT - 1) * s_qk_t + i_k * BK + tl.arange(0, BK)
        p_v = tl.make_block_ptr(v + i_bh * s_vo_h, (T, DV), (s_vo_t, s_vo_d), (T - i * BT, i_v * BV), (BT, BV), (1, 0))
        p_do = tl.make_block_ptr(do + i_bh * s_vo_h, (T, DV), (s_vo_t, s_vo_d), (T - i * BT, i_v * BV), (BT, BV), (1, 0))
        p_dk = tl.make_block_ptr(dk + (i_bh + i_v * B * H) * s_qk_h, (T, DK),
                                 (s_qk_t, s_qk_d), (T - i * BT, i_k * BK), (BT, BK), (1, 0))
        p_dv = tl.make_block_ptr(dv + (i_bh + i_k * B * H) * s_vo_h, (T, DV),
                                 (s_vo_t, s_vo_d), (T - i * BT, i_v * BV), (BT, BV), (1, 0))
        # [DK, BT]
        b_q = tl.load(p_q, boundary_check=(0, 1))
        # [BT, DK]
        b_k = tl.load(p_k, boundary_check=(0, 1))
        # [BT, DV]
        b_v = tl.load(p_v, boundary_check=(0, 1))
        b_do = tl.load(p_do, boundary_check=(0, 1))
        b_db = tl.load(p_db, mask=mask, other=0).to(tl.float32)

        # inter-chunk
        # [DK, DV]
        if CHECK and i == 1:
            b_dk = tl.trans(tl.dot(b_dh.to(b_v.dtype), tl.trans(b_v), allow_tf32=False))
            b_dv = tl.dot((b_k).to(b_v.dtype), b_dh.to(b_v.dtype), allow_tf32=False)
            b_dh = b_dh * tl.math.exp2(b_db)[:, None] + tl.dot(b_q.to(b_do.dtype), b_do, allow_tf32=False)
        else:
            b_dk = tl.trans(tl.dot(b_dh.to(b_v.dtype), tl.trans(b_v), allow_tf32=False))
            b_dv = tl.dot((b_k).to(b_v.dtype), b_dh.to(b_v.dtype), allow_tf32=False)
            b_dh = b_dh * tl.math.exp2(b_db)[:, None] + tl.dot(b_q.to(b_do.dtype), b_do, allow_tf32=False)

        tl.store(p_dk, b_dk.to(p_dk.dtype.element_ty), boundary_check=(0, 1))
        tl.store(p_dv, b_dv.to(p_dv.dtype.element_ty), boundary_check=(0, 1))


@triton.jit
def fwd_inner_chunk(
    q, k, g, A,
    s_qk_h,  # stride size: L * D_head_K
    s_qk_t,  # stride size: D_head_K
    s_qk_d,  # stride size: 1
    B,  # batch_size
    H,  # n_heads
    T,  # seq_len
    scale,  # D_head_K ** -0.5
    # clamp_min,  # minimum log value of the gate for numerical stability. default: -5
    BT: tl.constexpr,  # BLOCK SIZE along the sequence dimension, a.k.a. chunk size
    BK: tl.constexpr,  # BLOCK SIZE along the K dimension
    DK: tl.constexpr,  # D_head_K
):

    i_k, i_t, i_bh = tl.program_id(0), tl.program_id(1), tl.program_id(2)

    p_k = tl.make_block_ptr(k + i_bh * s_qk_h, (T, DK), (s_qk_t, s_qk_d), (i_t * BT, i_k * BK), (BT, BK), (1, 0))

    b_k = tl.load(p_k, boundary_check=(0, 1))

    p_g = tl.make_block_ptr(g + i_bh * s_qk_h, (T, DK), (s_qk_t, s_qk_d), (i_t * BT, i_k * BK), (BT, BK), (1, 0))

    b_g = tl.load(p_g, boundary_check=(0, 1)).to(tl.float32)

    mask = (i_k * BK + tl.arange(0, BK)) < DK
    o_i = tl.arange(0, BT)

    p_q = q + i_bh * s_qk_h + i_k * BK + i_t * BT * DK + tl.arange(0, BK)
    p_gq = g + i_bh * s_qk_h + i_k * BK + i_t * BT * DK + tl.arange(0, BK)
    p_A = A + (i_bh + (i_k * B * H)) * (tl.cdiv(T, BT) * BT * BT) + i_t * BT * BT + tl.arange(0, BT)

    for i in range(BT):
        _q = tl.load(p_q, mask=mask, other=0) * scale
        gq = tl.load(p_gq, mask=mask, other=0).to(tl.float32)
        s = _q[None, :] * b_k * tl.math.exp2(gq[None, :] - b_g)
        score = tl.sum(s, axis=1)
        score = tl.where(o_i <= i, score, 0)
        tl.store(p_A, score.to(p_A.dtype.element_ty))
        p_q += DK
        p_gq += DK
        p_A += BT


@triton.jit
def bwd_inner_chunk(
    q,
    k,
    g,
    dA,
    dq,
    dk,
    s_qk_h,  # stride size: L * D_head_K
    s_qk_t,  # stride size: D_head_K
    s_qk_d,  # stride size: 1
    B,  # batch_size
    H,  # n_heads
    T,  # seq_len
    scale,  # D_head_K ** -0.5
    # clamp_min,  # minimum log value of the gate for numerical stability. default: -5
    BT: tl.constexpr,  # BLOCK SIZE along the sequence dimension, a.k.a. chunk size
    BK: tl.constexpr,  # BLOCK SIZE along the K dimension
    DK: tl.constexpr,  # D_head_K
):
    i_k, i_t, i_bh = tl.program_id(0), tl.program_id(1), tl.program_id(2)
    p_k = tl.make_block_ptr(k + i_bh * s_qk_h, (T, DK), (s_qk_t, s_qk_d), (i_t * BT, i_k * BK), (BT, BK), (1, 0))
    b_k = tl.load(p_k, boundary_check=(0, 1))
    p_g = tl.make_block_ptr(g + i_bh * s_qk_h, (T, DK), (s_qk_t, s_qk_d), (i_t * BT, i_k * BK), (BT, BK), (1, 0))
    b_g = tl.load(p_g, boundary_check=(0, 1)).to(tl.float32)

    mask = (i_k * BK + tl.arange(0, BK)) < DK
    o_i = tl.arange(0, BT)

    p_q = q + i_bh * s_qk_h + i_k * BK + i_t * BT * DK + tl.arange(0, BK)
    p_dq = dq + (i_bh) * s_qk_h + i_k * BK + i_t * BT * DK + tl.arange(0, BK)
    p_gq = g + i_bh * s_qk_h + i_k * BK + i_t * BT * DK + tl.arange(0, BK)
    p_dA = dA + i_bh * (tl.cdiv(T, BT) * BT * BT) + i_t * BT * BT + tl.arange(0, BT)

    b_dk = tl.zeros([BT, BK], dtype=tl.float32)

    for i in range(BT):
        _q = tl.load(p_q, mask=mask, other=0)
        gq = tl.load(p_gq, mask=mask, other=0).to(tl.float32)
        score = tl.math.exp2(gq[None, :] - b_g)
        score = tl.where(o_i[:, None] <= i, score, 0)
        _dA = tl.load(p_dA)
        _dA = tl.where(o_i <= i, _dA, 0)
        b_dk += (_dA[:, None] * score * _q[None, :])
        b_dq = tl.sum(_dA[:, None] * score * b_k, axis=0)
        tl.store(p_dq, b_dq, mask=mask)
        p_q += DK
        p_dq += DK
        p_gq += DK
        p_dA += BT

    p_dk = tl.make_block_ptr(dk + i_bh * s_qk_h, (T, DK), (s_qk_t, s_qk_d), (i_t * BT, i_k * BK), (BT, BK), (1, 0))
    tl.store(p_dk, b_dk.to(dk.dtype.element_ty), boundary_check=(0, 1))


class FusedChunkGLAFunction(torch.autograd.Function):

    @staticmethod
    @contiguous
    @custom_fwd
    def forward(ctx, q, k, v, g, scale, initial_state, output_final_state):
        ctx.g_dtype = g.dtype
        g_original = g
        # cumulative decay should be in float32, otherwise the err will be accumulated and amplified.
        g = torch.empty_like(g, dtype=torch.float32)
        batch_size, n_heads, seq_len, d_head_qk = q.shape
        d_head_v = v.shape[-1]
        ctx.scale = scale

        # inter-chunk
        BT = 16  # chunk_size
        BK, BV = min(d_head_qk, 64), min(d_head_v, 64)
        num_stages = 1
        num_warps = 2

        NK, NV = triton.cdiv(d_head_qk, BK), triton.cdiv(d_head_v, BV)
        o = q.new_empty(NK, batch_size, n_heads, seq_len, d_head_v)
        q_g = torch.empty_like(q)
        k_g = torch.empty_like(k)
        grid = (NK, triton.cdiv(seq_len, BT), batch_size * n_heads)
        fwd_decay_cumsum[grid](
            g_original,
            g,
            q.stride(1), q.stride(2), q.stride(3),
            batch_size, n_heads, seq_len, scale,
            BT=BT, BK=BK, DK=d_head_qk, num_warps=1
        )
        prepare_qg_kg[grid](
            q, k, g, q_g, k_g,
            q.stride(1), q.stride(2), q.stride(3),
            batch_size, n_heads, seq_len, scale,
            BT=BT, BK=BK, DK=d_head_qk, num_warps=1
        )

        if output_final_state:
            final_state = q.new_empty(batch_size, n_heads, d_head_qk, d_head_v, dtype=torch.float, requires_grad=False)
        else:
            final_state = None
        # the bug still exists even for Triton 2.2 on H100 GPUs
        # so we always enable initial checks
        CHECK = True
        if version.parse(triton.__version__) < version.parse('2.2.0'):
            import warnings
            warnings.warn(
                "Triton<2.2.0 detected for running this kernel, "
                "which is known to have some weird compiler issues (refer to https://github.com/openai/triton/issues/2852) "
                "that lead to significant precision loss. "
                "We've add some initial condition checks to resolve this, sadly at the sacrifice of the speed. "
                "For optimal performance, it is recommended to install Triton>=2.2.0 (if possible)."
            )
            CHECK = True

        grid = (NV, NK, batch_size * n_heads)
        fused_chunk_gla_fwd_kernel[grid](
            q_g, k_g, v, g, o, initial_state, final_state,
            q.stride(1), q.stride(2), q.stride(3),
            v.stride(1), v.stride(2), v.stride(3),
            batch_size, n_heads, seq_len, scale,
            BT=BT, DK=d_head_qk, DV=d_head_v, BK=BK, BV=BV,
            USE_INITIAL_STATE=initial_state is not None,
            STORE_FINAL_STATE=output_final_state,
            CHECK=CHECK,
            num_warps=num_warps,
            num_stages=num_stages
        )

        o = o.sum(0)

        # intra-chunk
        chunk_size = 16
        num_chunk = seq_len // chunk_size
        v2 = rearrange(v, 'b h (n c) d -> b h n c d', n=num_chunk)
        BK = min(d_head_qk, 64)
        NK = triton.cdiv(d_head_qk, BK)
        A = q.new_empty(NK, batch_size, n_heads, triton.cdiv(seq_len, BT), BT, BT)
        grid = (NK, triton.cdiv(seq_len, BT), batch_size * n_heads)
        fwd_inner_chunk[grid](
            q, k, g, A,
            q.stride(1), q.stride(2), q.stride(3),
            batch_size, n_heads, seq_len, scale,  BT=BT, BK=BK, DK=d_head_qk, num_stages=3,
            num_warps=4
        )
        A = A.sum(0)
        o2 = A @ v2
        o2 = rearrange(o2, 'b h n c d -> b h (n c) d')
        # combine inner and inter
        o.add_(o2)
        ctx.save_for_backward(q, k, v, g_original, A, initial_state)
        ctx.CHECK = CHECK
        return o.to(v), final_state

    @staticmethod
    @contiguous
    @custom_bwd
    def backward(ctx, do, d_final_state=None):
        q, k, v, g_origin, A, initial_state = ctx.saved_tensors
        batch_size, n_heads, seq_len, d_head_qk = q.shape
        d_head_v = v.shape[-1]
        scale = ctx.scale

        # recomputation
        # inter-chunk
        BT = 16  # chunk_size
        g = torch.empty_like(g_origin, dtype=torch.float32)
        BK, BV = min(d_head_qk, 64), min(d_head_v, 64)
        NK, NV = triton.cdiv(d_head_qk, BK), triton.cdiv(d_head_v, BV)
        q_g = torch.empty_like(q)
        k_g = torch.empty_like(k)
        grid = (NK, triton.cdiv(seq_len, BT), batch_size * n_heads)
        fwd_decay_cumsum[grid](
            g_origin,
            g,
            q.stride(1), q.stride(2), q.stride(3),
            batch_size, n_heads, seq_len, scale,
            BT=BT, BK=BK, DK=d_head_qk, num_warps=1
        )
        prepare_qg_kg[grid](
            q, k, g, q_g, k_g,
            q.stride(1), q.stride(2), q.stride(3),
            batch_size, n_heads, seq_len, scale,
            BT=BT, BK=BK, DK=d_head_qk, num_warps=1
        )

        # inter-chunk
        BT = 16
        BK, BV = min(triton.next_power_of_2(d_head_qk), 64), min(triton.next_power_of_2(d_head_v), 64)
        NK, NV = triton.cdiv(d_head_qk, BK), triton.cdiv(d_head_v, BV)
        num_stages = 1
        num_warps = 2
        dq = q.new_empty(NV, batch_size, n_heads,  seq_len, d_head_qk)
        dk = q.new_empty(NV, batch_size, n_heads,  seq_len, d_head_qk)
        dv = q.new_empty(NK, batch_size, n_heads,  seq_len, d_head_v)

        grid = (NV, NK, batch_size * n_heads)

        fused_chunk_gla_bwd_kernel[grid](
            q_g, k_g, v, g, do, dq, dk, dv, initial_state,
            q.stride(1), q.stride(2), q.stride(3),
            v.stride(1), v.stride(2), v.stride(3),
            batch_size, n_heads, seq_len, scale,
            # clamp_min=-3,
            BT=BT, DK=d_head_qk, DV=d_head_v, BK=BK, BV=BV,
            USE_INITIAL_STATE=initial_state is not None,
            CHECK=ctx.CHECK,
            num_warps=num_warps,
            num_stages=num_stages,
        )
        dq = dq.sum(0)
        dk = dk.sum(0)
        dv = dv.sum(0)

        # intra chunk
        num_chunk = seq_len // BT
        v2 = rearrange(v, 'b h (n c) d -> b h n c d', n=num_chunk)
        do2 = rearrange(do, 'b h (n c) d -> b h n c d', n=num_chunk)
        dA2 = (do2 @ v2.transpose(-2, -1)) * scale
        dv2 = A.transpose(-1, -2) @ do2
        dv2 = rearrange(dv2, 'b h n c d -> b h (n c) d', n=num_chunk)

        BK = min(triton.next_power_of_2(d_head_qk), 16)
        NK = triton.cdiv(d_head_qk, BK)
        dk2 = torch.empty_like(k)
        dq2 = torch.empty_like(q)

        grid = (NK, triton.cdiv(seq_len, BT), batch_size * n_heads)
        bwd_inner_chunk[grid](
            q, k, g,
            dA2, dq2, dk2,
            q.stride(1), q.stride(2), q.stride(3),
            batch_size, n_heads, seq_len, scale,
            BT=BT, DK=d_head_qk, BK=BK,
            num_warps=1,
            num_stages=3
        )

        BK = min(triton.next_power_of_2(d_head_qk), 32)
        NK = triton.cdiv(d_head_qk, BK)
        dg = torch.empty_like(g, dtype=torch.float32)
        grid = (NK, triton.cdiv(seq_len, BT), batch_size * n_heads)
        bwd_decay_global_cumsum[grid](
            dq2, dq, dk2, dk, q, k, g, dg,
            q.stride(1), q.stride(2), q.stride(3),
            batch_size, n_heads, seq_len, scale,
            BT=BT, DK=d_head_qk, BK=BK,
            num_warps=1,
            num_stages=1
        )
        dg = rearrange(dg, 'b h (n c) d -> b h n c d', c=BT)

        def rev_cumsum_exclusive(x):
            cumsum_x = x.cumsum(-2)
            rev_cumsum_x = cumsum_x[..., -1, None, :] - cumsum_x
            return rev_cumsum_x

        rev_cumsum_dg = rev_cumsum_exclusive(dg[..., 0, :])
        dg.add_(rev_cumsum_dg.unsqueeze(-2))
        dv.add_(dv2)
        dg = rearrange(dg, 'b h n c d -> b h (n c) d')

        return dq.to(q), dk.to(k), dv.to(v), dg.to(ctx.g_dtype), None, None, None


def pad(x, chunk_size=16):
    seq_len = x.shape[-2]
    padded_seq_len = ceildiv(seq_len, chunk_size) * chunk_size
    if x.shape[-2] % chunk_size != 0:
        x = F.pad(x, (0, 0, 0, padded_seq_len - seq_len))
    
    return x


def ceildiv(a, b):
    return -(a // -b)


def fused_chunk_gla(
    q: torch.Tensor,
    k: torch.Tensor,
    v: torch.Tensor,
    g: torch.Tensor,
    scale: int = -1,
    initial_state: torch.Tensor = None,
    output_final_state: bool = False
) -> Tuple[torch.Tensor, torch.Tensor]:
    if scale == -1:
        scale = q.shape[-1] ** -0.5
    if initial_state is not None:
        initial_state = initial_state.detach()
    seq_len = q.shape[-2]
    q, k, v, g = map(lambda x: pad(x), [q, k, v, g])
    o, final_state = FusedChunkGLAFunction.apply(
        q, k, v, g, scale, initial_state, output_final_state)
    o = o[..., :seq_len, :]
    return o, final_state