fla/ops/attn/parallel.py

# Copyright (c) 2023-2025, Songlin Yang, Yu Zhang

import warnings

import torch
import triton
import triton.language as tl
from einops import reduce

from fla.ops.utils import prepare_chunk_indices
from fla.ops.utils.cumsum import chunk_global_cumsum
from fla.ops.utils.op import exp2, log2
from fla.utils import autocast_custom_bwd, autocast_custom_fwd, check_shared_mem, contiguous


@triton.heuristics({
    'USE_G': lambda args: args['g_cumsum'] is not None,
    'IS_VARLEN': lambda args: args['cu_seqlens'] is not None,
})
@triton.jit
def parallel_attn_fwd_kernel(
    q,
    k,
    v,
    o,
    g_cumsum,
    lse,
    scale,
    cu_seqlens,
    chunk_indices,
    T,
    B: tl.constexpr,
    H: tl.constexpr,
    HQ: tl.constexpr,
    G: tl.constexpr,
    K: tl.constexpr,
    V: tl.constexpr,
    BT: tl.constexpr,
    BS: tl.constexpr,
    BK: tl.constexpr,
    BV: tl.constexpr,
    USE_G: tl.constexpr,
    IS_VARLEN: tl.constexpr,
):
    i_v, i_t, i_bh = tl.program_id(0), tl.program_id(1), tl.program_id(2)
    i_b, i_hq = i_bh // HQ, i_bh % HQ
    i_h = i_hq // G

    if IS_VARLEN:
        i_n, i_t = tl.load(chunk_indices + i_t * 2).to(tl.int32), tl.load(chunk_indices + i_t * 2 + 1).to(tl.int32)
        bos, eos = tl.load(cu_seqlens + i_n).to(tl.int32), tl.load(cu_seqlens + i_n + 1).to(tl.int32)
        T = eos - bos
    else:
        i_n = i_b
        bos, eos = i_n * T, i_n * T + T
    RCP_LN2: tl.constexpr = 1.4426950216

    p_q = tl.make_block_ptr(q + (bos * HQ + i_hq) * K, (T, K), (HQ*K, 1), (i_t * BT, 0), (BT, BK), (1, 0))
    p_o = tl.make_block_ptr(o + (bos * HQ + i_hq) * V, (T, V), (HQ*V, 1), (i_t * BT, i_v * BV), (BT, BV), (1, 0))
    p_lse = tl.make_block_ptr(lse + bos * HQ + i_hq, (T,), (HQ,), (i_t * BT,), (BT,), (0,))

    # the Q block is kept in the shared memory throughout the whole kernel
    # [BT, BK]
    b_q = tl.load(p_q, boundary_check=(0, 1))
    # [BT, BV]
    b_o = tl.zeros([BT, BV], dtype=tl.float32)

    b_m = tl.full([BT], float('-inf'), dtype=tl.float32)
    b_acc = tl.zeros([BT], dtype=tl.float32)

    if USE_G:
        p_g = tl.make_block_ptr(g_cumsum + bos * HQ + i_hq, (T,), (HQ,), (i_t * BT,), (BT,), (0,))
        b_gq = tl.load(p_g, boundary_check=(0,)).to(tl.float32)
    else:
        b_gq = None

    for i_s in range(0, i_t * BT, BS):
        p_k = tl.make_block_ptr(k + (bos * H + i_h) * K, (K, T), (1, H*K), (0, i_s), (BK, BS), (0, 1))
        p_v = tl.make_block_ptr(v + (bos * H + i_h) * V, (T, V), (H*V, 1), (i_s, i_v * BV), (BS, BV), (1, 0))
        # [BK, BS]
        b_k = tl.load(p_k, boundary_check=(0, 1))
        # [BS, BV]
        b_v = tl.load(p_v, boundary_check=(0, 1))
        # [BT, BS]
        b_s = tl.dot(b_q, b_k) * scale * RCP_LN2

        if USE_G:
            o_k = i_s + tl.arange(0, BS)
            m_k = o_k < T
            b_gk = tl.load(g_cumsum + (bos + o_k) * HQ + i_hq, mask=m_k, other=0).to(tl.float32)
            b_s += b_gq[:, None] - b_gk[None, :]

        # [BT, BS]
        b_m, b_mp = tl.maximum(b_m, tl.max(b_s, 1)), b_m
        b_r = exp2(b_mp - b_m)
        # [BT, BS]
        b_p = exp2(b_s - b_m[:, None])
        # [BT]
        b_acc = b_acc * b_r + tl.sum(b_p, 1)
        # [BT, BV]
        b_o = b_o * b_r[:, None] + tl.dot(b_p.to(b_q.dtype), b_v)

        b_mp = b_m

    # [BT]
    o_q = i_t * BT + tl.arange(0, BT)
    for i_s in range(i_t * BT, min((i_t + 1) * BT, T), BS):
        p_k = tl.make_block_ptr(k + (bos * H + i_h) * K, (K, T), (1, H*K), (0, i_s), (BK, BS), (0, 1))
        p_v = tl.make_block_ptr(v + (bos * H + i_h) * V, (T, V), (H*V, 1), (i_s, i_v * BV), (BS, BV), (1, 0))

        # [BS]
        o_k = i_s + tl.arange(0, BS)
        m_k = o_k < T
        # [BK, BS]
        b_k = tl.load(p_k, boundary_check=(0, 1))
        # [BS, BV]
        b_v = tl.load(p_v, boundary_check=(0, 1))
        # [BT, BS]
        b_s = tl.dot(b_q, b_k) * scale * RCP_LN2

        if USE_G:
            b_gk = tl.load(g_cumsum + (bos + o_k) * HQ + i_hq, mask=m_k, other=0).to(tl.float32)
            b_s += b_gq[:, None] - b_gk[None, :]

        b_s = tl.where((o_q[:, None] >= o_k[None, :]) & m_k[None, :], b_s, float('-inf'))

        # [BT]
        b_m, b_mp = tl.maximum(b_m, tl.max(b_s, 1)), b_m
        b_r = exp2(b_mp - b_m)
        # [BT, BS]
        b_p = exp2(b_s - b_m[:, None])
        # [BT]
        b_acc = b_acc * b_r + tl.sum(b_p, 1)
        # [BT, BV]
        b_o = b_o * b_r[:, None] + tl.dot(b_p.to(b_q.dtype), b_v)
        b_mp = b_m

    b_o = b_o / b_acc[:, None]
    b_m += log2(b_acc)
    tl.store(p_o, b_o.to(p_o.dtype.element_ty), boundary_check=(0, 1))
    tl.store(p_lse, b_m.to(p_lse.dtype.element_ty), boundary_check=(0,))


@triton.jit
def parallel_attn_bwd_kernel_preprocess(
    o,
    do,
    delta,
    B: tl.constexpr,
    V: tl.constexpr,
):
    i_n = tl.program_id(0)
    o_d = tl.arange(0, B)
    m_d = o_d < V

    b_o = tl.load(o + i_n * V + o_d, mask=m_d, other=0)
    b_do = tl.load(do + i_n * V + o_d, mask=m_d, other=0).to(tl.float32)
    b_delta = tl.sum(b_o * b_do)

    tl.store(delta + i_n, b_delta.to(delta.dtype.element_ty))


@triton.heuristics({
    'USE_G': lambda args: args['g_cumsum'] is not None,
    'IS_VARLEN': lambda args: args['cu_seqlens'] is not None,
})
@triton.jit(do_not_specialize=['T'])
def parallel_attn_bwd_kernel_dq(
    q,
    k,
    v,
    lse,
    delta,
    do,
    dq,
    dg_cumsum,
    g_cumsum,
    scale,
    cu_seqlens,
    chunk_indices,
    T,
    B: tl.constexpr,
    H: tl.constexpr,
    HQ: tl.constexpr,
    G: tl.constexpr,
    K: tl.constexpr,
    V: tl.constexpr,
    BT: tl.constexpr,
    BS: tl.constexpr,
    BK: tl.constexpr,
    BV: tl.constexpr,
    IS_VARLEN: tl.constexpr,
    USE_G: tl.constexpr,
):
    i_v, i_t, i_bh = tl.program_id(0), tl.program_id(1), tl.program_id(2)
    i_b, i_hq = i_bh // HQ, i_bh % HQ
    i_h = i_hq // G

    if IS_VARLEN:
        i_n, i_t = tl.load(chunk_indices + i_t * 2).to(tl.int32), tl.load(chunk_indices + i_t * 2 + 1).to(tl.int32)
        bos, eos = tl.load(cu_seqlens + i_n).to(tl.int32), tl.load(cu_seqlens + i_n + 1).to(tl.int32)
        T = eos - bos
    else:
        i_n = i_b
        bos, eos = i_n * T, i_n * T + T
    # NOTE: we must multiply RCP_LN2 after tl.dot for high precision
    RCP_LN2: tl.constexpr = 1.4426950216

    p_q = tl.make_block_ptr(q + (bos * HQ + i_hq) * K, (T, K), (HQ*K, 1), (i_t * BT, 0), (BT, BK), (1, 0))
    p_dq = tl.make_block_ptr(dq + (bos * HQ + i_hq) * K, (T, K), (HQ*K, 1), (i_t * BT, 0), (BT, BK), (1, 0))
    p_do = tl.make_block_ptr(do + (bos * HQ + i_hq) * V, (T, V), (HQ*V, 1), (i_t * BT, i_v * BV), (BT, BV), (1, 0))
    p_lse = tl.make_block_ptr(lse + bos * HQ + i_hq, (T,), (HQ,), (i_t * BT,), (BT,), (0,))
    p_delta = tl.make_block_ptr(delta + bos * HQ + i_hq, (T,), (HQ,), (i_t * BT,), (BT,), (0,))

    # [BT, BK]
    b_q = tl.load(p_q, boundary_check=(0, 1))
    # [BT, BV]
    b_do = tl.load(p_do, boundary_check=(0, 1))
    # [BT]
    b_lse = tl.load(p_lse, boundary_check=(0,))
    b_delta = tl.load(p_delta, boundary_check=(0,))

    # [BT, BK]
    b_dq = tl.zeros([BT, BK], dtype=tl.float32)
    if USE_G:
        b_dg = tl.zeros([BT], dtype=tl.float32)
        p_gq = tl.make_block_ptr(g_cumsum + bos * HQ + i_hq, (T,), (HQ,), (i_t * BT,), (BT,), (0,))
        b_gq = tl.load(p_gq, boundary_check=(0,)).to(tl.float32)
    else:
        b_gq = None
        b_dg = None

    o_q = i_t * BT + tl.arange(0, BT)
    for i_s in range(0, i_t * BT, BS):
        p_k = tl.make_block_ptr(k + (bos * H + i_h) * K, (K, T), (1, H*K), (0, i_s), (BK, BS), (0, 1))
        p_v = tl.make_block_ptr(v + (bos * H + i_h) * V, (V, T), (1, H*V), (i_v * BV, i_s), (BV, BS), (0, 1))

        o_k = i_s + tl.arange(0, BS)
        m_k = o_k < T
        # [BK, BS]
        b_k = tl.load(p_k, boundary_check=(0, 1))
        # [BV, BS]
        b_v = tl.load(p_v, boundary_check=(0, 1))
        # [BT, BS]
        b_s = tl.dot(b_q, b_k) * scale * RCP_LN2
        if USE_G:
            b_gk = tl.load(g_cumsum + (bos + o_k) * HQ + i_hq, mask=m_k, other=0).to(tl.float32)
            b_s += b_gq[:, None] - b_gk[None, :]

        b_s = tl.where((o_q[:, None] >= o_k[None, :]) & m_k[None, :], b_s, float('-inf'))
        b_p = exp2(b_s - b_lse[:, None])
        # [BT, BV] @ [BV, BS] -> [BT, BS]
        b_dp = tl.dot(b_do, b_v)
        b_ds = b_p * (b_dp.to(tl.float32) - b_delta[:, None])
        # [BT, BS] @ [BS, BK] -> [BT, BK]
        b_dq += tl.dot(b_ds.to(b_k.dtype), tl.trans(b_k))
        if USE_G:
            b_dg += tl.sum(b_ds, 1)

    # [BT]
    o_q = i_t * BT + tl.arange(0, BT)
    for i_s in range(i_t * BT, min((i_t + 1) * BT, T), BS):
        p_k = tl.make_block_ptr(k + (bos * H + i_h) * K, (K, T), (1, H*K), (0, i_s), (BK, BS), (0, 1))
        p_v = tl.make_block_ptr(v + (bos * H + i_h) * V, (V, T), (1, H*V), (i_v * BV, i_s), (BV, BS), (0, 1))

        # [BS]
        o_k = i_s + tl.arange(0, BS)
        m_k = o_k < T
        # [BK, BS]
        b_k = tl.load(p_k, boundary_check=(0, 1))
        # [BV, BS]
        b_v = tl.load(p_v, boundary_check=(0, 1))
        # [BT, BS]
        b_s = tl.dot(b_q, b_k) * scale * RCP_LN2

        if USE_G:
            p_gk = tl.make_block_ptr(g_cumsum + bos * HQ + i_hq, (T,), (HQ,), (i_s,), (BS,), (0,))
            b_gk = tl.load(p_gk, boundary_check=(0,)).to(tl.float32)
            b_s += b_gq[:, None] - b_gk[None, :]
        b_p = tl.where((o_q[:, None] >= o_k[None, :]) & m_k[None, :], exp2(b_s - b_lse[:, None]), 0)

        # [BT, BV] @ [BV, BS] -> [BT, BS]
        b_dp = tl.dot(b_do, b_v)
        b_ds = b_p * (b_dp.to(tl.float32) - b_delta[:, None])
        # [BT, BS] @ [BS, BK] -> [BT, BK]
        b_dq += tl.dot(b_ds.to(b_k.dtype), tl.trans(b_k))
        if USE_G:
            b_dg += tl.sum(b_ds, 1)

    b_dq *= scale
    tl.store(p_dq, b_dq.to(p_dq.dtype.element_ty), boundary_check=(0, 1))
    if USE_G:
        p_dg = tl.make_block_ptr(dg_cumsum + bos * HQ + i_hq, (T,), (HQ,), (i_t * BT,), (BT,), (0,))
        tl.store(p_dg, b_dg.to(p_dg.dtype.element_ty), boundary_check=(0,))


@triton.heuristics({
    'USE_G': lambda args: args['g_cumsum'] is not None,
    'IS_VARLEN': lambda args: args['cu_seqlens'] is not None,
})
@triton.jit(do_not_specialize=['T'])
def parallel_attn_bwd_kernel_dkv(
    q,
    k,
    v,
    g_cumsum,
    lse,
    delta,
    do,
    dk,
    dv,
    dg_cumsum,
    cu_seqlens,
    chunk_indices,
    scale,
    T,
    B: tl.constexpr,
    H: tl.constexpr,
    HQ: tl.constexpr,
    G: tl.constexpr,
    K: tl.constexpr,
    V: tl.constexpr,
    BT: tl.constexpr,
    BS: tl.constexpr,
    BK: tl.constexpr,
    BV: tl.constexpr,
    USE_G: tl.constexpr,
    IS_VARLEN: tl.constexpr,
):
    i_v, i_t, i_bh = tl.program_id(0), tl.program_id(1), tl.program_id(2)
    i_b, i_hq = i_bh // HQ, i_bh % HQ
    i_h = i_hq // G

    if IS_VARLEN:
        i_n, i_t = tl.load(chunk_indices + i_t * 2).to(tl.int32), tl.load(chunk_indices + i_t * 2 + 1).to(tl.int32)
        bos, eos = tl.load(cu_seqlens + i_n).to(tl.int32), tl.load(cu_seqlens + i_n + 1).to(tl.int32)
        T = eos - bos
    else:
        i_n = i_b
        bos, eos = i_n * T, i_n * T + T
    RCP_LN2: tl.constexpr = 1.4426950216

    p_k = tl.make_block_ptr(k + (bos * H + i_h) * K, (T, K), (H*K, 1), (i_t * BT, 0), (BT, BK), (1, 0))
    p_v = tl.make_block_ptr(v + (bos * H + i_h) * V, (T, V), (H*V, 1), (i_t * BT, i_v * BV), (BT, BV), (1, 0))
    p_dk = tl.make_block_ptr(dk + (bos * HQ + i_hq) * K, (T, K), (HQ*K, 1), (i_t * BT, 0), (BT, BK), (1, 0))
    p_dv = tl.make_block_ptr(dv + (bos * HQ + i_hq) * V, (T, V), (HQ*V, 1), (i_t * BT, i_v * BV), (BT, BV), (1, 0))

    # [BT, BK]
    b_k = tl.load(p_k, boundary_check=(0, 1))
    b_dk = tl.zeros([BT, BK], dtype=tl.float32)
    # [BT, BV]
    b_v = tl.load(p_v, boundary_check=(0, 1))
    b_dv = tl.zeros([BT, BV], dtype=tl.float32)

    o_k = i_t * BT + tl.arange(0, BT)

    if USE_G:
        p_gk = tl.make_block_ptr(g_cumsum + bos * HQ + i_hq, (T,), (HQ,), (i_t * BT,), (BT,), (0,))
        b_gk = tl.load(p_gk, boundary_check=(0,)).to(tl.float32)
        b_dg = tl.zeros([BT], dtype=tl.float32)
    else:
        b_gk = None
        b_dg = None

    for i_s in range(i_t * BT, min((i_t + 1) * BT, T), BS):
        p_q = tl.make_block_ptr(q + (bos * HQ + i_hq) * K, (T, K), (HQ*K, 1), (i_s, 0), (BS, BK), (1, 0))
        p_do = tl.make_block_ptr(do + (bos * HQ + i_hq) * V, (T, V), (HQ*V, 1), (i_s, i_v * BV), (BS, BV), (1, 0))
        p_lse = tl.make_block_ptr(lse + bos * HQ + i_hq, (T,), (HQ,), (i_s,), (BS,), (0,))
        p_delta = tl.make_block_ptr(delta + bos * HQ + i_hq, (T,), (HQ,), (i_s,), (BS,), (0,))

        # [BS]
        o_q = i_s + tl.arange(0, BS)
        m_q = o_q < T
        # [BS, BK]
        b_q = tl.load(p_q, boundary_check=(0, 1))
        # [BS, BV]
        b_do = tl.load(p_do, boundary_check=(0, 1))
        # [BS]
        b_lse = tl.load(p_lse, boundary_check=(0,))
        b_delta = tl.load(p_delta, boundary_check=(0,))
        # [BT, BS]
        b_s = tl.dot(b_k, tl.trans(b_q)) * scale * RCP_LN2
        if USE_G:
            p_gq = tl.make_block_ptr(g_cumsum + bos * HQ + i_hq, (T,), (HQ,), (i_s,), (BS,), (0,))
            b_gq = tl.load(p_gq, boundary_check=(0,)).to(tl.float32)
            b_s += b_gq[None, :] - b_gk[:, None]
        b_p = tl.where((o_k[:, None] <= o_q[None, :]) & m_q[None, :], exp2(b_s - b_lse[None, :]), 0)
        # [BT, BS] @ [BS, BV] -> [BT, BV]
        b_dv += tl.dot(b_p.to(b_do.dtype), b_do)
        # [BT, BV] @ [BV, BS] -> [BT, BS]
        b_dp = tl.dot(b_v, tl.trans(b_do))
        # [BT, BS]
        b_ds = b_p * (b_dp - b_delta[None, :])
        # [BT, BS] @ [BS, BK] -> [BT, BK]
        b_dk += tl.dot(b_ds.to(b_q.dtype), b_q)
        if USE_G:
            b_dg -= tl.sum(b_ds, 1)

    for i_s in range((i_t + 1) * BT, tl.cdiv(T, BS) * BS, BS):
        p_q = tl.make_block_ptr(q + (bos * HQ + i_hq) * K, (T, K), (HQ*K, 1), (i_s, 0), (BS, BK), (1, 0))
        p_do = tl.make_block_ptr(do + (bos * HQ + i_hq) * V, (T, V), (HQ*V, 1), (i_s, i_v * BV), (BS, BV), (1, 0))
        p_lse = tl.make_block_ptr(lse + bos * HQ + i_hq, (T,), (HQ,), (i_s,), (BS,), (0,))
        p_delta = tl.make_block_ptr(delta + bos * HQ + i_hq, (T,), (HQ,), (i_s,), (BS,), (0,))

        # [BS]
        o_q = i_s + tl.arange(0, BS)
        m_q = o_q < T
        # [BS, BK]
        b_q = tl.load(p_q, boundary_check=(0, 1))
        # [BS, BV]
        b_do = tl.load(p_do, boundary_check=(0, 1))
        # [BS]
        b_lse = tl.load(p_lse, boundary_check=(0,))
        b_delta = tl.load(p_delta, boundary_check=(0,))
        # [BT, BS]
        b_s = tl.dot(b_k, tl.trans(b_q)) * scale * RCP_LN2
        if USE_G:
            p_gq = tl.make_block_ptr(g_cumsum + bos * HQ + i_hq, (T,), (HQ,), (i_s,), (BS,), (0,))
            b_gq = tl.load(p_gq, boundary_check=(0,)).to(tl.float32)
            b_s += b_gq[None, :] - b_gk[:, None]
        b_p = tl.where(m_q[None, :], exp2(b_s - b_lse[None, :]), 0)
        # [BT, BS] @ [BS, BV] -> [BT, BV]
        b_dv += tl.dot(b_p.to(b_do.dtype), b_do)
        # [BT, BV] @ [BV, BS] -> [BT, BS]
        b_dp = tl.dot(b_v, tl.trans(b_do))
        # [BT, BS]
        b_ds = b_p * (b_dp - b_delta[None, :])
        # [BT, BS] @ [BS, BK] -> [BT, BK]
        b_dk += tl.dot(b_ds.to(b_q.dtype), b_q)
        if USE_G:
            b_dg -= tl.sum(b_ds, 1)

    b_dk = b_dk * scale
    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))
    if USE_G:
        p_dg = tl.make_block_ptr(dg_cumsum + bos * HQ + i_hq, (T,), (HQ,), (i_t * BT,), (BT,), (0,))
        tl.store(p_dg, b_dg.to(p_dg.dtype.element_ty), boundary_check=(0,))


def parallel_attn_fwd(
    q: torch.Tensor,
    k: torch.Tensor,
    v: torch.Tensor,
    g_cumsum: torch.Tensor,
    scale: float,
    cu_seqlens: torch.LongTensor | None = None,
    chunk_indices: torch.LongTensor | None = None,
):
    B, T, H, K, V = *k.shape, v.shape[-1]
    HQ = q.shape[2]
    G = HQ // H
    BT = 128
    if check_shared_mem('hopper', q.device.index):
        BS = min(64, max(16, triton.next_power_of_2(T)))
        BK = min(256, max(16, triton.next_power_of_2(K)))
        BV = min(256, max(16, triton.next_power_of_2(V)))
        num_warps = 8
    elif check_shared_mem('ampere', q.device.index):
        BS = min(32, max(16, triton.next_power_of_2(T)))
        BK = min(256, max(16, triton.next_power_of_2(K)))
        BV = min(128, max(16, triton.next_power_of_2(V)))
        num_warps = 4
    else:
        BS = min(32, max(16, triton.next_power_of_2(T)))
        BK = min(256, max(16, triton.next_power_of_2(K)))
        BV = min(64, max(16, triton.next_power_of_2(V)))
        num_warps = 2
    NK = triton.cdiv(K, BK)
    NV = triton.cdiv(V, BV)

    if chunk_indices is None and cu_seqlens is not None:
        chunk_indices = prepare_chunk_indices(cu_seqlens, BT)
    NT = triton.cdiv(T, BT) if cu_seqlens is None else len(chunk_indices)
    assert NK == 1, "The key dimension can not be larger than 256"

    o = torch.empty(B, T, HQ, V, dtype=v.dtype, device=q.device)
    lse = torch.empty(B, T, HQ, dtype=torch.float, device=q.device)
    grid = (NV, NT, B * HQ)
    parallel_attn_fwd_kernel[grid](
        q=q,
        k=k,
        v=v,
        o=o,
        g_cumsum=g_cumsum,
        lse=lse,
        scale=scale,
        cu_seqlens=cu_seqlens,
        chunk_indices=chunk_indices,
        B=B,
        T=T,
        H=H,
        HQ=HQ,
        G=G,
        K=K,
        V=V,
        BT=BT,
        BS=BS,
        BK=BK,
        BV=BV,
        num_warps=num_warps,
    )
    return o, lse


def parallel_attn_bwd_preprocess(
    o: torch.Tensor,
    do: torch.Tensor,
):
    V = o.shape[-1]
    delta = torch.empty_like(o[..., 0], dtype=torch.float)
    parallel_attn_bwd_kernel_preprocess[(delta.numel(),)](
        o=o,
        do=do,
        delta=delta,
        B=triton.next_power_of_2(V),
        V=V,
    )
    return delta


def parallel_attn_bwd(
    q: torch.Tensor,
    k: torch.Tensor,
    v: torch.Tensor,
    o: torch.Tensor,
    g_cumsum: torch.Tensor,
    lse: torch.Tensor,
    do: torch.Tensor,
    scale: float = None,
    chunk_size: int = 128,
    cu_seqlens: torch.LongTensor | None = None,
    chunk_indices: torch.LongTensor | None = None,
):
    B, T, H, K, V = *k.shape, v.shape[-1]
    HQ = q.shape[2]
    G = HQ // H
    if check_shared_mem('hopper'):
        BT = 128
        BS = 64
        BK = max(triton.next_power_of_2(K), 16)
        BV = max(triton.next_power_of_2(V), 16)
        num_warps = 8
    elif check_shared_mem('ampere'):
        BS = 32
        BK = max(triton.next_power_of_2(K), 16)
        BV = max(triton.next_power_of_2(V), 16)
        BT = 128 if K <= 64 else 64
        num_warps = 4
    else:
        BT = 64
        BS = 32
        BK = max(triton.next_power_of_2(K), 16)
        BV = min(max(triton.next_power_of_2(V), 16), 64)
        num_warps = 2

    if chunk_indices is None and cu_seqlens is not None:
        chunk_indices = prepare_chunk_indices(cu_seqlens, BT)
    NT = triton.cdiv(T, BT) if cu_seqlens is None else len(chunk_indices)
    NV = triton.cdiv(V, BV)

    delta = parallel_attn_bwd_preprocess(o, do)

    dq = torch.empty(B, T, HQ, K, dtype=k.dtype if H == HQ else torch.float, device=q.device)
    dk = torch.empty(B, T, HQ, K, dtype=k.dtype if H == HQ else torch.float, device=q.device)
    dv = torch.empty(B, T, HQ, V, dtype=v.dtype if H == HQ else torch.float, device=q.device)
    grid = (NV, NT, B * HQ)

    dg_cumsum, dg_cumsum_k = None, None
    if g_cumsum is not None:
        dg_cumsum = torch.empty(B, T, HQ, dtype=torch.float, device=q.device)
        dg_cumsum_k = torch.empty(B, T, HQ, dtype=torch.float, device=q.device)

    parallel_attn_bwd_kernel_dq[grid](
        q=q,
        k=k,
        v=v,
        g_cumsum=g_cumsum,
        lse=lse,
        delta=delta,
        do=do,
        dq=dq,
        dg_cumsum=dg_cumsum,
        cu_seqlens=cu_seqlens,
        chunk_indices=chunk_indices,
        scale=scale,
        T=T,
        B=B,
        H=H,
        HQ=HQ,
        G=G,
        K=K,
        V=V,
        BT=BT,
        BS=BS,
        BK=BK,
        BV=BV,
        num_warps=num_warps,
    )
    parallel_attn_bwd_kernel_dkv[grid](
        q=q,
        k=k,
        v=v,
        g_cumsum=g_cumsum,
        lse=lse,
        delta=delta,
        do=do,
        dk=dk,
        dv=dv,
        dg_cumsum=dg_cumsum_k,
        cu_seqlens=cu_seqlens,
        chunk_indices=chunk_indices,
        scale=scale,
        T=T,
        B=B,
        H=H,
        HQ=HQ,
        G=G,
        K=K,
        V=V,
        BT=BT,
        BS=BS,
        BK=BK,
        BV=BV,
        num_warps=num_warps,
    )
    dk = reduce(dk, 'b t (h g) k -> b t h k', g=G, reduction='sum')
    dv = reduce(dv, 'b t (h g) v -> b t h v', g=G, reduction='sum')
    if g_cumsum is not None:
        dg_cumsum.add_(dg_cumsum_k)
    return dq, dk, dv, dg_cumsum


@torch.compile
class ParallelAttentionFunction(torch.autograd.Function):

    @staticmethod
    @contiguous
    @autocast_custom_fwd
    def forward(ctx, q, k, v, g, scale, cu_seqlens, chunk_indices=None):
        ctx.dtype = q.dtype

        RCP_LN2: float = 1.4426950216
        g_cumsum = chunk_global_cumsum(g, cu_seqlens=cu_seqlens, scale=RCP_LN2) if g is not None else None
        o, lse = parallel_attn_fwd(
            q=q,
            k=k,
            v=v,
            g_cumsum=g_cumsum,
            scale=scale,
            cu_seqlens=cu_seqlens,
            chunk_indices=chunk_indices,
        )
        ctx.save_for_backward(q, k, v, o, g_cumsum, lse)
        ctx.cu_seqlens = cu_seqlens
        ctx.scale = scale
        return o.to(q.dtype)

    @staticmethod
    @contiguous
    @autocast_custom_bwd
    def backward(ctx, do):
        q, k, v, o, g_cumsum, lse = ctx.saved_tensors
        dq, dk, dv, dg = parallel_attn_bwd(
            q=q,
            k=k,
            v=v,
            o=o,
            g_cumsum=g_cumsum,
            lse=lse,
            do=do,
            scale=ctx.scale,
            cu_seqlens=ctx.cu_seqlens,
        )
        if dg is not None:
            dg = chunk_global_cumsum(dg, cu_seqlens=ctx.cu_seqlens, reverse=True)

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


def parallel_attn(
    q: torch.Tensor,
    k: torch.Tensor,
    v: torch.Tensor,
    g: torch.Tensor | None = None,
    scale: float | None = None,
    cu_seqlens: torch.LongTensor | None = None,
    head_first: bool = False,
    chunk_indices: torch.LongTensor | None = None,
) -> torch.Tensor:
    r"""
    Args:
        q (torch.Tensor):
            queries of shape `[B, T, HQ, K]`.
        k (torch.Tensor):
            keys of shape `[B, T, H, K]`.
            GQA will be applied if HQ is divisible by H.
        v (torch.Tensor):
            values of shape `[B, T, H, V]`.
        g (Optional[torch.Tensor]):
            log decay factors of shape `[B, T, H]`.
        scale (Optional[float]):
            Scale factor for attention scores.
            If not provided, it will default to `1 / sqrt(K)`. Default: `None`.
        cu_seqlens (torch.LongTensor):
            Cumulative sequence lengths of shape `[N+1]` used for variable-length training,
            consistent with the FlashAttention API.
        head_first (Optional[bool]):
            Whether the inputs are in the head-first format. Default: `False`.
            This argument has been deprecated.

    Returns:
        o (torch.Tensor):
            Outputs of shape `[B, T, HQ, V]`.
    """
    if head_first:
        raise DeprecationWarning(
            "head_first is deprecated and will be removed in a future version. "
            "Please use head_first=False for now instead.",
        )
    if not head_first and q.shape[1] < q.shape[2]:
        warnings.warn(
            f"Input tensor shape suggests potential format mismatch: seq_len ({q.shape[1]}) < num_heads ({q.shape[2]}). "
            "This may indicate the inputs were passed in head-first format [B, H, T, ...] "
            "when head_first=False was specified. "
            "Please verify your input tensor format matches the expected shape [B, T, H, ...].",
        )
    if scale is None:
        scale = k.shape[-1] ** -0.5
    if cu_seqlens is not None:
        assert q.shape[0] == 1, "batch size must be 1 when cu_seqlens are provided"

    o = ParallelAttentionFunction.apply(q, k, v, g, scale, cu_seqlens, chunk_indices)
    return o