flash_attn_triton.py

"""
Based on the FlashAttention implementation from Phil Tillet.
https://github.com/openai/triton/blob/master/python/tutorials/06-fused-attention.py

Changes:
- Implement both causal and non-causal attention.
- Implement cross-attention (not just self-attention).
- Support arbitrary seqlens (not just multiples of 128), for both forward and backward.
- [WIP] Support all head dimensions up to 128 (not just 16, 32, 64, 128), for both the forward pass
and backward pass. For the backward pass, head dims that are not 16, 32, 64, 128 will require
more testing since there seems to be some race conditions due to the Triton compiler.
- Speed up the forward pass a bit, and only store the LSE instead of m and l.
- Make the backward for d=128 much faster by reducing register spilling.
- Optionally parallelize the backward pass across seqlen_k, to deal with the case of
small batch size * nheads.
"""

import math

import torch

import triton
import triton.language as tl


@triton.autotune(
    configs=[
        triton.Config({"BLOCK_M": 128, "BLOCK_N": 128}, num_warps=8, num_stages=1),
        triton.Config({"BLOCK_M": 64, "BLOCK_N": 64}, num_warps=4, num_stages=1),
    ],
    key=['CACHE_KEY_SEQLEN_Q', 'CACHE_KEY_SEQLEN_K', 'IS_CAUSAL', 'BLOCK_HEADDIM']
)
@triton.heuristics(
    {
        "EVEN_M": lambda args: args["seqlen_q"] % args["BLOCK_M"] == 0,
        "EVEN_N": lambda args: args["seqlen_k"] % args["BLOCK_N"] == 0,
        "EVEN_HEADDIM": lambda args: args["headdim"] == args["BLOCK_HEADDIM"],
    }
)
@triton.jit
def _fwd_kernel(
    Q, K, V, Out,
    Lse, TMP,  # NOTE: TMP is a scratchpad buffer to workaround a compiler bug
    softmax_scale,
    stride_qb, stride_qh, stride_qm,
    stride_kb, stride_kh, stride_kn,
    stride_vb, stride_vh, stride_vn,
    stride_ob, stride_oh, stride_om,
    nheads, seqlen_q, seqlen_k, seqlen_q_rounded, headdim,
    CACHE_KEY_SEQLEN_Q, CACHE_KEY_SEQLEN_K,
    IS_CAUSAL: tl.constexpr,
    BLOCK_HEADDIM: tl.constexpr,
    EVEN_M: tl.constexpr, EVEN_N: tl.constexpr, EVEN_HEADDIM: tl.constexpr,
    BLOCK_M: tl.constexpr, BLOCK_N: tl.constexpr,
):
    start_m = tl.program_id(0)
    off_hb = tl.program_id(1)
    off_b = off_hb // nheads
    off_h = off_hb % nheads
    # off_b = tl.program_id(1)
    # off_h = tl.program_id(2)
    # off_hb = off_b * nheads + off_h
    # initialize offsets
    offs_m = start_m * BLOCK_M + tl.arange(0, BLOCK_M)
    offs_n = tl.arange(0, BLOCK_N)
    offs_d = tl.arange(0, BLOCK_HEADDIM)
    # Initialize pointers to Q, K, V
    # Adding parenthesis around indexing might use int32 math instead of int64 math?
    # https://github.com/openai/triton/issues/741
    # I'm seeing a tiny bit of difference (5-7us)
    q_ptrs = Q + off_b * stride_qb + off_h * stride_qh + (offs_m[:, None] * stride_qm + offs_d[None, :])
    k_ptrs = K + off_b * stride_kb + off_h * stride_kh + (offs_n[:, None] * stride_kn + offs_d[None, :])
    v_ptrs = V + off_b * stride_vb + off_h * stride_vh + (offs_n[:, None] * stride_vn + offs_d[None, :])
    # initialize pointer to m and l
    t_ptrs = TMP + off_hb * seqlen_q_rounded + offs_m
    lse_i = tl.zeros([BLOCK_M], dtype=tl.float32) - float("inf")
    m_i = tl.zeros([BLOCK_M], dtype=tl.float32) - float("inf")
    acc_o = tl.zeros([BLOCK_M, BLOCK_HEADDIM], dtype=tl.float32)
    # load q: it will stay in SRAM throughout
    # [2022-10-30] TD: Idk why but in the case of EVEN_M=True and EVEN_N=False, if we just call
    # tl.load(q_ptrs), we get the wrong output! Could be a bug in the compiler?
    if EVEN_M & EVEN_N:
        if EVEN_HEADDIM:
            q = tl.load(q_ptrs)
        else:
            q = tl.load(q_ptrs, mask=offs_d[None, :] < headdim, other=0.0)
    else:
        if EVEN_HEADDIM:
            q = tl.load(q_ptrs, mask=offs_m[:, None] < seqlen_q, other=0.0)
        else:
            q = tl.load(q_ptrs, mask=(offs_m[:, None] < seqlen_q) & (offs_d[None, :] < headdim),
                        other=0.0)
    # loop over k, v and update accumulator
    end_n = seqlen_k if not IS_CAUSAL else tl.minimum((start_m + 1) * BLOCK_M, seqlen_k)
    for start_n in range(0, end_n, BLOCK_N):
        start_n = tl.multiple_of(start_n, BLOCK_N)
        # -- compute qk ----
        if EVEN_N:
            if EVEN_HEADDIM:
                k = tl.load(k_ptrs + start_n * stride_kn)
            else:
                k = tl.load(k_ptrs + start_n * stride_kn, mask=offs_d[None, :] < headdim, other=0.0)
        else:
            if EVEN_HEADDIM:
                k = tl.load(k_ptrs + start_n * stride_kn, mask=(start_n + offs_n)[:, None] < seqlen_k,
                            other=0.0)
            else:
                k = tl.load(k_ptrs + start_n * stride_kn,
                            mask=((start_n + offs_n)[:, None] < seqlen_k) & (offs_d[None, :] < headdim),
                            other=0.0)
        qk = tl.zeros([BLOCK_M, BLOCK_N], dtype=tl.float32)
        qk += tl.dot(q, k, trans_b=True)
        if not EVEN_N:
            qk += tl.where((start_n + offs_n)[None, :] < seqlen_k, 0, float("-inf"))
        if IS_CAUSAL:
            qk += tl.where(offs_m[:, None] >= (start_n + offs_n)[None, :], 0, float("-inf"))
        m_ij = tl.maximum(tl.max(qk, 1) * softmax_scale, lse_i)
        # Slightly faster to multiply the softmax_scale here since the compiler can then
        # fuse the mult and add into an fma instruction.
        p = tl.exp(qk * softmax_scale - m_ij[:, None])
        l_ij = tl.sum(p, 1)

        # scale acc_o
        acc_o_scale = tl.exp(m_i - m_ij)

        # # -- update output accumulator --
        # BUG: have to store and immediately load
        tl.store(t_ptrs, acc_o_scale)
        acc_o_scale = tl.load(t_ptrs)
        acc_o = acc_o * acc_o_scale[:, None]
        # update acc_o
        if EVEN_N:
            if EVEN_HEADDIM:
                v = tl.load(v_ptrs + start_n * stride_vn)
            else:
                v = tl.load(v_ptrs + start_n * stride_vn, mask=offs_d[None, :] < headdim, other=0.0)
        else:
            if EVEN_HEADDIM:
                v = tl.load(v_ptrs + start_n * stride_vn, mask=(start_n + offs_n)[:, None] < seqlen_k,
                            other=0.0)
            else:
                v = tl.load(v_ptrs + start_n * stride_vn,
                            mask=((start_n + offs_n)[:, None] < seqlen_k) & (offs_d[None, :] < headdim),
                            other=0.0)
        p = p.to(v.dtype)
        acc_o += tl.dot(p, v)

        # -- update statistics
        m_i = m_ij
        l_i_new = tl.exp(lse_i - m_ij) + l_ij
        lse_i = m_ij + tl.log(l_i_new)

    o_scale = tl.exp(m_i - lse_i)
    # BUG: have to store and immediately load
    tl.store(t_ptrs, o_scale)
    o_scale = tl.load(t_ptrs)
    acc_o = acc_o * o_scale[:, None]
    # rematerialize offsets to save registers
    start_m = tl.program_id(0)
    offs_m = start_m * BLOCK_M + tl.arange(0, BLOCK_M)
    # write back l and m
    lse_ptrs = Lse + off_hb * seqlen_q_rounded + offs_m
    tl.store(lse_ptrs, lse_i)
    # initialize pointers to output
    offs_n = tl.arange(0, BLOCK_HEADDIM)
    out_ptrs = Out + off_b * stride_ob + off_h * stride_oh + (offs_m[:, None] * stride_om + offs_n[None, :])
    if EVEN_M:
        if EVEN_HEADDIM:
            tl.store(out_ptrs, acc_o)
        else:
            tl.store(out_ptrs, acc_o, mask=offs_d[None, :] < headdim)
    else:
        if EVEN_HEADDIM:
            tl.store(out_ptrs, acc_o, mask=offs_m[:, None] < seqlen_q)
        else:
            tl.store(out_ptrs, acc_o,
                     mask=(offs_m[:, None] < seqlen_q) & (offs_d[None, :] < headdim))


@triton.jit
def _bwd_preprocess_do_o_dot(
    Out, DO, Delta,
    stride_ob, stride_oh, stride_om,
    stride_dob, stride_doh, stride_dom,
    nheads, seqlen_q, seqlen_q_rounded, headdim,
    BLOCK_M: tl.constexpr, BLOCK_HEADDIM: tl.constexpr,
):
    start_m = tl.program_id(0)
    off_hb = tl.program_id(1)
    off_b = off_hb // nheads
    off_h = off_hb % nheads
    # initialize offsets
    offs_m = start_m * BLOCK_M + tl.arange(0, BLOCK_M)
    offs_d = tl.arange(0, BLOCK_HEADDIM)
    # load
    o = tl.load(Out + off_b * stride_ob + off_h * stride_oh + offs_m[:, None] * stride_om + offs_d[None, :],
                mask=(offs_m[:, None] < seqlen_q) & (offs_d[None, :] < headdim), other=0.0).to(tl.float32)
    do = tl.load(DO + off_b * stride_dob + off_h * stride_doh + offs_m[:, None] * stride_dom + offs_d[None, :],
                 mask=(offs_m[:, None] < seqlen_q) & (offs_d[None, :] < headdim), other=0.0).to(tl.float32)
    delta = tl.sum(o * do, axis=1)
    # write-back
    tl.store(Delta + off_hb * seqlen_q_rounded + offs_m, delta)


@triton.jit
def _bwd_kernel_one_col_block(
    start_n,
    Q, K, V, softmax_scale,
    DO, DQ, DK, DV,
    LSE, D,
    stride_qm, stride_kn, stride_vn, stride_dom, stride_dqm, stride_dkn, stride_dvn,
    seqlen_q, seqlen_k, headdim,
    ATOMIC_ADD: tl.constexpr,
    IS_CAUSAL: tl.constexpr,
    BLOCK_HEADDIM: tl.constexpr,
    EVEN_M: tl.constexpr, EVEN_N: tl.constexpr, EVEN_HEADDIM: tl.constexpr,
    BLOCK_M: tl.constexpr, BLOCK_N: tl.constexpr,
):
    # We need to make sure begin_m is a multiple of BLOCK_M (not BLOCK_N)
    begin_m = 0 if not IS_CAUSAL else ((start_n * BLOCK_N) // BLOCK_M) * BLOCK_M
    # initialize row/col offsets
    offs_qm = begin_m + tl.arange(0, BLOCK_M)
    offs_n = start_n * BLOCK_N + tl.arange(0, BLOCK_N)
    offs_m = tl.arange(0, BLOCK_M)
    offs_d = tl.arange(0, BLOCK_HEADDIM)
    # initialize pointers to value-like data
    q_ptrs = Q + (offs_qm[:, None] * stride_qm + offs_d[None, :])
    k_ptrs = K + (offs_n[:, None] * stride_kn + offs_d[None, :])
    v_ptrs = V + (offs_n[:, None] * stride_vn + offs_d[None, :])
    do_ptrs = DO + (offs_qm[:, None] * stride_dom + offs_d[None, :])
    dq_ptrs = DQ + (offs_qm[:, None] * stride_dqm + offs_d[None, :])
    # initialize dv amd dk
    dv = tl.zeros([BLOCK_N, BLOCK_HEADDIM], dtype=tl.float32)
    dk = tl.zeros([BLOCK_N, BLOCK_HEADDIM], dtype=tl.float32)
    # k and v stay in SRAM throughout
    # [2022-10-30] TD: Same bug as the fwd. In the case of EVEN_N=True and EVEN_M=False,
    # if we just call tl.load(k_ptrs), we get the wrong output!
    if EVEN_N & EVEN_M:
        if EVEN_HEADDIM:
            k = tl.load(k_ptrs)
            v = tl.load(v_ptrs)
        else:
            k = tl.load(k_ptrs, mask=offs_d[None, :] < headdim, other=0.0)
            v = tl.load(v_ptrs, mask=offs_d[None, :] < headdim, other=0.0)
    else:
        if EVEN_HEADDIM:
            k = tl.load(k_ptrs, mask=offs_n[:, None] < seqlen_k, other=0.0)
            v = tl.load(v_ptrs, mask=offs_n[:, None] < seqlen_k, other=0.0)
        else:
            k = tl.load(k_ptrs, mask=(offs_n[:, None] < seqlen_k) & (offs_d[None, :] < headdim),
                        other=0.0)
            v = tl.load(v_ptrs, mask=(offs_n[:, None] < seqlen_k) & (offs_d[None, :] < headdim),
                        other=0.0)
    # loop over rows
    num_block_m = tl.cdiv(seqlen_q, BLOCK_M)
    for start_m in range(begin_m, num_block_m * BLOCK_M, BLOCK_M):
        start_m = tl.multiple_of(start_m, BLOCK_M)
        offs_m_curr = start_m + offs_m
        # load q, k, v, do on-chip
        if EVEN_M:
            if EVEN_HEADDIM:
                q = tl.load(q_ptrs)
            else:
                q = tl.load(q_ptrs, mask=(offs_d[None, :] < headdim))
        else:
            if EVEN_HEADDIM:
                q = tl.load(q_ptrs, mask=offs_m_curr[:, None] < seqlen_q, other=0.0)
            else:
                q = tl.load(q_ptrs, mask=(offs_m_curr[:, None] < seqlen_q)
                                         & (offs_d[None, :] < headdim), other=0.0)
        # recompute p = softmax(qk, dim=-1).T
        qk = tl.dot(q, k, trans_b=True)
        if not EVEN_N:  # Need to mask out otherwise the softmax is wrong
            qk = tl.where(offs_n[None, :] < seqlen_k, qk, float("-inf"))
        if IS_CAUSAL:
            qk = tl.where(offs_m_curr[:, None] >= (offs_n[None, :]), qk, float("-inf"))
        # There seems to be a race condition when headdim=48/96, and dq, dk, dv are wrong.
        if not EVEN_HEADDIM:
            tl.debug_barrier()
        lse_i = tl.load(LSE + offs_m_curr)
        p = tl.exp(qk * softmax_scale - lse_i[:, None])
        # compute dv
        # [2022-10-30] TD: A Triton bug: if EVEN_M=True and EVEN_HEADDIM=False, if we call
        # do = tl.load(do_ptrs, mask=offs_d[None, :] < headdim, other=0.0), we get wrong outputs
        # in the case of headdim=48/96, seqlen_q & seqlen_k >= 512. If headdim=40 or seqlen < 512,
        # the output is correct.
        if EVEN_M & EVEN_HEADDIM:
            do = tl.load(do_ptrs)
        # if EVEN_M:
        #     if EVEN_HEADDIM:
        #         do = tl.load(do_ptrs)
        #     else:
        #         do = tl.load(do_ptrs, mask=offs_d[None, :] < headdim, other=0.0)
        else:
            if EVEN_HEADDIM:
                do = tl.load(do_ptrs, mask=offs_m_curr[:, None] < seqlen_q, other=0.0)
            else:
                do = tl.load(do_ptrs, mask=(offs_m_curr[:, None] < seqlen_q)
                                           & (offs_d[None, :] < headdim), other=0.0)
        dv += tl.dot(p.to(do.dtype), do, trans_a=True)
        # compute dp = dot(v, do)
        # There seems to be a race condition when headdim=48/96, and dq, dk are wrong.
        if not EVEN_HEADDIM:
            tl.debug_barrier()
        dp = tl.dot(do, v, trans_b=True)
        # compute ds = p * (dp - delta[:, None])
        # Putting the subtraction after the dp matmul (instead of before) is slightly faster
        Di = tl.load(D + offs_m_curr)
        # Converting ds to q.dtype here reduces register pressure and makes it much faster
        # for BLOCK_HEADDIM=128
        ds = (p * (dp - Di[:, None]) * softmax_scale).to(q.dtype)
        # compute dk = dot(ds.T, q)
        dk += tl.dot(ds, q, trans_a=True)
        # compute dq
        if not ATOMIC_ADD:
            if EVEN_M:
                if EVEN_HEADDIM:
                    dq = tl.load(dq_ptrs, eviction_policy="evict_last")
                    dq += tl.dot(ds, k)
                    tl.store(dq_ptrs, dq, eviction_policy="evict_last")
                else:
                    dq = tl.load(dq_ptrs, mask=offs_d[None, :] < headdim, other=0.0,
                                 eviction_policy="evict_last")
                    dq += tl.dot(ds, k)
                    tl.store(dq_ptrs, dq, mask=offs_d[None, :] < headdim, eviction_policy="evict_last")
            else:
                if EVEN_HEADDIM:
                    dq = tl.load(dq_ptrs, mask=offs_m_curr[:, None] < seqlen_q, other=0.0,
                                eviction_policy="evict_last")
                    dq += tl.dot(ds, k)
                    tl.store(dq_ptrs, dq, mask=offs_m_curr[:, None] < seqlen_q,
                            eviction_policy="evict_last")
                else:
                    dq = tl.load(dq_ptrs,
                                 mask=(offs_m_curr[:, None] < seqlen_q) & (offs_d[None, :] < headdim),
                                 other=0.0, eviction_policy="evict_last")
                    dq += tl.dot(ds, k)
                    tl.store(dq_ptrs, dq,
                             mask=(offs_m_curr[:, None] < seqlen_q) & (offs_d[None, :] < headdim),
                             eviction_policy="evict_last")
        else:  # If we're parallelizing across the seqlen_k dimension
            dq = tl.dot(ds, k)
            if EVEN_M:
                if EVEN_HEADDIM:
                    tl.atomic_add(dq_ptrs, dq)
                else:
                    tl.atomic_add(dq_ptrs, dq, mask=offs_d[None, :] < headdim)
            else:
                if EVEN_HEADDIM:
                    tl.atomic_add(dq_ptrs, dq, mask=offs_m_curr[:, None] < seqlen_q)
                else:
                    tl.atomic_add(dq_ptrs, dq,
                                  mask=(offs_m_curr[:, None] < seqlen_q) & (offs_d[None, :] < headdim))
        # increment pointers
        dq_ptrs += BLOCK_M * stride_dqm
        q_ptrs += BLOCK_M * stride_qm
        do_ptrs += BLOCK_M * stride_dom
    # write-back
    dv_ptrs = DV + (offs_n[:, None] * stride_dvn + offs_d[None, :])
    dk_ptrs = DK + (offs_n[:, None] * stride_dkn + offs_d[None, :])
    if EVEN_N:
        if EVEN_HEADDIM:
            tl.store(dv_ptrs, dv)
            tl.store(dk_ptrs, dk)
        else:
            tl.store(dv_ptrs, dv, mask=offs_d[None, :] < headdim)
            tl.store(dk_ptrs, dk, mask=offs_d[None, :] < headdim)
    else:
        if EVEN_HEADDIM:
            tl.store(dv_ptrs, dv, mask=offs_n[:, None] < seqlen_k)
            tl.store(dk_ptrs, dk, mask=offs_n[:, None] < seqlen_k)
        else:
            tl.store(dv_ptrs, dv, mask=(offs_n[:, None] < seqlen_k) & (offs_d[None, :] < headdim))
            tl.store(dk_ptrs, dk, mask=(offs_n[:, None] < seqlen_k) & (offs_d[None, :] < headdim))


def init_to_zero(name):
    # def fn(nargs):
    #     with torch.no_grad():
    #         nargs[name].zero_()
    # return fn
    return lambda nargs: nargs[name].zero_()

@triton.autotune(
    configs=[
        triton.Config({"BLOCK_M": 128, "BLOCK_N": 128, "SEQUENCE_PARALLEL": False}, num_warps=8, num_stages=1, pre_hook=init_to_zero('DQ')),
        triton.Config({"BLOCK_M": 128, "BLOCK_N": 128, "SEQUENCE_PARALLEL": True}, num_warps=8, num_stages=1, pre_hook=init_to_zero('DQ')),
        # Other configs seem to give wrong results when seqlen_q % 128 != 0, disabling them for now
        # # Kernel is buggy (give wrong result) if we set BLOCK_m=128, BLOCK_n=64, num_warps=*4*
        # triton.Config({"BLOCK_M": 128, "BLOCK_N": 64, "SEQUENCE_PARALLEL": False}, num_warps=8, num_stages=1, pre_hook=init_to_zero('DQ')),
        # triton.Config({"BLOCK_M": 128, "BLOCK_N": 64, "SEQUENCE_PARALLEL": True}, num_warps=8, num_stages=1, pre_hook=init_to_zero('DQ')),
        # triton.Config({"BLOCK_M": 64, "BLOCK_N": 64, "SEQUENCE_PARALLEL": False}, num_warps=4, num_stages=1, pre_hook=init_to_zero('DQ')),
        # triton.Config({"BLOCK_M": 64, "BLOCK_N": 64, "SEQUENCE_PARALLEL": True}, num_warps=4, num_stages=1, pre_hook=init_to_zero('DQ')),
        # triton.Config({"BLOCK_M": 128, "BLOCK_N": 128, "SEQUENCE_PARALLEL": False}, num_warps=8, num_stages=1),
        # triton.Config({"BLOCK_M": 128, "BLOCK_N": 128, "SEQUENCE_PARALLEL": True}, num_warps=8, num_stages=1),
        # triton.Config({"BLOCK_M": 128, "BLOCK_N": 64, "SEQUENCE_PARALLEL": False}, num_warps=4, num_stages=1),
        # triton.Config({"BLOCK_M": 128, "BLOCK_N": 64, "SEQUENCE_PARALLEL": True}, num_warps=4, num_stages=1),
        # triton.Config({"BLOCK_M": 64, "BLOCK_N": 64, "SEQUENCE_PARALLEL": False}, num_warps=4, num_stages=1),
        # triton.Config({"BLOCK_M": 64, "BLOCK_N": 64, "SEQUENCE_PARALLEL": True}, num_warps=4, num_stages=1),
    ],
    key=['CACHE_KEY_SEQLEN_Q', 'CACHE_KEY_SEQLEN_K', 'IS_CAUSAL', 'BLOCK_HEADDIM'],
    # reset_to_zero=['DQ']
)
@triton.heuristics(
    {
        "EVEN_M": lambda args: args["seqlen_q"] % args["BLOCK_M"] == 0,
        "EVEN_N": lambda args: args["seqlen_k"] % args["BLOCK_N"] == 0,
        "EVEN_HEADDIM": lambda args: args["headdim"] == args["BLOCK_HEADDIM"],
    }
)
@triton.jit
def _bwd_kernel(
    Q, K, V,
    DO, DQ, DK, DV,
    LSE, D,
    softmax_scale,
    stride_qb, stride_qh, stride_qm,
    stride_kb, stride_kh, stride_kn,
    stride_vb, stride_vh, stride_vn,
    stride_dob, stride_doh, stride_dom,
    stride_dqb, stride_dqh, stride_dqm,
    stride_dkb, stride_dkh, stride_dkn,
    stride_dvb, stride_dvh, stride_dvn,
    nheads, seqlen_q, seqlen_k, seqlen_q_rounded, headdim,
    CACHE_KEY_SEQLEN_Q, CACHE_KEY_SEQLEN_K,
    IS_CAUSAL: tl.constexpr,
    BLOCK_HEADDIM: tl.constexpr,
    SEQUENCE_PARALLEL: tl.constexpr,
    EVEN_M: tl.constexpr, EVEN_N: tl.constexpr, EVEN_HEADDIM: tl.constexpr,
    BLOCK_M: tl.constexpr, BLOCK_N: tl.constexpr,
):
    off_hb = tl.program_id(1)
    off_b = off_hb // nheads
    off_h = off_hb % nheads
    # offset pointers for batch/head
    Q += off_b * stride_qb + off_h * stride_qh
    K += off_b * stride_kb + off_h * stride_kh
    V += off_b * stride_vb + off_h * stride_vh
    DO += off_b * stride_dob + off_h * stride_doh
    DQ += off_b * stride_dqb + off_h * stride_dqh
    DK += off_b * stride_dkb + off_h * stride_dkh
    DV += off_b * stride_dvb + off_h * stride_dvh
    # pointer to row-wise quantities in value-like data
    D += off_hb * seqlen_q_rounded
    LSE += off_hb * seqlen_q_rounded
    if not SEQUENCE_PARALLEL:
        num_block_n = tl.cdiv(seqlen_k, BLOCK_N)
        for start_n in range(0, num_block_n):
            _bwd_kernel_one_col_block(
                start_n,
                Q, K, V, softmax_scale,
                DO, DQ, DK, DV,
                LSE, D,
                stride_qm, stride_kn, stride_vn, stride_dom, stride_dqm, stride_dkn, stride_dvn,
                seqlen_q, seqlen_k, headdim,
                ATOMIC_ADD=False,
                IS_CAUSAL=IS_CAUSAL,
                BLOCK_HEADDIM=BLOCK_HEADDIM,
                EVEN_M=EVEN_M, EVEN_N=EVEN_N, EVEN_HEADDIM=EVEN_HEADDIM,
                BLOCK_M=BLOCK_M, BLOCK_N=BLOCK_N
            )
    else:
        start_n = tl.program_id(0)
        _bwd_kernel_one_col_block(
            start_n,
            Q, K, V, softmax_scale,
            DO, DQ, DK, DV,
            LSE, D,
            stride_qm, stride_kn, stride_vn, stride_dom, stride_dqm, stride_dkn, stride_dvn,
            seqlen_q, seqlen_k, headdim,
            ATOMIC_ADD=True,
            IS_CAUSAL=IS_CAUSAL,
            BLOCK_HEADDIM=BLOCK_HEADDIM,
            EVEN_M=EVEN_M, EVEN_N=EVEN_N, EVEN_HEADDIM=EVEN_HEADDIM,
            BLOCK_M=BLOCK_M, BLOCK_N=BLOCK_N
        )


def _flash_attn_forward(q, k, v, causal=False, softmax_scale=None):
    # shape constraints
    batch, seqlen_q, nheads, d = q.shape
    _, seqlen_k, _, _ = k.shape
    assert k.shape == (batch, seqlen_k, nheads, d)
    assert v.shape == (batch, seqlen_k, nheads, d)
    assert d <= 128, 'FlashAttention only support head dimensions up to 128'
    assert q.dtype == k.dtype == v.dtype, 'All tensors must have the same type'
    assert q.dtype in [torch.float16, torch.bfloat16], 'Only support fp16 and bf16'
    assert q.is_cuda and k.is_cuda and v.is_cuda
    softmax_scale = softmax_scale or 1.0 / math.sqrt(d)
    seqlen_q_rounded = math.ceil(seqlen_q / 128) * 128
    lse = torch.empty((batch, nheads, seqlen_q_rounded), device=q.device, dtype=torch.float32)
    # lse = torch.full((batch, nheads, seqlen_q_rounded), float('inf'), device=q.device,
                     # dtype=torch.float32)
    tmp = torch.empty((batch, nheads, seqlen_q_rounded), device=q.device, dtype=torch.float32)
    o = torch.empty_like(q)

    BLOCK_HEADDIM = max(triton.next_power_of_2(d), 16)
    # BLOCK = 128
    # num_warps = 4 if d <= 64 else 8
    grid = lambda META: (triton.cdiv(seqlen_q, META["BLOCK_M"]), batch * nheads)
    _fwd_kernel[grid](
        q, k, v, o,
        lse, tmp,
        softmax_scale,
        q.stride(0), q.stride(2), q.stride(1),
        k.stride(0), k.stride(2), k.stride(1),
        v.stride(0), v.stride(2), v.stride(1),
        o.stride(0), o.stride(2), o.stride(1),
        nheads, seqlen_q, seqlen_k, seqlen_q_rounded, d,
        seqlen_q // 32,  seqlen_k // 32, # key for triton cache (limit number of compilations)
        # Can't use kwargs here because triton autotune expects key to be args, not kwargs
        # IS_CAUSAL=causal, BLOCK_HEADDIM=d,
        causal, BLOCK_HEADDIM,
        # BLOCK_M=BLOCK, BLOCK_N=BLOCK,
        # num_warps=num_warps,
        # num_stages=1,
    )
    return o, lse, softmax_scale  # softmax_scale could have been updated


def _flash_attn_backward(do, q, k, v, o, lse, dq, dk, dv, causal=False, softmax_scale=None):
    # Make sure that the last dimension is contiguous
    if do.stride(-1) != 1:
        do = do.contiguous()
    batch, seqlen_q, nheads, d = q.shape
    _, seqlen_k, _, _ = k.shape
    # assert d in {16, 32, 64, 128}
    assert d <= 128
    seqlen_q_rounded = math.ceil(seqlen_q / 128) * 128
    assert lse.shape == (batch, nheads, seqlen_q_rounded)
    # dq_accum = torch.zeros_like(q, dtype=torch.float32)
    dq_accum = torch.empty_like(q, dtype=torch.float32)
    delta = torch.empty_like(lse)
    # delta = torch.zeros_like(lse)

    BLOCK_HEADDIM = max(triton.next_power_of_2(d), 16)
    grid = lambda META: (triton.cdiv(seqlen_q, META["BLOCK_M"]), batch * nheads)
    _bwd_preprocess_do_o_dot[grid](
        o, do, delta,
        o.stride(0), o.stride(2), o.stride(1),
        do.stride(0), do.stride(2), do.stride(1),
        nheads, seqlen_q, seqlen_q_rounded, d,
        BLOCK_M=128, BLOCK_HEADDIM=BLOCK_HEADDIM,
    )

    # TODO: There are 2 Memcpy DtoD when I use the autotuner.
    # BLOCK_M = 128
    # BLOCK_N = 64
    # num_warps = 4
    grid = lambda META: (triton.cdiv(seqlen_k, META["BLOCK_N"]) if META["SEQUENCE_PARALLEL"] else 1,
                    batch * nheads)
    _bwd_kernel[grid](
        q, k, v,
        do, dq_accum, dk, dv,
        lse, delta,
        softmax_scale,
        q.stride(0), q.stride(2), q.stride(1),
        k.stride(0), k.stride(2), k.stride(1),
        v.stride(0), v.stride(2), v.stride(1),
        do.stride(0), do.stride(2), do.stride(1),
        dq_accum.stride(0), dq_accum.stride(2), dq_accum.stride(1),
        dk.stride(0), dk.stride(2), dk.stride(1),
        dv.stride(0), dv.stride(2), dv.stride(1),
        nheads, seqlen_q, seqlen_k, seqlen_q_rounded, d,
        seqlen_q // 32,  seqlen_k // 32, # key for triton cache (limit number of compilations)
        # Can't use kwargs here because triton autotune expects key to be args, not kwargs
        # IS_CAUSAL=causal, BLOCK_HEADDIM=d,
        causal, BLOCK_HEADDIM,
        # SEQUENCE_PARALLEL=False,
        # BLOCK_M=BLOCK_M, BLOCK_N=BLOCK_N,
        # num_warps=num_warps,
        # num_stages=1,
    )
    dq.copy_(dq_accum)


class FlashAttnQKVPackedFunc(torch.autograd.Function):

    @staticmethod
    def forward(ctx, qkv, causal=False, softmax_scale=None):
        """
            qkv: (batch, seqlen, 3, nheads, headdim)
        """
        # Make sure that the last dimension is contiguous
        if qkv.stride(-1) != 1:
            qkv = qkv.contiguous()
        o, lse, ctx.softmax_scale = _flash_attn_forward(
            qkv[:, :, 0], qkv[:, :, 1], qkv[:, :, 2], causal=causal, softmax_scale=softmax_scale
        )
        ctx.save_for_backward(qkv, o, lse)
        ctx.causal = causal
        return o

    @staticmethod
    def backward(ctx, do):
        qkv, o, lse = ctx.saved_tensors
        dqkv = torch.empty_like(qkv)
        _flash_attn_backward(do, qkv[:, :, 0], qkv[:, :, 1], qkv[:, :, 2], o, lse,
                             dqkv[:, :, 0], dqkv[:, :, 1], dqkv[:, :, 2],
                             causal=ctx.causal, softmax_scale=ctx.softmax_scale)
        return dqkv, None, None


flash_attn_qkvpacked_func = FlashAttnQKVPackedFunc.apply


class FlashAttnKVPackedFunc(torch.autograd.Function):

    @staticmethod
    def forward(ctx, q, kv, causal=False, softmax_scale=None):
        """
            q: (batch, seqlen, nheads, headdim)
            kv: (batch, seqlen, 2, nheads, headdim)
        """
        # Make sure that the last dimension is contiguous
        q, kv = [x if x.stride(-1) == 1 else x.contiguous() for x in [q, kv]]
        o, lse, ctx.softmax_scale = _flash_attn_forward(
            q, kv[:, :, 0], kv[:, :, 1], causal=causal, softmax_scale=softmax_scale
        )
        ctx.save_for_backward(q, kv, o, lse)
        ctx.causal = causal
        return o

    @staticmethod
    def backward(ctx, do):
        q, kv, o, lse = ctx.saved_tensors
        dq = torch.empty_like(q)
        dkv = torch.empty_like(kv)
        _flash_attn_backward(do, q, qkv[:, :, 0], qkv[:, :, 1], o, lse,
                             dq, dkv[:, :, 0], dkv[:, :, 1],
                             causal=ctx.causal, softmax_scale=ctx.softmax_scale)
        return dq, dkv, None, None


flash_attn_kvpacked_func = FlashAttnKVPackedFunc.apply


class FlashAttnFunc(torch.autograd.Function):

    @staticmethod
    def forward(ctx, q, k, v, causal=False, softmax_scale=None):
        """
            q, k, v: (batch_size, seqlen, nheads, headdim)
        """
        # Make sure that the last dimension is contiguous
        q, k, v = [x if x.stride(-1) == 1 else x.contiguous() for x in [q, k, v]]
        o, lse, ctx.softmax_scale = _flash_attn_forward(q, k, v, causal=causal,
                                                        softmax_scale=softmax_scale)
        ctx.save_for_backward(q, k, v, o, lse)
        ctx.causal = causal
        return o

    @staticmethod
    def backward(ctx, do):
        q, k, v, o, lse = ctx.saved_tensors
        dq = torch.empty_like(q)
        dk = torch.empty_like(k)
        dv = torch.empty_like(v)
        _flash_attn_backward(do, q, k, v, o, lse, dq, dk, dv,
                             causal=ctx.causal, softmax_scale=ctx.softmax_scale)
        return dq, dk, dv, None, None


flash_attn_func = FlashAttnFunc.apply