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fla / build /torch-cuda /ops /attn /parallel.py

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	# Copyright (c) 2023-2026, Songlin Yang, Yu Zhang, Zhiyuan Li
	#
	# This source code is licensed under the MIT license found in the
	# LICENSE file in the root directory of this source tree.
	# For a list of all contributors, visit:
	# https://github.com/fla-org/flash-linear-attention/graphs/contributors

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

	from ...ops.backends import dispatch
	from ...ops.utils import prepare_chunk_indices
	from ...ops.utils.constant import RCP_LN2
	from ...ops.utils.cumsum import chunk_global_cumsum
	from ...ops.utils.op import exp2, log2
	from ...utils import autocast_custom_bwd, autocast_custom_fwd, check_shared_mem, contiguous


	@triton.heuristics({
	'USE_G': lambda args: args['g_cumsum'] is not None,
	'USE_SINK_BIAS': lambda args: args['sink_bias'] is not None,
	'USE_WINDOW': lambda args: args['W'] 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,
	sink_bias,
	lse,
	scale,
	cu_seqlens,
	chunk_indices,
	T,
	W: tl.constexpr,
	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,
	USE_SINK_BIAS: tl.constexpr,
	USE_WINDOW: 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.int64), tl.load(cu_seqlens + i_n + 1).to(tl.int64)
	T = (eos - bos).to(tl.int32)
	else:
	i_n = i_b
	bos, eos = (i_n * T).to(tl.int64), (i_n * T + T).to(tl.int64)
	RCP_LN2: tl.constexpr = 1.4426950216

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

	if USE_SINK_BIAS:
	b_sink_bias = tl.load(sink_bias + i_hq).to(tl.float32)
	else:
	b_sink_bias = None

	# [BT]
	o_q = i_t * BT + tl.arange(0, BT)

	# for sliding window, skip key blocks that are entirely outside the window.
	# the earliest key position any query in this block needs: max(0, i_t*BT - W + 1)
	i_start = tl.maximum((i_t * BT - W + 1) // BS * BS, 0) if USE_WINDOW else 0

	for i_s in range(i_start, 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), (HV, 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

	o_k = i_s + tl.arange(0, BS)
	m_k = o_k < T
	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, :]

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

	# [BT, BS]
	b_m, b_mp = tl.maximum(b_m, tl.max(b_s, 1)), b_m
	# keep the online softmax pivot finite for rows that still have no valid key.
	# this matches sglang's masked-row stabilization and avoids -inf - (-inf) = NaN.
	b_mw = tl.where(b_m == float('-inf'), 0., b_m)
	b_r = exp2(b_mp - b_mw)
	b_p = exp2(b_s - b_mw[:, 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

	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), (HV, 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, :]

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

	# [BT]
	b_m, b_mp = tl.maximum(b_m, tl.max(b_s, 1)), b_m
	b_mw = tl.where(b_m == float('-inf'), 0., b_m)
	b_r = exp2(b_mp - b_mw)
	b_p = exp2(b_s - b_mw[:, 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

	if USE_SINK_BIAS:
	# when a row has no valid key at all, b_m is still -inf here.
	# use a finite pivot before merging the sink-bias mass so lse becomes
	# the sink-bias logit instead of hitting the -inf + inf = NaN path.
	b_m = tl.where(b_m == float('-inf'), 0., b_m)
	# denominator-only sink-bias update (matches GPT-OSS / sglang):
	# the bias logit augments the softmax normalizer without contributing
	# to the value matmul.
	b_acc += exp2(b_sink_bias - 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).to(tl.int64)
	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,
	'USE_WINDOW': lambda args: args['W'] 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,
	W: tl.constexpr,
	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,
	USE_WINDOW: 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.int64), tl.load(cu_seqlens + i_n + 1).to(tl.int64)
	T = (eos - bos).to(tl.int32)
	else:
	i_n = i_b
	bos, eos = (i_n * T).to(tl.int64), (i_n * T + T).to(tl.int64)
	# 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), (HQK, 1), (i_t BT, 0), (BT, BK), (1, 0))
	p_dq = tl.make_block_ptr(dq + (bos * HQ + i_hq) * K, (T, K), (HQK, 1), (i_t BT, 0), (BT, BK), (1, 0))
	p_do = tl.make_block_ptr(do + (bos * HQ + i_hq) * V, (T, V), (HQV, 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)

	i_start = tl.maximum((i_t * BT - W + 1) // BS * BS, 0) if USE_WINDOW else 0

	for i_s in range(i_start, 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, HV), (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, :]

	if USE_WINDOW:
	b_s = tl.where((o_q[:, None] - o_k[None, :] < W) & 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)

	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, HV), (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, :]
	if USE_WINDOW:
	b_p = tl.where(
	(o_q[:, None] >= o_k[None, :]) & (o_q[:, None] - o_k[None, :] < W) & m_k[None, :],
	exp2(b_s - b_lse[:, None]), 0
	)
	else:
	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,
	'USE_WINDOW': lambda args: args['W'] 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,
	W: tl.constexpr,
	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,
	USE_WINDOW: 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.int64), tl.load(cu_seqlens + i_n + 1).to(tl.int64)
	T = (eos - bos).to(tl.int32)
	else:
	i_n = i_b
	bos, eos = (i_n * T).to(tl.int64), (i_n * T + T).to(tl.int64)
	RCP_LN2: tl.constexpr = 1.4426950216

	p_k = tl.make_block_ptr(k + (bos * H + i_h) * K, (T, K), (HK, 1), (i_t BT, 0), (BT, BK), (1, 0))
	p_v = tl.make_block_ptr(v + (bos * H + i_h) * V, (T, V), (HV, 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), (HQK, 1), (i_t BT, 0), (BT, BK), (1, 0))
	p_dv = tl.make_block_ptr(dv + (bos * HQ + i_hq) * V, (T, V), (HQV, 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), (HQV, 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]
	if USE_WINDOW:
	b_p = tl.where(
	(o_k[:, None] <= o_q[None, :]) & (o_q[None, :] - o_k[:, None] < W) & m_q[None, :],
	exp2(b_s - b_lse[None, :]), 0
	)
	else:
	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 sliding window, limit the range of future query blocks to process.
	# a key at position k_pos can only be attended to by queries at positions [k_pos, k_pos + W - 1].
	# so we only need queries up to (i_t + 1) * BT - 1 + W - 1.
	i_end = min(tl.cdiv(T, BS) * BS, (i_t + 1) * BT + W - 1) if USE_WINDOW else tl.cdiv(T, BS) * BS

	for i_s in range((i_t + 1) * BT, i_end, 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), (HQV, 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]
	if USE_WINDOW:
	b_p = tl.where((o_q[None, :] - o_k[:, None] < W) & m_q[None, :], exp2(b_s - b_lse[None, :]), 0)
	else:
	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,))


	@dispatch('attn')
	def parallel_attn_fwd(
	q: torch.Tensor,
	k: torch.Tensor,
	v: torch.Tensor,
	g_cumsum: torch.Tensor,
	sink_bias: torch.Tensor \| None,
	scale: float,
	window_size: int \| None = None,
	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,
	sink_bias=sink_bias,
	lse=lse,
	scale=scale,
	cu_seqlens=cu_seqlens,
	chunk_indices=chunk_indices,
	B=B,
	T=T,
	W=window_size,
	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


	@dispatch('attn')
	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,
	sink_bias: torch.Tensor \| None = None,
	scale: float = None,
	window_size: int \| None = 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
	# dq/dk are reduced over the full value dim in one program (no cross-program accumulation),
	# so BV must span all of V (NV == 1). Don't cap it here -- the forward can, the backward can't.
	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 = max(triton.next_power_of_2(V), 16)
	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,
	W=window_size,
	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,
	W=window_size,
	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)

	dsink_bias = None
	if sink_bias is not None:
	p_sink_bias = torch.exp2(sink_bias[None, None, :] - lse)
	dsink_bias = -(p_sink_bias * delta).sum((0, 1))

	return dq, dk, dv, dg_cumsum, dsink_bias


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

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

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

	@staticmethod
	@contiguous
	@autocast_custom_bwd
	def backward(ctx, do):
	q, k, v, o, g_cumsum, lse, sink_bias = ctx.saved_tensors
	dq, dk, dv, dg, dsink_bias = parallel_attn_bwd(
	q=q,
	k=k,
	v=v,
	o=o,
	g_cumsum=g_cumsum,
	lse=lse,
	do=do,
	sink_bias=sink_bias,
	scale=ctx.scale,
	window_size=ctx.window_size,
	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, dsink_bias, None, None, None, None


	def parallel_attn(
	q: torch.Tensor,
	k: torch.Tensor,
	v: torch.Tensor,
	g: torch.Tensor \| None = None,
	scale: float \| None = None,
	window_size: int \| None = None,
	cu_seqlens: torch.LongTensor \| None = None,
	chunk_indices: torch.LongTensor \| None = None,
	*,
	sink_bias: torch.Tensor \| None = None,
	**kwargs
	) -> 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, HQ]`.
	scale (Optional[float]):
	Scale factor for attention scores.
	If not provided, it will default to `1 / sqrt(K)`. Default: `None`.
	window_size (Optional[int]):
	Sliding window size. If provided, each query at position i only attends to
	keys in `[i - window_size + 1, i]`. If `None`, full causal attention is used.
	Default: `None`.
	cu_seqlens (torch.LongTensor):
	Cumulative sequence lengths of shape `[N+1]` used for variable-length training,
	consistent with the FlashAttention API.
	sink_bias (Optional[torch.Tensor]):
	Per-query-head attention-sink bias logits of shape `[HQ]` — one
	learnable scalar per query head, as introduced by GPT-OSS.

	Augments the softmax denominator with `exp(sink_bias[h])` without
	adding a corresponding key/value entry, so the model can route
	attention mass to a learnable "no-op" target:
	p_i = exp(s_i) / (sum_j exp(s_j) + exp(sink_bias[h]))
	o = sum_i p_i * v_i # sink slot contributes no value
	When `None`, standard softmax is used. Reserved name: the future
	`sink_tokens_*` kwargs will support Xiao 2024-style K/V sink tokens
	and may be combined with `sink_bias`.

	Returns:
	o (torch.Tensor):
	Outputs of shape `[B, T, HQ, V]`.
	"""
	if 'head_first' in kwargs:
	raise DeprecationWarning(
	"head_first has been removed. Inputs must be in `[B, T, H, ...]` format.",
	)
	if scale is None:
	scale = k.shape[-1] ** -0.5
	if cu_seqlens is not None and q.shape[0] != 1:
	raise ValueError(
	f"The batch size is expected to be 1 rather than {q.shape[0]} when using `cu_seqlens`. "
	f"Please flatten variable-length inputs before processing.",
	)
	if sink_bias is not None:
	assert sink_bias.shape == (q.shape[2],), "sink_bias must have shape [HQ]"

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