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fla/ops/attn/parallel.py

@@ -0,0 +1,629 @@
+# -*- coding: utf-8 -*-
+# Copyright (c) 2023-2025, Songlin Yang, Yu Zhang
+
+from typing import Optional
+
+import torch
+import triton
+import triton.language as tl
+from einops import rearrange, reduce
+
+from fla.ops.common.utils import prepare_chunk_indices
+from fla.ops.utils.op import exp, log
+from fla.utils import autocast_custom_bwd, autocast_custom_fwd, check_shared_mem, contiguous
+
+
+@triton.heuristics({
+    'USE_OFFSETS': lambda args: args['offsets'] is not None
+})
+@triton.autotune(
+    configs=[
+        triton.Config({}, num_warps=num_warps, num_stages=num_stages)
+        for num_warps in [1, 2, 4] + ([8] if check_shared_mem('hopper') else [])
+        for num_stages in [2, 3, 4, 5]
+    ],
+    key=['B', 'H', 'G', 'K', 'V', 'BK', 'BV'],
+)
+@triton.jit
+def parallel_attn_fwd_kernel(
+    q,
+    k,
+    v,
+    o,
+    lse,
+    scale,
+    offsets,
+    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_OFFSETS: 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 USE_OFFSETS:
+        i_n, i_t = tl.load(indices + i_t * 2).to(tl.int32), tl.load(indices + i_t * 2 + 1).to(tl.int32)
+        bos, eos = tl.load(offsets + i_n).to(tl.int32), tl.load(offsets + i_n + 1).to(tl.int32)
+        T = eos - bos
+    else:
+        i_n = i_b
+        bos, eos = i_n * T, i_n * T + T
+
+    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))
+    b_q = (b_q * scale).to(b_q.dtype)
+    # [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)
+    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)
+
+        # [BT, BS]
+        b_m, b_mp = tl.maximum(b_m, tl.max(b_s, 1)), b_m
+        b_r = exp(b_mp - b_m)
+        # [BT, BS]
+        b_p = exp(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)
+        # [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)
+        b_s = tl.where(o_q[:, None] >= o_k[None, :], b_s, float('-inf'))
+
+        # [BT]
+        b_m, b_mp = tl.maximum(b_m, tl.max(b_s, 1)), b_m
+        b_r = exp(b_mp - b_m)
+        # [BT, BS]
+        b_p = exp(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 += log(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_OFFSETS': lambda args: args['offsets'] is not None
+})
+@triton.autotune(
+    configs=[
+        triton.Config({}, num_warps=num_warps, num_stages=num_stages)
+        for num_warps in [1, 2, 4] + ([8] if check_shared_mem('hopper') else [])
+        for num_stages in [2, 3, 4, 5]
+    ],
+    key=['B', 'H', 'G', 'K', 'V', 'BK', 'BV'],
+)
+@triton.jit(do_not_specialize=['T'])
+def parallel_attn_bwd_kernel_dq(
+    q,
+    k,
+    v,
+    lse,
+    delta,
+    do,
+    dq,
+    scale,
+    offsets,
+    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_OFFSETS: 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 USE_OFFSETS:
+        i_n, i_t = tl.load(indices + i_t * 2).to(tl.int32), tl.load(indices + i_t * 2 + 1).to(tl.int32)
+        bos, eos = tl.load(offsets + i_n).to(tl.int32), tl.load(offsets + i_n + 1).to(tl.int32)
+        T = eos - bos
+    else:
+        i_n = i_b
+        bos, eos = i_n * T, i_n * T + T
+
+    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))
+    b_q = (b_q * scale).to(b_q.dtype)
+    # [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)
+    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))
+        # [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)
+        b_p = exp(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))
+
+    # [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)
+        # [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)
+        b_p = exp(b_s - b_lse[:, None])
+        b_p = tl.where(o_q[:, None] >= o_k[None, :], b_p, 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))
+
+    b_dq *= scale
+
+    tl.store(p_dq, b_dq.to(p_dq.dtype.element_ty), boundary_check=(0, 1))
+
+
+@triton.heuristics({
+    'USE_OFFSETS': lambda args: args['offsets'] is not None
+})
+@triton.autotune(
+    configs=[
+        triton.Config({}, num_warps=num_warps, num_stages=num_stages)
+        for num_warps in [1, 2, 4] + ([8] if check_shared_mem('hopper') else [])
+        for num_stages in [2, 3, 4, 5]
+    ],
+    key=['B', 'H', 'G', 'K', 'V', 'BK', 'BV'],
+)
+@triton.jit(do_not_specialize=['T'])
+def parallel_attn_bwd_kernel_dkv(
+    q,
+    k,
+    v,
+    lse,
+    delta,
+    do,
+    dk,
+    dv,
+    offsets,
+    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_OFFSETS: 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 USE_OFFSETS:
+        i_n, i_t = tl.load(indices + i_t * 2).to(tl.int32), tl.load(indices + i_t * 2 + 1).to(tl.int32)
+        bos, eos = tl.load(offsets + i_n).to(tl.int32), tl.load(offsets + i_n + 1).to(tl.int32)
+        T = eos - bos
+    else:
+        i_n = i_b
+        bos, eos = i_n * T, i_n * T + T
+
+    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)
+    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)
+        # [BS, BK]
+        b_q = tl.load(p_q, boundary_check=(0, 1))
+        b_q = (b_q * scale).to(b_q.dtype)
+        # [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))
+        b_p = exp(b_s - b_lse[None, :])
+        b_p = tl.where(o_k[:, None] <= o_q[None, :], b_p, 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)
+
+    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)
+        # [BS, BK]
+        b_q = tl.load(p_q, boundary_check=(0, 1))
+        b_q = (b_q * scale).to(b_q.dtype)
+        # [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))
+        b_p = exp(b_s - b_lse[None, :])
+        # [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)
+
+    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))
+
+
+def parallel_attn_fwd(
+    q: torch.Tensor,
+    k: torch.Tensor,
+    v: torch.Tensor,
+    scale: float,
+    chunk_size: int = 128,
+    offsets: Optional[torch.LongTensor] = None,
+    indices: Optional[torch.LongTensor] = None,
+):
+    B, T, H, K, V = *k.shape, v.shape[-1]
+    HQ = q.shape[2]
+    G = HQ // H
+    BT = chunk_size
+    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)))
+    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)))
+    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)))
+    NK = triton.cdiv(K, BK)
+    NV = triton.cdiv(V, BV)
+    NT = triton.cdiv(T, BT) if offsets is None else len(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,
+        lse=lse,
+        scale=scale,
+        offsets=offsets,
+        indices=indices,
+        B=B,
+        T=T,
+        H=H,
+        HQ=HQ,
+        G=G,
+        K=K,
+        V=V,
+        BT=BT,
+        BS=BS,
+        BK=BK,
+        BV=BV,
+    )
+    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.float32)
+    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,
+    lse: torch.Tensor,
+    do: torch.Tensor,
+    scale: float = None,
+    chunk_size: int = 128,
+    offsets: Optional[torch.LongTensor] = None,
+    indices: Optional[torch.LongTensor] = None,
+):
+    B, T, H, K, V = *k.shape, v.shape[-1]
+    HQ = q.shape[2]
+    G = HQ // H
+    BT = chunk_size
+    BS = max(16, triton.next_power_of_2(T))
+    BS = min(32, BS) if check_shared_mem('ampere') else min(16, BS)
+    BK = max(16, triton.next_power_of_2(K))
+    BV = max(16, triton.next_power_of_2(V))
+    NV = triton.cdiv(V, BV)
+    NT = triton.cdiv(T, BT) if offsets is None else len(indices)
+
+    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)
+    parallel_attn_bwd_kernel_dq[grid](
+        q=q,
+        k=k,
+        v=v,
+        lse=lse,
+        delta=delta,
+        do=do,
+        dq=dq,
+        offsets=offsets,
+        indices=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
+    )
+    parallel_attn_bwd_kernel_dkv[grid](
+        q=q,
+        k=k,
+        v=v,
+        lse=lse,
+        delta=delta,
+        do=do,
+        dk=dk,
+        dv=dv,
+        offsets=offsets,
+        indices=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
+    )
+    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')
+    return dq, dk, dv
+
+
+@torch.compile
+class ParallelAttentionFunction(torch.autograd.Function):
+
+    @staticmethod
+    @contiguous
+    @autocast_custom_fwd
+    def forward(ctx, q, k, v, scale, offsets):
+        ctx.dtype = q.dtype
+
+        chunk_size = min(128, max(16, triton.next_power_of_2(q.shape[1])))
+        # 2-d indices denoting the offsets of chunks in each sequence
+        # for example, if the passed `offsets` is [0, 100, 356] and `chunk_size` is 64,
+        # then there are 2 and 4 chunks in the 1st and 2nd sequences respectively, and `indices` will be
+        # [[0, 0], [0, 1], [1, 0], [1, 1], [1, 2], [1, 3]]
+        indices = prepare_chunk_indices(offsets, chunk_size) if offsets is not None else None
+
+        o, lse = parallel_attn_fwd(
+            q=q,
+            k=k,
+            v=v,
+            scale=scale,
+            chunk_size=chunk_size,
+            offsets=offsets,
+            indices=indices
+        )
+        ctx.save_for_backward(q, k, v, o, lse)
+        ctx.chunk_size = chunk_size
+        ctx.offsets = offsets
+        ctx.indices = indices
+        ctx.scale = scale
+        return o.to(q.dtype)
+
+    @staticmethod
+    @contiguous
+    @autocast_custom_bwd
+    def backward(ctx, do):
+        q, k, v, o, lse = ctx.saved_tensors
+        dq, dk, dv = parallel_attn_bwd(
+            q=q,
+            k=k,
+            v=v,
+            o=o,
+            lse=lse,
+            do=do,
+            scale=ctx.scale,
+            chunk_size=ctx.chunk_size,
+            offsets=ctx.offsets,
+            indices=ctx.indices
+        )
+        return dq.to(q), dk.to(k), dv.to(v), None, None, None, None, None, None, None, None
+
+
+def parallel_attn(
+    q: torch.Tensor,
+    k: torch.Tensor,
+    v: torch.Tensor,
+    scale: Optional[float] = None,
+    cu_seqlens: Optional[torch.LongTensor] = None,
+    head_first: bool = False
+) -> torch.Tensor:
+    r"""
+    Args:
+        q (torch.Tensor):
+            queries of shape `[B, T, HQ, K]` if `head_first=False` else `[B, HQ, T, K]`.
+        k (torch.Tensor):
+            keys of shape `[B, T, H, K]` if `head_first=False` else `[B, H, T, K]`.
+            GQA will be applied if HQ is divisible by H.
+        v (torch.Tensor):
+            values of shape `[B, T, H, V]` if `head_first=False` else `[B, H, T, V]`.
+        scale (Optional[int]):
+            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`.
+
+    Returns:
+        o (torch.Tensor):
+            Outputs of shape `[B, T, HQ, V]` if `head_first=False` else `[B, HQ, T, V]`.
+    """
+    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"
+    if head_first:
+        q, k, v = map(lambda x: rearrange(x, 'b h t d -> b t h d'), (q, k, v))
+    o = ParallelAttentionFunction.apply(q, k, v, scale, cu_seqlens)
+    if head_first:
+        o = rearrange(o, 'b t h d -> b h t d')
+    return o

fla/ops/common/chunk_delta_h.py

@@ -0,0 +1,399 @@
+# -*- coding: utf-8 -*-
+# Copyright (c) 2023-2025, Songlin Yang, Yu Zhang
+
+from typing import Optional, Tuple
+
+import torch
+import triton
+import triton.language as tl
+
+from fla.ops.common.utils import prepare_chunk_offsets
+from fla.ops.utils.op import exp
+from fla.utils import check_shared_mem, is_nvidia_hopper, use_cuda_graph
+
+NUM_WARPS = [2, 4] if is_nvidia_hopper else [2, 4, 8, 16]
+
+
+@triton.heuristics({
+    'USE_G': lambda args: args['g'] is not None,
+    'USE_INITIAL_STATE': lambda args: args['h0'] is not None,
+    'STORE_FINAL_STATE': lambda args: args['ht'] is not None,
+    'USE_OFFSETS': lambda args: args['offsets'] is not None,
+})
+@triton.autotune(
+    configs=[
+        triton.Config({}, num_warps=num_warps, num_stages=num_stages)
+        for num_warps in NUM_WARPS
+        for num_stages in [2, 3, 4]
+    ],
+    key=['H', 'K', 'V', 'BT', 'BK', 'BV', 'USE_G'],
+    use_cuda_graph=use_cuda_graph,
+)
+@triton.jit(do_not_specialize=['T'])
+def chunk_gated_delta_rule_fwd_kernel_h(
+    k,
+    v,
+    d,
+    v_new,
+    g,
+    h,
+    h0,
+    ht,
+    offsets,
+    chunk_offsets,
+    T,
+    H: tl.constexpr,
+    K: tl.constexpr,
+    V: tl.constexpr,
+    BT: tl.constexpr,
+    BC: tl.constexpr,
+    BK: tl.constexpr,
+    BV: tl.constexpr,
+    NT: tl.constexpr,
+    USE_G: tl.constexpr,
+    USE_INITIAL_STATE: tl.constexpr,
+    STORE_FINAL_STATE: tl.constexpr,
+    USE_OFFSETS: tl.constexpr,
+    HEAD_FIRST: tl.constexpr,
+):
+    i_k, i_v, i_nh = tl.program_id(0), tl.program_id(1), tl.program_id(2)
+    i_n, i_h = i_nh // H, i_nh % H
+    if USE_OFFSETS:
+        bos, eos = tl.load(offsets + i_n).to(tl.int32), tl.load(offsets + i_n + 1).to(tl.int32)
+        T = eos - bos
+        NT = tl.cdiv(T, BT)
+        boh = tl.load(chunk_offsets + i_n).to(tl.int32)
+    else:
+        bos, eos = i_n * T, i_n * T + T
+        NT = tl.cdiv(T, BT)
+        boh = i_n * NT
+
+    # [BK, BV]
+    b_h = tl.zeros([BK, BV], dtype=tl.float32)
+    if USE_INITIAL_STATE:
+        p_h0 = tl.make_block_ptr(h0 + i_nh * K*V, (K, V), (V, 1), (i_k * BK, i_v * BV), (BK, BV), (1, 0))
+        b_h = tl.load(p_h0, boundary_check=(0, 1)).to(tl.float32)
+
+    for i_t in range(NT):
+        if HEAD_FIRST:
+            p_h = tl.make_block_ptr(h + (i_nh * NT + i_t) * K*V, (K, V), (V, 1), (i_k * BK, i_v * BV), (BK, BV), (1, 0))
+        else:
+            p_h = tl.make_block_ptr(h + ((boh + i_t) * H + i_h) * K*V, (K, V), (V, 1), (i_k * BK, i_v * BV), (BK, BV), (1, 0))
+        tl.store(p_h, b_h.to(p_h.dtype.element_ty), boundary_check=(0, 1))
+        b_hc = tl.zeros([BK, BV], dtype=tl.float32)
+        if USE_G:
+            last_idx = min((i_t + 1) * BT, T) - 1
+            if HEAD_FIRST:
+                b_g_last = tl.load(g + i_nh * T + last_idx)
+            else:
+                b_g_last = tl.load(g + bos * H + last_idx * H + i_h)
+        else:
+            b_g_last = None
+            last_idx = None
+        # since we need to make all DK in the SRAM. we face serve SRAM memory burden. By subchunking we allievate such burden
+        for i_c in range(tl.cdiv(min(BT, T - i_t * BT), BC)):
+            if HEAD_FIRST:
+                p_k = tl.make_block_ptr(k + i_nh * T*K, (K, T), (1, K), (i_k * BK, i_t * BT + i_c * BC), (BK, BC), (0, 1))
+                p_d = tl.make_block_ptr(d + i_nh * T*K, (T, K), (K, 1), (i_t * BT + i_c * BC, i_k * BK), (BC, BK), (1, 0))
+                p_v = tl.make_block_ptr(v + i_nh * T*V, (T, V), (V, 1), (i_t * BT + i_c * BC, i_v * BV), (BC, BV), (1, 0))
+                p_v_new = tl.make_block_ptr(v_new+i_nh*T*V, (T, V), (V, 1), (i_t * BT + i_c * BC, i_v * BV), (BC, BV), (1, 0))
+                p_g = tl.make_block_ptr(g + i_nh * T, (T,), (1,), (i_t * BT + i_c * BC,), (BC,), (0,)) if USE_G else None
+            else:
+                p_k = tl.make_block_ptr(k+(bos*H+i_h)*K, (K, T), (1, H*K), (i_k * BK, i_t * BT + i_c * BC), (BK, BC), (0, 1))
+                p_d = tl.make_block_ptr(d+(bos*H+i_h)*K, (T, K), (H*K, 1), (i_t * BT + i_c * BC, i_k * BK), (BC, BK), (1, 0))
+                p_v = tl.make_block_ptr(v+(bos*H+i_h)*V, (T, V), (H*V, 1), (i_t * BT + i_c * BC, i_v * BV), (BC, BV), (1, 0))
+                p_v_new = tl.make_block_ptr(v_new+(bos*H+i_h)*V, (T, V), (H*V, 1), (i_t*BT+i_c*BC, i_v * BV), (BC, BV), (1, 0))
+                p_g = tl.make_block_ptr(g+bos*H+i_h, (T,), (H,), (i_t*BT+i_c*BC, ), (BC,), (0,)) if USE_G else None
+            b_g = tl.load(p_g, boundary_check=(0, )) if USE_G else None
+            # [BK, BC]
+            b_k = tl.load(p_k, boundary_check=(0, 1))
+            b_k = (b_k * exp(b_g_last - b_g)[None, :]).to(b_k.dtype) if USE_G else b_k
+            # [BC, BK]
+            b_d = tl.load(p_d, boundary_check=(0, 1))
+            b_d = (b_d * exp(b_g)[:, None]).to(b_d.dtype) if USE_G else b_d
+            # [BC, BV]
+            b_v = tl.load(p_v, boundary_check=(0, 1))
+            b_v2 = b_v - tl.dot(b_d, b_h.to(b_d.dtype))
+            # [BK, BV]
+            tl.store(p_v_new, b_v2.to(p_v_new.dtype.element_ty), boundary_check=(0, 1))
+            b_hc += tl.dot(b_k, b_v2.to(b_k.dtype), allow_tf32=False)
+        b_h *= exp(b_g_last) if USE_G else 1
+        b_h += b_hc
+
+    if STORE_FINAL_STATE:
+        p_ht = tl.make_block_ptr(ht + i_nh * K*V, (K, V), (V, 1), (i_k * BK, i_v * BV), (BK, BV), (1, 0))
+        tl.store(p_ht, b_h.to(p_ht.dtype.element_ty), boundary_check=(0, 1))
+
+
+@triton.heuristics({
+    'USE_G': lambda args: args['g'] is not None,
+    'USE_INITIAL_STATE': lambda args: args['dh0'] is not None,
+    'USE_FINAL_STATE_GRADIENT': lambda args: args['dht'] is not None,
+    'USE_OFFSETS': lambda args: args['offsets'] is not None,
+})
+@triton.autotune(
+    configs=[
+        triton.Config({}, num_warps=num_warps, num_stages=num_stages)
+        for num_warps in NUM_WARPS
+        for num_stages in [2, 3, 4]
+    ],
+    key=['BT', 'BK', 'BV', 'USE_G'],
+    use_cuda_graph=use_cuda_graph,
+)
+@triton.jit(do_not_specialize=['T'])
+def chunk_gated_delta_rule_bwd_kernel_dhu(
+    q,
+    k,
+    d,
+    g,
+    dht,
+    dh0,
+    do,
+    dh,
+    dv,
+    dv2,
+    offsets,
+    chunk_offsets,
+    scale,
+    T,
+    H: tl.constexpr,
+    K: tl.constexpr,
+    V: tl.constexpr,
+    BT: tl.constexpr,
+    BC: tl.constexpr,
+    BK: tl.constexpr,
+    BV: tl.constexpr,
+    USE_G: tl.constexpr,
+    USE_INITIAL_STATE: tl.constexpr,
+    USE_FINAL_STATE_GRADIENT: tl.constexpr,
+    USE_OFFSETS: tl.constexpr,
+    HEAD_FIRST: tl.constexpr
+):
+    i_k, i_v, i_nh = tl.program_id(0), tl.program_id(1), tl.program_id(2)
+    i_n, i_h = i_nh // H, i_nh % H
+    if USE_OFFSETS:
+        bos, eos = tl.load(offsets + i_n).to(tl.int32), tl.load(offsets + i_n + 1).to(tl.int32)
+        T = eos - bos
+        NT = tl.cdiv(T, BT)
+        boh = tl.load(chunk_offsets + i_n).to(tl.int32)
+    else:
+        bos, eos = i_n * T, i_n * T + T
+        NT = tl.cdiv(T, BT)
+        boh = i_n * NT
+
+    # [BK, BV]
+    b_dh = tl.zeros([BK, BV], dtype=tl.float32)
+    if USE_FINAL_STATE_GRADIENT:
+        p_dht = tl.make_block_ptr(dht + i_nh * K*V, (K, V), (V, 1), (i_k * BK, i_v * BV), (BK, BV), (1, 0))
+        b_dh += tl.load(p_dht, boundary_check=(0, 1))
+
+    for i_t in range(NT - 1, -1, -1):
+        if HEAD_FIRST:
+            p_dh = tl.make_block_ptr(dh + (i_nh * NT + i_t) * K*V, (K, V), (V, 1), (i_k * BK, i_v * BV), (BK, BV), (1, 0))
+        else:
+            p_dh = tl.make_block_ptr(dh + ((boh+i_t) * H + i_h) * K*V, (K, V), (V, 1), (i_k * BK, i_v * BV), (BK, BV), (1, 0))
+        tl.store(p_dh, b_dh.to(p_dh.dtype.element_ty), boundary_check=(0, 1))
+        b_dh_tmp = tl.zeros([BK, BV], dtype=tl.float32)
+        if USE_G:
+            last_idx = min((i_t + 1) * BT, T) - 1
+            if HEAD_FIRST:
+                bg_last = tl.load(g + i_nh * T + last_idx)
+            else:
+                bg_last = tl.load(g + (bos + last_idx) * H + i_h)
+        else:
+            bg_last = None
+            last_idx = None
+        for i_c in range(tl.cdiv(BT, BC) - 1, -1, -1):
+            if HEAD_FIRST:
+                p_q = tl.make_block_ptr(q + i_nh * T*K, (K, T), (1, K), (i_k * BK, i_t * BT + i_c * BC), (BK, BC), (0, 1))
+                p_k = tl.make_block_ptr(k + i_nh * T*K, (T, K), (K, 1), (i_t * BT + i_c * BC, i_k * BK), (BC, BK), (1, 0))
+                p_d = tl.make_block_ptr(d + i_nh * T*K, (K, T), (1, K), (i_k * BK, i_t * BT + i_c * BC), (BK, BC), (0, 1))
+                p_dv = tl.make_block_ptr(dv + i_nh * T*V, (T, V), (V, 1), (i_t * BT + i_c * BC, i_v * BV), (BC, BV), (1, 0))
+                p_do = tl.make_block_ptr(do + i_nh * T*V, (T, V), (V, 1), (i_t * BT + i_c * BC, i_v * BV), (BC, BV), (1, 0))
+                p_g = tl.make_block_ptr(g + i_nh * T, (T,), (1,), (i_t * BT + i_c * BC,), (BC,), (0,)) if USE_G else None
+                p_dv2 = tl.make_block_ptr(dv2 + i_nh * T*V, (T, V), (V, 1), (i_t * BT + i_c * BC, i_v * BV), (BC, BV), (1, 0))
+            else:
+                p_q = tl.make_block_ptr(q+(bos*H+i_h)*K, (K, T), (1, H*K), (i_k * BK, i_t * BT + i_c * BC), (BK, BC), (0, 1))
+                p_k = tl.make_block_ptr(k+(bos*H+i_h)*K, (T, K), (H*K, 1), (i_t * BT + i_c * BC, i_k * BK), (BC, BK), (1, 0))
+                p_d = tl.make_block_ptr(d+(bos*H+i_h)*K, (K, T), (1, H*K), (i_k * BK, i_t * BT + i_c * BC), (BK, BC), (0, 1))
+                p_dv = tl.make_block_ptr(dv+(bos*H+i_h)*V, (T, V), (H*V, 1), (i_t*BT + i_c * BC, i_v * BV), (BC, BV), (1, 0))
+                p_do = tl.make_block_ptr(do+(bos*H+i_h)*V, (T, V), (H*V, 1), (i_t*BT + i_c * BC, i_v * BV), (BC, BV), (1, 0))
+                p_g = tl.make_block_ptr(g+bos*H+i_h, (T,), (H,), (i_t*BT + i_c * BC,), (BC,), (0,)) if USE_G else None
+                p_dv2 = tl.make_block_ptr(dv2+(bos*H+i_h)*V, (T, V), (H*V, 1), (i_t*BT + i_c * BC, i_v * BV), (BC, BV), (1, 0))
+            b_g = tl.load(p_g, boundary_check=(0,)) if USE_G else None
+            # [BK, BT]
+            b_q = tl.load(p_q, boundary_check=(0, 1))
+            b_q = (b_q * scale * exp(b_g)[None, :]).to(b_q.dtype) if USE_G else (b_q * scale).to(b_q.dtype)
+            # [BT, BK]
+            b_k = tl.load(p_k, boundary_check=(0, 1))
+            b_d = tl.load(p_d, boundary_check=(0, 1))
+            b_k = (b_k * exp(bg_last - b_g)[:, None]).to(b_k.dtype) if USE_G else b_k
+            b_d = (b_d * exp(b_g)[None, :]).to(b_d.dtype) if USE_G else b_d
+            # [BT, V]
+            b_do = tl.load(p_do, boundary_check=(0, 1))
+            b_dv = tl.load(p_dv, boundary_check=(0, 1))
+            b_dv2 = b_dv + tl.dot(b_k, b_dh.to(b_k.dtype), allow_tf32=False)
+            tl.store(p_dv2, b_dv2.to(p_dv.dtype.element_ty), boundary_check=(0, 1))
+            # [BK, BV]
+            b_dh_tmp += tl.dot(b_q, b_do.to(b_q.dtype), allow_tf32=False)
+            b_dh_tmp -= tl.dot(b_d, b_dv2.to(b_q.dtype), allow_tf32=False)
+        b_dh *= exp(bg_last) if USE_G else 1
+        b_dh += b_dh_tmp
+
+    if USE_INITIAL_STATE:
+        p_dh0 = tl.make_block_ptr(dh0 + i_nh * K*V, (K, V), (V, 1), (i_k * BK, i_v * BV), (BK, BV), (1, 0))
+        tl.store(p_dh0, b_dh.to(p_dh0.dtype.element_ty), boundary_check=(0, 1))
+
+
+def chunk_gated_delta_rule_fwd_h(
+    k: torch.Tensor,
+    w: torch.Tensor,
+    u: torch.Tensor,
+    g: Optional[torch.Tensor] = None,
+    initial_state: Optional[torch.Tensor] = None,
+    output_final_state: bool = False,
+    offsets: Optional[torch.LongTensor] = None,
+    indices: Optional[torch.LongTensor] = None,
+    head_first: bool = True,
+    chunk_size: int = 64
+) -> Tuple[torch.Tensor, torch.Tensor]:
+    if head_first:
+        B, H, T, K, V = *k.shape, u.shape[-1]
+    else:
+        B, T, H, K, V = *k.shape, u.shape[-1]
+    BT = min(chunk_size, max(triton.next_power_of_2(T), 16))
+    # N: the actual number of sequences in the batch with either equal or variable lengths
+    if offsets is None:
+        N, NT, chunk_offsets = B, triton.cdiv(T, BT), None
+    else:
+        N, NT, chunk_offsets = len(offsets) - 1, len(indices), prepare_chunk_offsets(offsets, BT)
+    BK = triton.next_power_of_2(K)
+    assert BK <= 256, "current kernel does not support head dimension larger than 256."
+    # H100 can have larger block size
+    if check_shared_mem('hopper', k.device.index):
+        BV = 64
+        BC = 64 if K <= 128 else 32
+    # A100
+    elif check_shared_mem('ampere', k.device.index):
+        BV = 32
+        BC = 64
+    else:
+        BV = 32
+        BC = 32 if K <= 128 else 16
+    BC = min(BT, BC)
+    NK = triton.cdiv(K, BK)
+    NV = triton.cdiv(V, BV)
+    assert NK == 1, 'NK > 1 is not supported because it involves time-consuming synchronization'
+
+    if head_first:
+        h = k.new_empty(B, H, NT, K, V)
+    else:
+        h = k.new_empty(B, NT, H, K, V)
+    final_state = k.new_empty(N, H, K, V, dtype=torch.float32) if output_final_state else None
+
+    v_new = torch.empty_like(u)
+    grid = (NK, NV, N * H)
+
+    chunk_gated_delta_rule_fwd_kernel_h[grid](
+        k=k,
+        v=u,
+        d=w,
+        v_new=v_new,
+        g=g,
+        h=h,
+        h0=initial_state,
+        ht=final_state,
+        offsets=offsets,
+        chunk_offsets=chunk_offsets,
+        T=T,
+        H=H,
+        K=K,
+        V=V,
+        BT=BT,
+        BC=BC,
+        BK=BK,
+        BV=BV,
+        NT=NT,
+        HEAD_FIRST=head_first
+    )
+    return h, v_new, final_state
+
+
+def chunk_gated_delta_rule_bwd_dhu(
+    q: torch.Tensor,
+    k: torch.Tensor,
+    w: torch.Tensor,
+    g: torch.Tensor,
+    h0: torch.Tensor,
+    dht: Optional[torch.Tensor],
+    do: torch.Tensor,
+    dv: torch.Tensor,
+    scale: float,
+    offsets: Optional[torch.LongTensor] = None,
+    indices: Optional[torch.LongTensor] = None,
+    head_first: bool = True,
+    chunk_size: int = 64
+) -> Tuple[torch.Tensor, torch.Tensor, torch.Tensor]:
+    if head_first:
+        B, H, T, K, V = *q.shape, do.shape[-1]
+    else:
+        B, T, H, K, V = *q.shape, do.shape[-1]
+    BT = min(chunk_size, max(triton.next_power_of_2(T), 16))
+    # N: the actual number of sequences in the batch with either equal or variable lengths
+    if offsets is None:
+        N, NT, chunk_offsets = B, triton.cdiv(T, BT), None
+    else:
+        N, NT, chunk_offsets = len(offsets) - 1, len(indices), prepare_chunk_offsets(offsets, BT)
+
+    BK = triton.next_power_of_2(K)
+    assert BK <= 256, "current kernel does not support head dimension being larger than 256."
+
+    # H100
+    if check_shared_mem('hopper', q.device.index):
+        BV = 64
+        BC = 64 if K <= 128 else 32
+    # A100
+    elif check_shared_mem('ampere', q.device.index):
+        BV = 32
+        BC = 64 if K <= 128 else 32
+    else:
+        BV = 32 if K <= 128 else 16
+        BC = 16
+
+    BC = min(BT, BC)
+    NK, NV = triton.cdiv(K, BK), triton.cdiv(V, BV)
+    assert NK == 1, 'NK > 1 is not supported because it involves time-consuming synchronization'
+
+    if head_first:
+        dh = q.new_empty(B, H, NT, K, V)
+    else:
+        dh = q.new_empty(B, NT, H, K, V)
+    dh0 = torch.empty_like(h0, dtype=torch.float32) if h0 is not None else None
+    dv2 = torch.empty_like(dv)
+
+    grid = (NK, NV, N * H)
+    chunk_gated_delta_rule_bwd_kernel_dhu[grid](
+        q=q,
+        k=k,
+        d=w,
+        g=g,
+        dht=dht,
+        dh0=dh0,
+        do=do,
+        dh=dh,
+        dv=dv,
+        dv2=dv2,
+        offsets=offsets,
+        chunk_offsets=chunk_offsets,
+        scale=scale,
+        T=T,
+        H=H,
+        K=K,
+        V=V,
+        BT=BT,
+        BC=BC,
+        BK=BK,
+        BV=BV,
+        HEAD_FIRST=head_first
+    )
+    return dh, dh0, dv2

fla/ops/common/chunk_h_parallel.py

@@ -0,0 +1,650 @@
+# -*- coding: utf-8 -*-
+# Copyright (c) 2023-2025, Songlin Yang, Yu Zhang
+
+"""
+Fully parallelized state passing.
+"""
+
+from typing import Optional, Tuple
+
+import torch
+import triton
+import triton.language as tl
+
+from fla.ops.utils.op import exp
+
+
+@triton.heuristics({
+    'USE_INITIAL_STATE': lambda args: args['h0'] is not None,
+    'STORE_FINAL_STATE': lambda args: args['ht'] is not None,
+    'USE_OFFSETS': lambda args: args['offsets'] is not None
+})
+@triton.autotune(
+    configs=[
+        triton.Config({'BK': BK, 'BV': BV}, num_warps=num_warps, num_stages=num_stages)
+        for BK in [32, 64, 128]
+        for BV in [32, 64, 128]
+        for num_warps in [2, 4, 8]
+        for num_stages in [2, 3, 4]
+    ],
+    key=['BT', 'USE_G', 'USE_GK', 'USE_GV']
+)
+@triton.jit(do_not_specialize=['T'])
+def chunk_fwd_kernel_h_parallel(
+    k,
+    v,
+    h,
+    g,
+    gk,
+    gv,
+    h0,
+    ht,
+    offsets,
+    indices,
+    T,
+    H: tl.constexpr,
+    K: tl.constexpr,
+    V: tl.constexpr,
+    BT: tl.constexpr,
+    BK: tl.constexpr,
+    BV: tl.constexpr,
+    USE_G: tl.constexpr,
+    USE_GK: tl.constexpr,
+    USE_GV: tl.constexpr,
+    USE_INITIAL_STATE: tl.constexpr,
+    STORE_FINAL_STATE: tl.constexpr,
+    USE_OFFSETS: tl.constexpr,
+    HEAD_FIRST: tl.constexpr
+):
+    i_kv, i_t, i_bh = tl.program_id(0), tl.program_id(1), tl.program_id(2)
+
+    NV = tl.cdiv(V, BV)
+    # i_b: batch index
+    # i_h: head index
+    # i_n: sequence index
+    # i_t: chunk index within current sequence
+    # i_tg: (global) chunk index across all sequences
+    i_k, i_v = i_kv // NV, i_kv % NV
+    i_b, i_h = i_bh // H, i_bh % H
+    if USE_OFFSETS:
+        i_tg = i_t
+        i_n, i_t = tl.load(indices + i_t * 2).to(tl.int32), tl.load(indices + i_t * 2 + 1).to(tl.int32)
+        bos, eos = tl.load(offsets + i_n).to(tl.int32), tl.load(offsets + i_n + 1).to(tl.int32)
+        T = eos - bos
+        NT = tl.cdiv(T, BT)
+    else:
+        bos, eos = i_b * T, i_b * T + T
+        NT = tl.cdiv(T, BT)
+        i_n, i_tg = i_b, i_b * NT + i_t
+    i_nh = i_n * H + i_h
+
+    if HEAD_FIRST:
+        p_k = tl.make_block_ptr(k + i_bh * T*K, (K, T), (1, K), (i_k * BK, i_t * BT), (BK, BT), (0, 1))
+        p_v = tl.make_block_ptr(v + i_bh * T*V, (T, V), (V, 1), (i_t * BT, i_v * BV), (BT, BV), (1, 0))
+        p_h = tl.make_block_ptr(h + (i_bh * NT + i_t) * K*V, (K, V), (V, 1), (i_k * BK, i_v * BV), (BK, BV), (1, 0))
+    else:
+        p_k = tl.make_block_ptr(k + (bos*H + i_h) * K, (K, T), (1, H*K), (i_k * BK, i_t * BT), (BK, BT), (0, 1))
+        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_h = tl.make_block_ptr(h + (i_tg * H + i_h) * K*V, (K, V), (V, 1), (i_k * BK, i_v * BV), (BK, BV), (1, 0))
+
+    if i_t == 0:
+        if USE_INITIAL_STATE:
+            p_h0 = tl.make_block_ptr(h0 + i_nh * K*V, (K, V), (V, 1), (i_k * BK, i_v * BV), (BK, BV), (1, 0))
+            b_h = tl.load(p_h0, boundary_check=(0, 1)).to(tl.float32)
+        else:
+            b_h = tl.zeros([BK, BV], dtype=tl.float32)
+        tl.store(p_h, b_h.to(p_h.dtype.element_ty), boundary_check=(0, 1))
+
+    # [BK, BT]
+    b_k = tl.load(p_k, boundary_check=(0, 1))
+    # [BT, BV]
+    b_v = tl.load(p_v, boundary_check=(0, 1))
+
+    last_idx = min(i_t * BT + BT, T) - 1
+    # scalar decay
+    if USE_G:
+        if HEAD_FIRST:
+            b_g_last = tl.load(g + i_bh * T + last_idx)
+            p_g = g + i_bh * T + i_t * BT + tl.arange(0, BT)
+            p_g = tl.max_contiguous(tl.multiple_of(p_g, BT), BT)
+        else:
+            b_g_last = tl.load(g + bos * H + last_idx * H + i_h)
+            p_g = g + bos*H + (i_t * BT + tl.arange(0, BT)) * H + i_h
+        b_g = tl.load(p_g, mask=(i_t * BT + tl.arange(0, BT) < T), other=0.)
+        b_v = (b_v * exp(b_g_last - b_g)[:, None]).to(b_v.dtype)
+
+    # vector decay, h = Diag(gk) @ h
+    if USE_GK:
+        if HEAD_FIRST:
+            p_gk = tl.make_block_ptr(gk + i_bh * T*K, (K, T), (1, K), (i_k * BK, i_t * BT), (BK, BT), (0, 1))
+            p_gk_last = gk + i_bh * T*K + last_idx * K + i_k * BK + tl.arange(0, BK)
+            p_gk_last = tl.max_contiguous(tl.multiple_of(p_gk_last, BK), BK)
+        else:
+            p_gk = tl.make_block_ptr(gk + (bos*H + i_h) * K, (K, T), (1, H*K), (i_k * BK, i_t * BT), (BK, BT), (0, 1))
+            p_gk_last = gk + (bos + last_idx) * H*K + i_h * K + i_k * BK + tl.arange(0, BK)
+
+        b_gk_last = tl.load(p_gk_last, mask=(i_k * BK + tl.arange(0, BK) < K), other=0.)
+
+        b_gk = tl.load(p_gk, boundary_check=(0, 1))
+        b_k = (b_k * exp(b_gk_last[:, None] - b_gk)).to(b_k.dtype)
+
+    # vector decay, h = h @ Diag(gv)
+    if USE_GV:
+        if HEAD_FIRST:
+            p_gv = tl.make_block_ptr(gv + i_bh * T*V, (T, V), (V, 1), (i_t * BT, i_v * BV), (BT, BV), (1, 0))
+            p_gv_last = gv + i_bh * T*V + last_idx * V + i_v * BV + tl.arange(0, BV)
+            p_gv_last = tl.max_contiguous(tl.multiple_of(p_gv_last, BV), BV)
+        else:
+            p_gv = tl.make_block_ptr(gv + (bos*H + i_h) * V, (T, V), (H*V, 1), (i_t * BT, i_v * BV), (BT, BV), (1, 0))
+            p_gv_last = gv + (bos + last_idx) * H*V + i_h * V + i_v * BV + tl.arange(0, BV)
+
+        b_gv_last = tl.load(p_gv_last, mask=(i_v * BV + tl.arange(0, BV) < V), other=0.)
+
+        b_gv = tl.load(p_gv, boundary_check=(0, 1))
+        b_v = (b_v * exp(b_gv_last[None, :] - b_gv)).to(b_v.dtype)
+
+    b_h = tl.dot(b_k, b_v)
+    if i_t < NT - 1:
+        if HEAD_FIRST:
+            p_h = tl.make_block_ptr(h + (i_bh * NT + i_t + 1) * K*V, (K, V), (V, 1), (i_k * BK, i_v * BV), (BK, BV), (1, 0))
+        else:
+            p_h = tl.make_block_ptr(h + ((i_tg + 1) * H + i_h) * K*V, (K, V), (V, 1), (i_k * BK, i_v * BV), (BK, BV), (1, 0))
+        tl.store(p_h, b_h.to(p_h.dtype.element_ty), boundary_check=(0, 1))
+    elif STORE_FINAL_STATE:
+        p_ht = tl.make_block_ptr(ht + i_nh * K*V, (K, V), (V, 1), (i_k * BK, i_v * BV), (BK, BV), (1, 0))
+        tl.store(p_ht, b_h.to(p_ht.dtype.element_ty), boundary_check=(0, 1))
+
+
+@triton.heuristics({
+    'STORE_FINAL_STATE': lambda args: args['ht'] is not None,
+    'USE_OFFSETS': lambda args: args['offsets'] is not None
+})
+@triton.autotune(
+    configs=[
+        triton.Config({'BK': BK, 'BV': BV}, num_warps=num_warps, num_stages=num_stages)
+        for BK in [32, 64, 128]
+        for BV in [32, 64, 128]
+        for num_warps in [2, 4, 8, 16]
+        for num_stages in [2, 3]
+    ],
+    key=['BT', 'USE_G', 'USE_GK', 'USE_GV']
+)
+@triton.jit(do_not_specialize=['T'])
+def chunk_fwd_kernel_h_reduction(
+    h,
+    g,
+    gk,
+    gv,
+    kvt,
+    ht,
+    offsets,
+    chunk_offsets,
+    T,
+    H: tl.constexpr,
+    K: tl.constexpr,
+    V: tl.constexpr,
+    BT: tl.constexpr,
+    BK: tl.constexpr,
+    BV: tl.constexpr,
+    USE_G: tl.constexpr,
+    USE_GK: tl.constexpr,
+    USE_GV: tl.constexpr,
+    STORE_FINAL_STATE: tl.constexpr,
+    USE_OFFSETS: tl.constexpr,
+    HEAD_FIRST: tl.constexpr
+):
+    i_k, i_v, i_nh = tl.program_id(0), tl.program_id(1), tl.program_id(2)
+    i_n, i_h = i_nh // H, i_nh % H
+    if USE_OFFSETS:
+        bos, eos = tl.load(offsets + i_n).to(tl.int32), tl.load(offsets + i_n + 1).to(tl.int32)
+        T = eos - bos
+        NT = tl.cdiv(T, BT)
+        boh = tl.load(chunk_offsets + i_n).to(tl.int32)
+    else:
+        bos, eos = i_n * T, i_n * T + T
+        NT = tl.cdiv(T, BT)
+        boh = i_n * NT
+
+    # [BK, BV]
+    b_h = tl.zeros([BK, BV], dtype=tl.float32)
+    for i_t in range(NT):
+        if HEAD_FIRST:
+            p_h = tl.make_block_ptr(h + (i_nh * NT + i_t) * K*V, (K, V), (V, 1), (i_k * BK, i_v * BV), (BK, BV), (1, 0))
+        else:
+            p_h = tl.make_block_ptr(h + ((boh + i_t) * H + i_h) * K*V, (K, V), (V, 1), (i_k * BK, i_v * BV), (BK, BV), (1, 0))
+        b_h += tl.load(p_h, boundary_check=(0, 1)).to(tl.float32)
+        if i_t > 0:
+            tl.store(p_h, b_h.to(p_h.dtype.element_ty), boundary_check=(0, 1))
+
+        last_idx = min(i_t * BT + BT, T) - 1
+        # scalar decay
+        if USE_G:
+            if HEAD_FIRST:
+                b_g_last = tl.load(g + i_nh * T + last_idx)
+            else:
+                b_g_last = tl.load(g + bos * H + last_idx * H + i_h)
+            b_h *= exp(b_g_last)
+
+        # vector decay, h = Diag(gk) @ h
+        if USE_GK:
+            if HEAD_FIRST:
+                p_gk_last = gk + i_nh * T*K + last_idx * K + i_k * BK + tl.arange(0, BK)
+                p_gk_last = tl.max_contiguous(tl.multiple_of(p_gk_last, BK), BK)
+            else:
+                p_gk_last = gk + (bos + last_idx) * H*K + i_h * K + i_k * BK + tl.arange(0, BK)
+
+            b_gk_last = tl.load(p_gk_last, mask=(i_k * BK + tl.arange(0, BK) < K), other=0.)
+            b_h *= exp(b_gk_last)[:, None]
+
+        # vector decay, h = h @ Diag(gv)
+        if USE_GV:
+            if HEAD_FIRST:
+                p_gv_last = gv + i_nh * T*V + last_idx * V + i_v * BV + tl.arange(0, BV)
+                p_gv_last = tl.max_contiguous(tl.multiple_of(p_gv_last, BV), BV)
+            else:
+                p_gv_last = gv + (bos + last_idx) * H*V + i_h * V + i_v * BV + tl.arange(0, BV)
+
+            b_gv_last = tl.load(p_gv_last, mask=(i_v * BV + tl.arange(0, BV) < V), other=0.)
+            b_h *= exp(b_gv_last)[None, :]
+
+    if STORE_FINAL_STATE:
+        p_kvt = tl.make_block_ptr(kvt + i_nh * K*V, (K, V), (V, 1), (i_k * BK, i_v * BV), (BK, BV), (1, 0))
+        p_ht = tl.make_block_ptr(ht + i_nh * K*V, (K, V), (V, 1), (i_k * BK, i_v * BV), (BK, BV), (1, 0))
+        b_h += tl.load(p_kvt, boundary_check=(0, 1)).to(tl.float32)
+        tl.store(p_ht, b_h.to(p_ht.dtype.element_ty), boundary_check=(0, 1))
+
+
+@triton.heuristics({
+    'STORE_INITIAL_STATE_GRADIENT': lambda args: args['dh0'] is not None,
+    'USE_FINAL_STATE_GRADIENT': lambda args: args['dht'] is not None,
+    'USE_OFFSETS': lambda args: args['offsets'] is not None
+})
+@triton.autotune(
+    configs=[
+        triton.Config({'BK': BK, 'BV': BV}, num_warps=num_warps, num_stages=num_stages)
+        for BK in [32, 64, 128]
+        for BV in [32, 64, 128]
+        for num_warps in [2, 4, 8]
+        for num_stages in [2, 3, 4]
+    ],
+    key=['BT', 'USE_G', 'USE_GK', 'USE_GV']
+)
+@triton.jit(do_not_specialize=['T'])
+def chunk_bwd_kernel_dh_parallel(
+    q,
+    g,
+    gk,
+    gv,
+    do,
+    dh,
+    dht,
+    dh0,
+    offsets,
+    indices,
+    scale,
+    T,
+    HQ: tl.constexpr,
+    H: tl.constexpr,
+    K: tl.constexpr,
+    V: tl.constexpr,
+    BT: tl.constexpr,
+    BK: tl.constexpr,
+    BV: tl.constexpr,
+    NG: tl.constexpr,
+    USE_G: tl.constexpr,
+    USE_GK: tl.constexpr,
+    USE_GV: tl.constexpr,
+    STORE_INITIAL_STATE_GRADIENT: tl.constexpr,
+    USE_FINAL_STATE_GRADIENT: tl.constexpr,
+    USE_OFFSETS: tl.constexpr,
+    HEAD_FIRST: tl.constexpr
+):
+    i_kv, i_t, i_bh = tl.program_id(0), tl.program_id(1), tl.program_id(2)
+
+    NV = tl.cdiv(V, BV)
+    i_k, i_v = i_kv // NV, i_kv % NV
+    i_b, i_hq, i_bg = i_bh // HQ, i_bh % HQ, i_bh // NG
+    i_h = i_hq // NG
+    if USE_OFFSETS:
+        i_tg = i_t
+        i_n, i_t = tl.load(indices + i_t * 2).to(tl.int32), tl.load(indices + i_t * 2 + 1).to(tl.int32)
+        bos, eos = tl.load(offsets + i_n).to(tl.int32), tl.load(offsets + i_n + 1).to(tl.int32)
+        T = eos - bos
+        NT = tl.cdiv(T, BT)
+    else:
+        bos, eos = i_b * T, i_b * T + T
+        NT = tl.cdiv(T, BT)
+        i_n, i_tg = i_b, i_b * NT + i_t
+    i_nh = i_n * HQ + i_hq
+
+    if HEAD_FIRST:
+        p_q = tl.make_block_ptr(q + i_bh * T*K, (K, T), (1, K), (i_k * BK, i_t * BT), (BK, BT), (0, 1))
+        p_do = tl.make_block_ptr(do + i_bh * T*V, (T, V), (V, 1), (i_t * BT, i_v * BV), (BT, BV), (1, 0))
+        p_dh = tl.make_block_ptr(dh + (i_bh * NT + i_t) * K*V, (K, V), (V, 1), (i_k * BK, i_v * BV), (BK, BV), (1, 0))
+    else:
+        p_q = tl.make_block_ptr(q + (bos*HQ + i_hq) * K, (K, T), (1, HQ*K), (i_k * BK, i_t * BT), (BK, BT), (0, 1))
+        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_dh = tl.make_block_ptr(dh + (i_tg * H + i_h) * K*V, (K, V), (V, 1), (i_k * BK, i_v * BV), (BK, BV), (1, 0))
+
+    if i_t == NT - 1:
+        if USE_FINAL_STATE_GRADIENT:
+            p_dht = tl.make_block_ptr(dht + i_nh * K*V, (K, V), (V, 1), (i_k * BK, i_v * BV), (BK, BV), (1, 0))
+            b_dh = tl.load(p_dht, boundary_check=(0, 1)).to(tl.float32)
+        else:
+            b_dh = tl.zeros([BK, BV], dtype=tl.float32)
+        tl.store(p_dh, b_dh.to(p_dh.dtype.element_ty), boundary_check=(0, 1))
+
+    # [BK, BT]
+    b_q = tl.load(p_q, boundary_check=(0, 1))
+    b_q = (b_q * scale).to(b_q.dtype)
+    # [BT, BV]
+    b_do = tl.load(p_do, boundary_check=(0, 1))
+
+    if USE_G:
+        if HEAD_FIRST:
+            p_g = g + i_bg * T + i_t * BT + tl.arange(0, BT)
+            p_g = tl.max_contiguous(tl.multiple_of(p_g, BT), BT)
+        else:
+            p_g = g + (bos + i_t * BT + tl.arange(0, BT)) * H + i_h
+        b_g = tl.load(p_g, mask=(i_t * BT + tl.arange(0, BT) < T), other=0.)
+        b_q = (b_q * exp(b_g)[None, :]).to(b_q.dtype)
+
+    if USE_GK:
+        if HEAD_FIRST:
+            p_gk = tl.make_block_ptr(gk + i_bg * T*K, (K, T), (1, K), (i_k * BK, i_t * BT), (BK, BT), (0, 1))
+        else:
+            p_gk = tl.make_block_ptr(gk + (bos*H + i_h) * K, (K, T), (1, H*K), (i_k * BK, i_t * BT), (BK, BT), (0, 1))
+        b_gk = tl.load(p_gk, boundary_check=(0, 1))
+        b_q = (b_q * exp(b_gk)).to(b_q.dtype)
+
+    if USE_GV:
+        if HEAD_FIRST:
+            p_gv = tl.make_block_ptr(gv + i_bg * T*V, (T, V), (V, 1), (i_t * BT, i_v * BV), (BT, BV), (1, 0))
+        else:
+            p_gv = tl.make_block_ptr(gv + (bos*H + i_h) * V, (T, V), (H*V, 1), (i_t * BT, i_v * BV), (BT, BV), (1, 0))
+        b_gv = tl.load(p_gv, boundary_check=(0, 1))
+        b_do = (b_do * exp(b_gv)).to(b_do.dtype)
+
+    b_dh = tl.dot(b_q, b_do)
+    if i_t > 0:
+        if HEAD_FIRST:
+            p_dh = tl.make_block_ptr(dh + (i_bh * NT + i_t - 1) * K*V, (K, V), (V, 1), (i_k * BK, i_v * BV), (BK, BV), (1, 0))
+        else:
+            p_dh = tl.make_block_ptr(dh + ((i_tg - 1) * H + i_h) * K*V, (K, V), (V, 1), (i_k * BK, i_v * BV), (BK, BV), (1, 0))
+        tl.store(p_dh, b_dh.to(p_dh.dtype.element_ty), boundary_check=(0, 1))
+    elif STORE_INITIAL_STATE_GRADIENT:
+        p_dh0 = tl.make_block_ptr(dh0 + i_nh * K*V, (K, V), (V, 1), (i_k * BK, i_v * BV), (BK, BV), (1, 0))
+        tl.store(p_dh0, b_dh.to(p_dh0.dtype.element_ty), boundary_check=(0, 1))
+
+
+@triton.heuristics({
+    'STORE_INITIAL_STATE_GRADIENT': lambda args: args['dh0'] is not None,
+    'USE_OFFSETS': lambda args: args['offsets'] is not None
+})
+@triton.autotune(
+    configs=[
+        triton.Config({'BK': BK, 'BV': BV}, num_warps=num_warps, num_stages=num_stages)
+        for BK in [32, 64, 128]
+        for BV in [32, 64, 128]
+        for num_warps in [2, 4, 8, 16]
+        for num_stages in [2, 3]
+    ],
+    key=['BT', 'USE_G', 'USE_GK', 'USE_GV']
+)
+@triton.jit(do_not_specialize=['T'])
+def chunk_bwd_kernel_dh_reduction(
+    g,
+    gk,
+    gv,
+    dh,
+    doq0,
+    dh0,
+    offsets,
+    chunk_offsets,
+    T,
+    HQ: tl.constexpr,
+    H: tl.constexpr,
+    K: tl.constexpr,
+    V: tl.constexpr,
+    BT: tl.constexpr,
+    BK: tl.constexpr,
+    BV: tl.constexpr,
+    NG: tl.constexpr,
+    USE_G: tl.constexpr,
+    USE_GK: tl.constexpr,
+    USE_GV: tl.constexpr,
+    STORE_INITIAL_STATE_GRADIENT: tl.constexpr,
+    USE_OFFSETS: tl.constexpr,
+    HEAD_FIRST: tl.constexpr
+):
+    i_k, i_v, i_nh = tl.program_id(0), tl.program_id(1), tl.program_id(2)
+    i_bg = i_nh // NG
+    i_n, i_hq = i_nh // HQ, i_nh % HQ
+    i_h = i_hq // NG
+    if USE_OFFSETS:
+        bos, eos = tl.load(offsets + i_n).to(tl.int32), tl.load(offsets + i_n + 1).to(tl.int32)
+        T = eos - bos
+        NT = tl.cdiv(T, BT)
+        boh = tl.load(chunk_offsets + i_n).to(tl.int32)
+    else:
+        bos, eos = i_n * T, i_n * T + T
+        NT = tl.cdiv(T, BT)
+        boh = i_n * NT
+
+    # [BK, BV]
+    b_dh = tl.zeros([BK, BV], dtype=tl.float32)
+    for i_t in range(NT - 1, -1, -1):
+        if HEAD_FIRST:
+            p_dh = tl.make_block_ptr(dh + (i_nh * NT + i_t) * K*V, (K, V), (V, 1), (i_k * BK, i_v * BV), (BK, BV), (1, 0))
+        else:
+            p_dh = tl.make_block_ptr(dh + ((boh+i_t) * H + i_h) * K*V, (K, V), (V, 1), (i_k * BK, i_v * BV), (BK, BV), (1, 0))
+        b_dh += tl.load(p_dh, boundary_check=(0, 1)).to(tl.float32)
+        if i_t < NT - 1:
+            tl.store(p_dh, b_dh.to(p_dh.dtype.element_ty), boundary_check=(0, 1))
+
+        last_idx = min(i_t * BT + BT, T) - 1
+        if USE_G:
+            if HEAD_FIRST:
+                b_g_last = tl.load(g + i_bg * T + last_idx)
+            else:
+                b_g_last = tl.load(g + (bos + last_idx) * H + i_h)
+            b_dh *= exp(b_g_last)
+
+        if USE_GK:
+            if HEAD_FIRST:
+                p_gk_last = gk + (i_bg * T + last_idx) * K + i_k * BK + tl.arange(0, BK)
+                p_gk_last = tl.max_contiguous(tl.multiple_of(p_gk_last, BK), BK)
+            else:
+                p_gk_last = gk + (bos + last_idx) * H*K + i_h * K + i_k * BK + tl.arange(0, BK)
+
+            b_gk_last = tl.load(p_gk_last, mask=(i_k * BK + tl.arange(0, BK) < K), other=0.)
+            b_dh *= exp(b_gk_last)[:, None]
+
+        if USE_GV:
+            if HEAD_FIRST:
+                p_gv_last = gv + (i_bg * T + last_idx) * V + i_v * BV + tl.arange(0, BV)
+                p_gv_last = tl.max_contiguous(tl.multiple_of(p_gv_last, BV), BV)
+            else:
+                p_gv_last = gv + (bos + last_idx) * H*V + i_h * V + i_v * BV + tl.arange(0, BV)
+
+            b_gv_last = tl.load(p_gv_last, mask=(i_v * BV + tl.arange(0, BV) < V), other=0.)
+            b_dh *= exp(b_gv_last)[None, :]
+
+    if STORE_INITIAL_STATE_GRADIENT:
+        p_doq0 = tl.make_block_ptr(doq0 + i_nh * K*V, (K, V), (V, 1), (i_k * BK, i_v * BV), (BK, BV), (1, 0))
+        p_dh0 = tl.make_block_ptr(dh0 + i_nh * K*V, (K, V), (V, 1), (i_k * BK, i_v * BV), (BK, BV), (1, 0))
+        b_dh += tl.load(p_doq0, boundary_check=(0, 1)).to(tl.float32)
+        tl.store(p_dh0, b_dh.to(p_dh0.dtype.element_ty), boundary_check=(0, 1))
+
+
+def chunk_fwd_h(
+    k: torch.Tensor,
+    v: torch.Tensor,
+    g: torch.Tensor,
+    gk: torch.Tensor,
+    gv: torch.Tensor,
+    h0: torch.Tensor,
+    output_final_state: bool,
+    states_in_fp32: bool = False,
+    offsets: Optional[torch.Tensor] = None,
+    indices: Optional[torch.Tensor] = None,
+    head_first: bool = True,
+    chunk_size: int = 64
+) -> Tuple[torch.Tensor, torch.Tensor]:
+    if head_first:
+        B, H, T, K, V = *k.shape, v.shape[-1]
+    else:
+        B, T, H, K, V = *k.shape, v.shape[-1]
+    BT = min(chunk_size, max(16, triton.next_power_of_2(T)))
+    # N: the actual number of sequences in the batch with either equal or variable lengths
+    if offsets is None:
+        N, NT, chunk_offsets = B, triton.cdiv(T, BT), None
+    else:
+        if indices is None:
+            indices = torch.cat([torch.arange(n) for n in triton.cdiv(offsets[1:] - offsets[:-1], BT).tolist()])
+            indices = torch.stack([indices.eq(0).cumsum(0) - 1, indices], 1).to(offsets)
+        N, NT = len(offsets) - 1, len(indices)
+        chunk_offsets = torch.cat([offsets.new_tensor([0]), triton.cdiv(offsets[1:] - offsets[:-1], BT)]).cumsum(-1)
+
+    h = k.new_empty(B, H, NT, K, V, dtype=torch.float) if head_first else k.new_empty(B, NT, H, K, V, dtype=torch.float)
+    ht = k.new_empty(N, H, K, V, dtype=torch.float) if output_final_state else None
+    def grid(meta): return (triton.cdiv(K, meta['BK']) * triton.cdiv(V, meta['BV']), NT, B * H)
+    chunk_fwd_kernel_h_parallel[grid](
+        k=k,
+        v=v,
+        h=h,
+        g=g,
+        gk=gk,
+        gv=gv,
+        h0=h0,
+        ht=ht,
+        offsets=offsets,
+        indices=indices,
+        T=T,
+        H=H,
+        K=K,
+        V=V,
+        BT=BT,
+        USE_G=g is not None,
+        USE_GK=gk is not None,
+        USE_GV=gv is not None,
+        HEAD_FIRST=head_first
+    )
+    kvt, ht = ht, (torch.empty_like(ht) if output_final_state else None)
+    def grid(meta): return (triton.cdiv(K, meta['BK']), triton.cdiv(V, meta['BV']), N * H)
+    chunk_fwd_kernel_h_reduction[grid](
+        h=h,
+        g=g,
+        gk=gk,
+        gv=gv,
+        kvt=kvt,
+        ht=ht,
+        offsets=offsets,
+        chunk_offsets=chunk_offsets,
+        T=T,
+        H=H,
+        K=K,
+        V=V,
+        BT=BT,
+        USE_G=g is not None,
+        USE_GK=gk is not None,
+        USE_GV=gv is not None,
+        HEAD_FIRST=head_first
+    )
+    h = h.to(k.dtype) if not states_in_fp32 else h
+    return h, ht
+
+
+def chunk_bwd_dh(
+    q: torch.Tensor,
+    k: torch.Tensor,
+    v: torch.Tensor,
+    g: torch.Tensor,
+    gk: torch.Tensor,
+    gv: torch.Tensor,
+    do: torch.Tensor,
+    h0: torch.Tensor,
+    dht: torch.Tensor,
+    scale: float,
+    states_in_fp32: bool = False,
+    offsets: Optional[torch.Tensor] = None,
+    indices: Optional[torch.Tensor] = None,
+    head_first: bool = True,
+    chunk_size: int = 64
+) -> Tuple[torch.Tensor, torch.Tensor]:
+    if head_first:
+        B, H, T, K, V = *k.shape, v.shape[-1]
+        HQ = q.shape[1]
+    else:
+        B, T, H, K, V = *k.shape, v.shape[-1]
+        HQ = q.shape[2]
+    BT = min(chunk_size, max(16, triton.next_power_of_2(T)))
+    # N: the actual number of sequences in the batch with either equal or variable lengths
+    # NG: number of groups in GQA
+    if offsets is None:
+        N, NT, chunk_offsets = B, triton.cdiv(T, BT), None
+    else:
+        if indices is None:
+            indices = torch.cat([torch.arange(n) for n in triton.cdiv(offsets[1:] - offsets[:-1], BT).tolist()])
+            indices = torch.stack([indices.eq(0).cumsum(0) - 1, indices], 1).to(offsets)
+        N, NT = len(offsets) - 1, len(indices)
+        chunk_offsets = torch.cat([offsets.new_tensor([0]), triton.cdiv(offsets[1:] - offsets[:-1], BT)]).cumsum(-1)
+    NG = HQ // H
+
+    if head_first:
+        dh = k.new_empty(B, HQ, NT, K, V, dtype=k.dtype if not states_in_fp32 else torch.float)
+    else:
+        dh = k.new_empty(B, NT, HQ, K, V, dtype=k.dtype if not states_in_fp32 else torch.float)
+    dh0 = torch.empty_like(h0, dtype=torch.float) if h0 is not None else None
+
+    def grid(meta): return (triton.cdiv(K, meta['BK']) * triton.cdiv(V, meta['BV']), NT, B * HQ)
+    chunk_bwd_kernel_dh_parallel[grid](
+        q=q,
+        g=g,
+        gk=gk,
+        gv=gv,
+        do=do,
+        dh=dh,
+        dht=dht,
+        dh0=dh0,
+        offsets=offsets,
+        indices=indices,
+        scale=scale,
+        T=T,
+        HQ=HQ,
+        H=H,
+        K=K,
+        V=V,
+        BT=BT,
+        NG=NG,
+        USE_G=g is not None,
+        USE_GK=gk is not None,
+        USE_GV=gv is not None,
+        HEAD_FIRST=head_first
+    )
+
+    doq0, dh0 = dh0, (torch.empty_like(dh0) if dh0 is not None else None)
+    def grid(meta): return (triton.cdiv(K, meta['BK']), triton.cdiv(V, meta['BV']), N * HQ)
+    chunk_bwd_kernel_dh_reduction[grid](
+        g=g,
+        gk=gk,
+        gv=gv,
+        dh=dh,
+        doq0=doq0,
+        dh0=dh0,
+        offsets=offsets,
+        chunk_offsets=chunk_offsets,
+        T=T,
+        HQ=HQ,
+        H=H,
+        K=K,
+        V=V,
+        BT=BT,
+        NG=NG,
+        USE_G=g is not None,
+        USE_GK=gk is not None,
+        USE_GV=gv is not None,
+        HEAD_FIRST=head_first
+    )
+    dh = dh.to(q.dtype) if not states_in_fp32 else dh
+    return dh, dh0

fla/ops/common/fused_recurrent.py

@@ -0,0 +1,575 @@
+# -*- coding: utf-8 -*-
+# Copyright (c) 2023-2025, Songlin Yang, Yu Zhang
+
+from typing import Optional
+
+import torch
+import triton
+import triton.language as tl
+
+from fla.ops.utils import chunk_global_cumsum
+from fla.ops.utils.op import exp
+from fla.utils import autocast_custom_bwd, autocast_custom_fwd, input_guard
+
+
+@triton.heuristics({
+    'USE_INITIAL_STATE': lambda args: args['h0'] is not None,
+    'STORE_FINAL_STATE': lambda args: args['ht'] is not None,
+    'USE_OFFSETS': lambda args: args['offsets'] is not None
+})
+@triton.autotune(
+    configs=[
+        triton.Config({}, num_warps=num_warps)
+        for num_warps in [1, 2, 4]
+    ],
+    key=["BK", "BV", "USE_GK", "USE_GV", "USE_G"],
+)
+@triton.jit(do_not_specialize=['T'])
+def fused_recurrent_fwd_kernel(
+    q,
+    k,
+    v,
+    g,
+    gk,
+    gv,
+    o,
+    h0,
+    ht,
+    offsets,
+    scale,
+    T,
+    B: tl.constexpr,
+    H: tl.constexpr,
+    K: tl.constexpr,
+    V: tl.constexpr,
+    BK: tl.constexpr,
+    BV: tl.constexpr,
+    REVERSE: tl.constexpr,
+    USE_G: tl.constexpr,
+    USE_GK: tl.constexpr,
+    USE_GV: tl.constexpr,
+    USE_INITIAL_STATE: tl.constexpr,
+    STORE_FINAL_STATE: tl.constexpr,
+    USE_OFFSETS: tl.constexpr,
+    HEAD_FIRST: tl.constexpr
+):
+    # indices
+    i_v, i_k, i_nh = tl.program_id(0).to(tl.int64), tl.program_id(1).to(tl.int64), tl.program_id(2).to(tl.int64)
+    i_n, i_h = i_nh // H, i_nh % H
+    if USE_OFFSETS:
+        bos, eos = tl.load(offsets + i_n).to(tl.int64), tl.load(offsets + i_n + 1).to(tl.int64)
+        all = T
+        T = eos - bos
+    else:
+        bos, eos = i_n * T, i_n * T + T
+        all = B * T
+
+    if HEAD_FIRST:
+        p_q = q + i_nh * T*K + ((T-1) * K if REVERSE else 0) + i_k * BK + tl.arange(0, BK)
+        p_k = k + i_nh * T*K + ((T-1) * K if REVERSE else 0) + i_k * BK + tl.arange(0, BK)
+        p_v = v + i_nh * T*V + ((T-1) * V if REVERSE else 0) + i_v * BV + tl.arange(0, BV)
+        p_o = o + (i_k * B*H + i_nh) * T*V + ((T-1) * V if REVERSE else 0) + i_v * BV + tl.arange(0, BV)
+        if USE_G:
+            p_g = g + i_nh * T + ((T-1) if REVERSE else 0)
+        if USE_GK:
+            p_gk = gk + i_nh * T*K + ((T-1) * K if REVERSE else 0) + i_k * BK + tl.arange(0, BK)
+        if USE_GV:
+            p_gv = gv + i_nh * T*V + ((T-1) * V if REVERSE else 0) + i_v * BV + tl.arange(0, BV)
+    else:
+        p_q = q + (bos + ((T-1) if REVERSE else 0)) * H*K + i_h * K + i_k * BK + tl.arange(0, BK)
+        p_k = k + (bos + ((T-1) if REVERSE else 0)) * H*K + i_h * K + i_k * BK + tl.arange(0, BK)
+        p_v = v + (bos + ((T-1) if REVERSE else 0)) * H*V + i_h * V + i_v * BV + tl.arange(0, BV)
+        p_o = o + ((i_k * all + bos) + ((T-1) if REVERSE else 0)) * H*V + i_h * V + i_v * BV + tl.arange(0, BV)
+        if USE_G:
+            p_g = g + (bos + ((T-1) if REVERSE else 0)) * H + i_h
+        if USE_GK:
+            p_gk = gk + (bos + ((T-1) if REVERSE else 0)) * H*K + i_h * K + i_k * BK + tl.arange(0, BK)
+        if USE_GV:
+            p_gv = gv + (bos + ((T-1) if REVERSE else 0)) * H*V + i_h * V + i_v * BV + tl.arange(0, BV)
+
+    mask_k = (i_k * BK + tl.arange(0, BK)) < K
+    mask_v = (i_v * BV + tl.arange(0, BV)) < V
+    mask_h = mask_k[None, :] & mask_v[:, None]
+    b_h = tl.zeros([BV, BK], dtype=tl.float32)
+
+    if USE_INITIAL_STATE:
+        p_h0 = h0 + i_nh * K*V + (i_k * BK + tl.arange(0, BK)[None, :]) * V + (i_v * BV + tl.arange(0, BV)[:, None])
+        b_h += tl.load(p_h0, mask=mask_h, other=0).to(tl.float32)
+
+    for _ in range(0, T):
+        b_q = tl.load(p_q, mask=mask_k, other=0).to(tl.float32) * scale
+        b_k = tl.load(p_k, mask=mask_k, other=0).to(tl.float32)
+        b_v = tl.load(p_v, mask=mask_v, other=0).to(tl.float32)
+        if USE_GK:
+            b_gk = tl.load(p_gk, mask=mask_k, other=0).to(tl.float32)
+            b_h = b_h * exp(b_gk[None, :])
+        if USE_GV:
+            b_gv = tl.load(p_gv, mask=mask_v, other=0).to(tl.float32)
+            b_h = b_h * exp(b_gv[:, None])
+        if USE_G:
+            b_g = tl.load(p_g).to(tl.float32)
+            b_h = b_h * exp(b_g)
+        b_h += b_k[None, :] * b_v[:, None]
+        b_o = b_h * b_q[None, :]
+        b_o = tl.sum(b_o, axis=1)
+        tl.store(p_o, b_o.to(p_o.dtype.element_ty), mask=mask_v)
+        p_q += (-1 if REVERSE else 1) * (1 if HEAD_FIRST else H) * K
+        p_k += (-1 if REVERSE else 1) * (1 if HEAD_FIRST else H) * K
+        p_v += (-1 if REVERSE else 1) * (1 if HEAD_FIRST else H) * V
+        p_o += (-1 if REVERSE else 1) * (1 if HEAD_FIRST else H) * V
+        if USE_GK:
+            p_gk += (-1 if REVERSE else 1) * (1 if HEAD_FIRST else H) * K
+        if USE_GV:
+            p_gv += (-1 if REVERSE else 1) * (1 if HEAD_FIRST else H) * V
+        if USE_G:
+            p_g += (-1 if REVERSE else 1) * (1 if HEAD_FIRST else H)
+
+    if STORE_FINAL_STATE:
+        p_ht = ht + i_nh * K*V + (i_k * BK + tl.arange(0, BK)[None, :]) * V + (i_v * BV + tl.arange(0, BV)[:, None])
+        tl.store(p_ht, b_h.to(p_ht.dtype.element_ty), mask=mask_h)
+
+
+@triton.heuristics({
+    'USE_INITIAL_STATE': lambda args: args['h0'] is not None,
+    'STORE_INITIAL_STATE_GRADIENT': lambda args: args['dh0'] is not None,
+    'USE_FINAL_STATE_GRADIENT': lambda args: args['dht'] is not None,
+    'USE_OFFSETS': lambda args: args['offsets'] is not None
+})
+@triton.autotune(
+    configs=[
+        triton.Config({}, num_warps=num_warps)
+        for num_warps in [1, 2, 4]
+    ],
+    key=['BK', 'BV', 'USE_GK', 'USE_GV', 'USE_G'],
+)
+@triton.jit(do_not_specialize=['T'])
+def fused_recurrent_bwd_kernel(
+    q,
+    k,
+    v,
+    g,
+    gk,
+    gv,
+    h0,
+    do,
+    dq,
+    dk,
+    dv,
+    dht,
+    dh0,
+    offsets,
+    scale,
+    T,
+    B: tl.constexpr,
+    H: tl.constexpr,
+    K: tl.constexpr,
+    V: tl.constexpr,
+    BK: tl.constexpr,
+    BV: tl.constexpr,
+    REVERSE: tl.constexpr,
+    USE_G: tl.constexpr,
+    USE_GK: tl.constexpr,
+    USE_GV: tl.constexpr,
+    USE_INITIAL_STATE: tl.constexpr,
+    STORE_INITIAL_STATE_GRADIENT: tl.constexpr,
+    USE_FINAL_STATE_GRADIENT: tl.constexpr,
+    USE_OFFSETS: tl.constexpr,
+    HEAD_FIRST: tl.constexpr
+):
+    i_v, i_k, i_nh = tl.program_id(0).to(tl.int64), tl.program_id(1).to(tl.int64), tl.program_id(2).to(tl.int64)
+    i_n, i_h = i_nh // H, i_nh % H
+    if USE_OFFSETS:
+        bos, eos = tl.load(offsets + i_n).to(tl.int64), tl.load(offsets + i_n + 1).to(tl.int64)
+        all = T
+        T = eos - bos
+    else:
+        bos, eos = i_n * T, i_n * T + T
+        all = B * T
+
+    if HEAD_FIRST:
+        p_k = k + i_nh * T*K + ((T-1) * K if REVERSE else 0) + i_k * BK + tl.arange(0, BK)
+        p_v = v + i_nh * T*V + ((T-1) * V if REVERSE else 0) + i_v * BV + tl.arange(0, BV)
+        p_do = do + i_nh * T*V + ((T-1) * V if REVERSE else 0) + i_v * BV + tl.arange(0, BV)
+        p_dq = dq + (i_v * B*H + i_nh) * T*K + ((T-1) * K if REVERSE else 0) + i_k * BK + tl.arange(0, BK)
+        if USE_G:
+            p_g = g + i_nh * T + ((T-1) if REVERSE else 0)
+        if USE_GK:
+            p_gk = gk + i_nh * T*K + ((T-1) * K if REVERSE else 0) + i_k * BK + tl.arange(0, BK)
+        if USE_GV:
+            p_gv = gv + i_nh * T*V + ((T-1) * V if REVERSE else 0) + i_v * BV + tl.arange(0, BV)
+    else:
+        p_k = k + (bos + ((T-1) if REVERSE else 0)) * H*K + i_h * K + i_k * BK + tl.arange(0, BK)
+        p_v = v + (bos + ((T-1) if REVERSE else 0)) * H*V + i_h * V + i_v * BV + tl.arange(0, BV)
+        p_do = do + (bos + ((T-1) if REVERSE else 0)) * H*V + i_h * V + i_v * BV + tl.arange(0, BV)
+        p_dq = dq + ((i_v * all + bos) + ((T-1) if REVERSE else 0)) * H*K + i_h * K + i_k * BK + tl.arange(0, BK)
+        if USE_G:
+            p_g = g + (bos + ((T-1) if REVERSE else 0)) * H + i_h
+        if USE_GK:
+            p_gk = gk + (bos + ((T-1) if REVERSE else 0)) * H*K + i_h * K + i_k * BK + tl.arange(0, BK)
+        if USE_GV:
+            p_gv = gv + (bos + ((T-1) if REVERSE else 0)) * H*V + i_h * V + i_v * BV + tl.arange(0, BV)
+
+    mask_k = i_k * BK + tl.arange(0, BK) < K
+    mask_v = i_v * BV + tl.arange(0, BV) < V
+    mask_h = mask_k[:, None] & mask_v[None, :]
+
+    b_h = tl.zeros([BK, BV], dtype=tl.float32)
+    if USE_INITIAL_STATE:
+        p_h0 = h0 + i_nh * K*V + (i_k * BK + tl.arange(0, BK)[:, None]) * V + (i_v * BV + tl.arange(0, BV)[None, :])
+        b_h += tl.load(p_h0, mask=mask_h, other=0).to(tl.float32)
+
+    for _ in range(0, T):
+        b_k = tl.load(p_k, mask=mask_k, other=0).to(tl.float32)
+        b_v = tl.load(p_v, mask=mask_v, other=0).to(tl.float32)
+        b_do = tl.load(p_do, mask=mask_v, other=0).to(tl.float32)
+        if USE_G:
+            b_g = tl.load(p_g).to(tl.float32)
+            b_h = b_h * exp(b_g)
+        if USE_GK:
+            b_gk = tl.load(p_gk, mask=mask_k, other=0).to(tl.float32)
+            b_h = b_h * exp(b_gk[:, None])
+        if USE_GV:
+            b_gv = tl.load(p_gv, mask=mask_v, other=0).to(tl.float32)
+            b_h = b_h * exp(b_gv[None, :])
+        b_h += b_k[:, None] * b_v[None, :]
+        b_dq = b_h * b_do[None, :]
+        b_dq = tl.sum(b_dq, axis=1) * scale
+        tl.store(p_dq, b_dq.to(p_dq.dtype.element_ty), mask=mask_k)
+
+        p_k += (-1 if REVERSE else 1) * (1 if HEAD_FIRST else H) * K
+        p_v += (-1 if REVERSE else 1) * (1 if HEAD_FIRST else H) * V
+        p_do += (-1 if REVERSE else 1) * (1 if HEAD_FIRST else H) * V
+        p_dq += (-1 if REVERSE else 1) * (1 if HEAD_FIRST else H) * K
+        if USE_G:
+            p_g += (-1 if REVERSE else 1) * (1 if HEAD_FIRST else H)
+        if USE_GK:
+            p_gk += (-1 if REVERSE else 1) * (1 if HEAD_FIRST else H) * K
+        if USE_GV:
+            p_gv += (-1 if REVERSE else 1) * (1 if HEAD_FIRST else H) * V
+
+    # sync threads
+    tl.debug_barrier()
+
+    if HEAD_FIRST:
+        p_q = q + i_nh * T*K + ((T - 1) * K if not REVERSE else 0) + i_k * BK + tl.arange(0, BK)
+        p_k = k + i_nh * T*K + ((T - 1) * K if not REVERSE else 0) + i_k * BK + tl.arange(0, BK)
+        p_v = v + i_nh * T*V + ((T - 1) * V if not REVERSE else 0) + i_v * BV + tl.arange(0, BV)
+        p_do = do + i_nh * T*V + ((T - 1) * V if not REVERSE else 0) + i_v * BV + tl.arange(0, BV)
+        p_dk = dk + (i_v * B*H + i_nh) * T*K + ((T - 1) * K if not REVERSE else 0) + i_k * BK + tl.arange(0, BK)
+        p_dv = dv + (i_k * B*H + i_nh) * T*V + ((T - 1) * V if not REVERSE else 0) + i_v * BV + tl.arange(0, BV)
+        if USE_G:
+            p_g = g + i_nh * T + ((T - 1) if not REVERSE else 0)
+        if USE_GK:
+            p_gk = gk + i_nh * T*K + ((T - 1) * K if not REVERSE else 0) + i_k * BK + tl.arange(0, BK)
+        if USE_GV:
+            p_gv = gv + i_nh * T*V + ((T - 1) * V if not REVERSE else 0) + i_v * BV + tl.arange(0, BV)
+    else:
+        p_q = q + (bos + ((T - 1) if not REVERSE else 0)) * H*K + i_h * K + i_k * BK + tl.arange(0, BK)
+        p_k = k + (bos + ((T - 1) if not REVERSE else 0)) * H*K + i_h * K + i_k * BK + tl.arange(0, BK)
+        p_v = v + (bos + ((T - 1) if not REVERSE else 0)) * H*V + i_h * V + i_v * BV + tl.arange(0, BV)
+        p_do = do + (bos + ((T - 1) if not REVERSE else 0)) * H*V + i_h * V + i_v * BV + tl.arange(0, BV)
+        p_dk = dk + ((i_v * all + bos) + ((T - 1) if not REVERSE else 0)) * H*K + i_h * K + i_k * BK + tl.arange(0, BK)
+        p_dv = dv + ((i_k * all + bos) + ((T - 1) if not REVERSE else 0)) * H*V + i_h * V + i_v * BV + tl.arange(0, BV)
+        if USE_G:
+            p_g = g + (bos + ((T - 1) if not REVERSE else 0)) * H + i_h
+        if USE_GK:
+            p_gk = gk + (bos + ((T - 1) if not REVERSE else 0)) * H*K + i_h * K + i_k * BK + tl.arange(0, BK)
+        if USE_GV:
+            p_gv = gv + (bos + ((T - 1) if not REVERSE else 0)) * H*V + i_h * V + i_v * BV + tl.arange(0, BV)
+
+    b_dh = tl.zeros([BK, BV], dtype=tl.float32)
+    if USE_FINAL_STATE_GRADIENT:
+        p_dht = dht + i_nh * K*V + (i_k * BK + tl.arange(0, BK)[:, None]) * V + (i_v * BV + tl.arange(0, BV)[None, :])
+        b_dh += tl.load(p_dht, mask=mask_h, other=0).to(tl.float32)
+
+    for _ in range(T):
+        b_q = tl.load(p_q, mask=mask_k, other=0).to(tl.float32) * scale
+        b_k = tl.load(p_k, mask=mask_k, other=0).to(tl.float32)
+        b_v = tl.load(p_v, mask=mask_v, other=0).to(tl.float32)
+        b_do = tl.load(p_do, mask=mask_v, other=0).to(tl.float32)
+        b_dh += b_q[:, None] * b_do[None, :]
+        b_dk = tl.sum(b_dh * b_v[None, :], axis=1)
+        b_dv = tl.sum(b_dh * b_k[:, None], axis=0)
+        if USE_G:
+            b_g = tl.load(p_g).to(tl.float32)
+            b_dh *= exp(b_g)
+        if USE_GK:
+            b_gk = tl.load(p_gk, mask=mask_k, other=0).to(tl.float32)
+            b_dh *= exp(b_gk)[:, None]
+        if USE_GV:
+            b_gv = tl.load(p_gv, mask=mask_v, other=0).to(tl.float32)
+            b_dh *= exp(b_gv)[None, :]
+        tl.store(p_dk, b_dk.to(p_dk.dtype.element_ty), mask=mask_k)
+        tl.store(p_dv, b_dv.to(p_dv.dtype.element_ty), mask=mask_v)
+
+        p_q += (1 if REVERSE else -1) * (1 if HEAD_FIRST else H) * K
+        p_k += (1 if REVERSE else -1) * (1 if HEAD_FIRST else H) * K
+        p_v += (1 if REVERSE else -1) * (1 if HEAD_FIRST else H) * V
+        p_do += (1 if REVERSE else -1) * (1 if HEAD_FIRST else H) * V
+        p_dk += (1 if REVERSE else -1) * (1 if HEAD_FIRST else H) * K
+        p_dv += (1 if REVERSE else -1) * (1 if HEAD_FIRST else H) * V
+        if USE_G:
+            p_g += (1 if REVERSE else -1) * (1 if HEAD_FIRST else H)
+        if USE_GK:
+            p_gk += (1 if REVERSE else -1) * (1 if HEAD_FIRST else H) * K
+        if USE_GV:
+            p_gv += (1 if REVERSE else -1) * (1 if HEAD_FIRST else H) * V
+
+    if STORE_INITIAL_STATE_GRADIENT:
+        p_dh0 = dh0 + i_nh * K*V + (i_k * BK + tl.arange(0, BK)[:, None]) * V + (i_v * BV + tl.arange(0, BV)[None, :])
+        tl.store(p_dh0, b_dh.to(p_dh0.dtype.element_ty), mask=mask_h)
+
+
+def fused_recurrent_fwd(
+    q: torch.Tensor,
+    k: torch.Tensor,
+    v: torch.Tensor,
+    g: Optional[torch.Tensor] = None,
+    gk: Optional[torch.Tensor] = None,
+    gv: Optional[torch.Tensor] = None,
+    scale: Optional[float] = None,
+    initial_state: Optional[torch.Tensor] = None,
+    output_final_state: bool = False,
+    reverse: bool = False,
+    offsets: Optional[torch.LongTensor] = None,
+    head_first: bool = True
+):
+    if head_first:
+        B, H, T, K, V = *k.shape, v.shape[-1]
+    else:
+        B, T, H, K, V = *k.shape, v.shape[-1]
+    N = B if offsets is None else len(offsets) - 1
+    BK, BV = min(K, 64), min(V, 64)
+    NK, NV = triton.cdiv(K, BK), triton.cdiv(V, BV)
+
+    h0 = initial_state
+    if output_final_state:
+        ht = q.new_empty(N, H, K, V, dtype=torch.float32)
+    else:
+        ht = None
+    o = q.new_empty(NK, *v.shape, dtype=torch.float32)
+
+    grid = (NV, NK, N * H)
+    fused_recurrent_fwd_kernel[grid](
+        q,
+        k,
+        v,
+        g,
+        gk,
+        gv,
+        o,
+        h0,
+        ht,
+        offsets,
+        scale,
+        T=T,
+        B=B,
+        H=H,
+        K=K,
+        V=V,
+        BK=BK,
+        BV=BV,
+        USE_G=g is not None,
+        USE_GK=gk is not None,
+        USE_GV=gv is not None,
+        REVERSE=reverse,
+        HEAD_FIRST=head_first
+    )
+    o = o.sum(0)
+    return o, ht
+
+
+def fused_recurrent_bwd(
+    q: torch.Tensor,
+    k: torch.Tensor,
+    v: torch.Tensor,
+    g: Optional[torch.Tensor] = None,
+    gk: Optional[torch.Tensor] = None,
+    gv: Optional[torch.Tensor] = None,
+    o: Optional[torch.Tensor] = None,
+    do: Optional[torch.Tensor] = None,
+    dht: Optional[torch.Tensor] = None,
+    scale: Optional[float] = None,
+    initial_state: Optional[torch.Tensor] = None,
+    reverse: bool = False,
+    offsets: Optional[torch.LongTensor] = None,
+    head_first: bool = True
+):
+    if head_first:
+        B, H, T, K, V = *k.shape, v.shape[-1]
+    else:
+        B, T, H, K, V = *k.shape, v.shape[-1]
+    N = B if offsets is None else len(offsets) - 1
+
+    BK, BV = min(K, 64), min(V, 64)
+    NK, NV = triton.cdiv(K, BK), triton.cdiv(V, BV)
+
+    dq = q.new_empty(NV, *q.shape, dtype=torch.float32)
+    dk = q.new_empty(NV, *k.shape, dtype=torch.float32)
+    dv = q.new_empty(NK, *v.shape, dtype=torch.float32)
+    h0 = initial_state
+    dh0 = torch.empty_like(initial_state) if initial_state is not None else None
+
+    grid = (NV, NK, N * H)
+    fused_recurrent_bwd_kernel[grid](
+        q,
+        k,
+        v,
+        g,
+        gk,
+        gv,
+        h0,
+        do,
+        dq,
+        dk,
+        dv,
+        dht,
+        dh0,
+        offsets,
+        scale,
+        B=B,
+        T=T,
+        H=H,
+        K=K,
+        V=V,
+        BK=BK,
+        BV=BV,
+        USE_G=g is not None,
+        USE_GK=gk is not None,
+        USE_GV=gv is not None,
+        REVERSE=reverse,
+        HEAD_FIRST=head_first
+    )
+    dq = dq.sum(0)
+    dk = dk.sum(0)
+    dv = dv.sum(0)
+    dg, dgk, dgv = None, None, None
+    if g is not None:
+        dg = chunk_global_cumsum(
+            (dq * q.float() - dk * k.float()).sum(-1),
+            reverse=not reverse,
+            offsets=offsets,
+            head_first=head_first
+        )
+    if gk is not None:
+        dgk = chunk_global_cumsum(
+            dq * q.float() - dk * k.float(),
+            reverse=not reverse,
+            offsets=offsets,
+            head_first=head_first
+        )
+    if gv is not None:
+        dgv = chunk_global_cumsum(
+            do.float() * o.float() - dv * v.float(),
+            reverse=not reverse,
+            offsets=offsets,
+            head_first=head_first
+        )
+
+    return dq, dk, dv, dg, dgk, dgv, dh0
+
+
+class FusedRecurrentFunction(torch.autograd.Function):
+
+    @staticmethod
+    @input_guard
+    @autocast_custom_fwd
+    def forward(
+        ctx,
+        q: torch.Tensor,
+        k: torch.Tensor,
+        v: torch.Tensor,
+        g: Optional[torch.Tensor] = None,
+        gk: Optional[torch.Tensor] = None,
+        gv: Optional[torch.Tensor] = None,
+        scale: Optional[float] = None,
+        initial_state: Optional[torch.Tensor] = None,
+        output_final_state: bool = False,
+        reverse: bool = False,
+        offsets: Optional[torch.LongTensor] = None,
+        head_first: bool = True
+    ):
+        o, ht = fused_recurrent_fwd(
+            q=q,
+            k=k,
+            v=v,
+            g=g,
+            gk=gk,
+            gv=gv,
+            scale=scale,
+            initial_state=initial_state,
+            output_final_state=output_final_state,
+            reverse=reverse,
+            offsets=offsets,
+            head_first=head_first
+        )
+        ctx.save_for_backward(q, k, v, g, gk, gv, initial_state, o)
+        ctx.scale = scale
+        ctx.reverse = reverse
+        ctx.offsets = offsets
+        ctx.head_first = head_first
+        return o.to(q.dtype), ht
+
+    @staticmethod
+    @input_guard
+    @autocast_custom_bwd
+    def backward(ctx, do, dht):
+        q, k, v, g, gk, gv, initial_state, o = ctx.saved_tensors
+        # not supported yet.
+        if dht is not None:
+            if not dht.eq(0).all():
+                if g is not None:
+                    assert g.requires_grad is False, "Cannot load final state gradient and use gates at the same time"
+                if gk is not None:
+                    assert gk.requires_grad is False, "Cannot load final state gradient and use gates at the same time"
+                if gv is not None:
+                    assert gv.requires_grad is False, "Cannot load final state gradient and use gates at the same time"
+        dq, dk, dv, dg, dgk, dgv, dh0 = fused_recurrent_bwd(
+            q=q,
+            k=k,
+            v=v,
+            g=g,
+            gk=gk,
+            gv=gv,
+            o=o,
+            do=do,
+            dht=dht,
+            scale=ctx.scale,
+            initial_state=initial_state,
+            reverse=ctx.reverse,
+            offsets=ctx.offsets,
+            head_first=ctx.head_first
+        )
+        return dq.to(q.dtype), dk.to(k.dtype), dv.to(v.dtype), dg, dgk, dgv, None, dh0, None, None, None, None
+
+
+def fused_recurrent(
+    q: torch.Tensor,
+    k: torch.Tensor,
+    v: torch.Tensor,
+    g: Optional[torch.Tensor] = None,
+    gk: Optional[torch.Tensor] = None,
+    gv: Optional[torch.Tensor] = None,
+    scale: Optional[float] = None,
+    initial_state: Optional[torch.Tensor] = None,
+    output_final_state: bool = False,
+    reverse: bool = False,
+    cu_seqlens: Optional[torch.LongTensor] = None,
+    head_first: bool = True
+):
+    if scale is None:
+        scale = k.shape[-1] ** -0.5
+    return FusedRecurrentFunction.apply(
+        q,
+        k,
+        v,
+        g,
+        gk,
+        gv,
+        scale,
+        initial_state,
+        output_final_state,
+        reverse,
+        cu_seqlens,
+        head_first
+    )

fla/ops/common/utils.py

@@ -0,0 +1,69 @@
+# -*- coding: utf-8 -*-
+# Copyright (c) 2023-2025, Songlin Yang, Yu Zhang
+
+import torch
+import triton
+import triton.language as tl
+
+from fla.utils import tensor_cache
+
+
+@triton.autotune(
+    configs=[
+        triton.Config({}, num_warps=num_warps)
+        for num_warps in [4, 8, 16, 32]
+    ],
+    key=['B'],
+)
+@triton.jit
+def prepare_position_ids_kernel(
+    y,
+    offsets,
+    B: tl.constexpr
+):
+    i_n = tl.program_id(0)
+    bos, eos = tl.load(offsets + i_n).to(tl.int32), tl.load(offsets + i_n + 1).to(tl.int32)
+    T = eos - bos
+
+    o = tl.arange(0, B)
+    for i in range(0, tl.cdiv(T, B) * B, B):
+        o_i = o + i
+        tl.store(y + bos + o_i, o_i, o_i < T)
+
+
+@tensor_cache
+def prepare_lens(offsets: torch.LongTensor) -> torch.LongTensor:
+    return offsets[1:] - offsets[:-1]
+
+
+@tensor_cache
+def prepare_position_ids(offsets: torch.LongTensor) -> torch.LongTensor:
+    return torch.cat([torch.arange(n, dtype=offsets.dtype, device=offsets.device) for n in prepare_lens(offsets).unbind()])
+
+
+@tensor_cache
+def prepare_sequence_ids(position_ids: torch.LongTensor) -> torch.LongTensor:
+    return position_ids.eq(0).cumsum(0) - 1
+
+
+@tensor_cache
+def prepare_token_indices(offsets: torch.LongTensor) -> torch.LongTensor:
+    position_ids = prepare_position_ids(offsets)
+    return torch.stack([prepare_sequence_ids(position_ids), position_ids], 1).to(offsets)
+
+
+@tensor_cache
+def prepare_chunk_indices(
+    offsets: torch.LongTensor,
+    chunk_size: int
+) -> torch.LongTensor:
+    indices = torch.cat([torch.arange(n) for n in triton.cdiv(prepare_lens(offsets), chunk_size).tolist()])
+    return torch.stack([prepare_sequence_ids(indices), indices], 1).to(offsets)
+
+
+@tensor_cache
+def prepare_chunk_offsets(
+    offsets: torch.LongTensor,
+    chunk_size: int
+) -> torch.LongTensor:
+    return torch.cat([offsets.new_tensor([0]), triton.cdiv(prepare_lens(offsets), chunk_size)]).cumsum(-1)

fla/ops/delta_rule/README.md

@@ -0,0 +1,90 @@
+# Chunkwise-form Parallelism of DeltaNet
+
+This section expands on the formulation presented in Appendix B of the DeltaNet paper.[^1]
+
+To reduce notational clutter, we focus on the first chunk, denoting $\mathbf{S}^r=\mathbf{S}_{[1]}^r$. By partially expanding the recurrence, we have:
+```math
+\begin{equation}
+\begin{aligned}
+\mathbf{S}^r &= \underbrace{\left(\prod_{i=1}^r \mathbf{I} - \beta^i \boldsymbol{k}^i \boldsymbol{k}^{i\top} \right)}_{:= \mathbf{P}^r} \cdot\mathbf{S}^{0} + \overbrace{\sum_{i=1}^{r} \underbrace{\left(\prod_{j=i+1}^r \mathbf{I} - \beta^j \boldsymbol{k}^j \boldsymbol{k}^{j\top} \right)}_{:= \mathbf{P}_{i+1}^r}\beta^i \boldsymbol{k}^i\boldsymbol{v}^{i\top}}^{:=\mathbf{H}^r} \\
+&=\mathbf{P}^r \cdot \mathbf{S}^{0} + \mathbf{H}^r
+\end{aligned}
+\end{equation}
+```
+
+where $\mathbf{P}_i^r$ involves cumulative products of generalized Householder matrices.
+We abbreviate $\mathbf{P}_1^r$ as $\mathbf{P}^r$.
+This can be optimized using the classical WY representation:
+```math
+\begin{equation}
+\mathbf{P}^{r} = \mathbf{I} - \sum_{i=1}^{r}\boldsymbol{k}^i\boldsymbol{w}^{i\top}  \in \mathbb{R}^{d_k \times d_k};\qquad
+\boldsymbol{w}^r = \beta^r \left(\boldsymbol{k}^r -  \sum_{i=1}^{r-1} \left(\boldsymbol{k}^{r\top}\boldsymbol{k}^i \right)\boldsymbol{w}^i  \right) \in \mathbb{R}^{d_k}
+\end{equation}
+```
+
+We prove this by induction:
+```math
+\begin{align*}
+\mathbf{P}^{r} &= \prod_{i=1}^r \mathbf{I} - \beta^i \boldsymbol{k}^i \boldsymbol{k}^{i\top} \\
+&= \left(\mathbf{I} - \beta^r \boldsymbol{k}^r \boldsymbol{k}^{r\top}\right)\mathbf{P}^{r-1} \\
+&= \left(\mathbf{I} - \beta^r \boldsymbol{k}^r \boldsymbol{k}^{r\top}\right)\left(\mathbf{I} - \sum_{i=1}^{r-1}\boldsymbol{k}^i\boldsymbol{w}^{i\top}\right) \\
+&= \mathbf{I} - \sum_{i=1}^{r-1}\boldsymbol{k}^i\boldsymbol{w}^{i\top} - \beta^r \boldsymbol{k}^r \boldsymbol{k}^{r\top} + \beta^r\boldsymbol{k}^r \boldsymbol{k}^{r\top} \left(\sum_{i=1}^{r-1}\boldsymbol{k}^i\boldsymbol{w}^{i\top}\right) \\
+&= \mathbf{I} - \sum_{i=1}^{r-1}\boldsymbol{k}^i\boldsymbol{w}^{i\top} - \beta^r \boldsymbol{k}^r \left(\boldsymbol{k}^{r} - \left(\sum_{i=1}^{r-1}\left(\boldsymbol{k}^{r\top} \boldsymbol{k}^i\right)\boldsymbol{w}^{i}\right) \right)^\top \\
+&= \mathbf{I} - \sum_{i=1}^{r}\boldsymbol{k}^i\boldsymbol{w}^{i\top}
+\end{align*}
+```
+
+Similarly, $\mathbf{H}^r$ can be represented as:
+```math
+\begin{equation}
+\mathbf{H}^{r} = \sum_{i=1}^{r} \boldsymbol{k}^i \boldsymbol{u}^{i\top}  \in \mathbb{R}^{d_k \times d_v};\qquad \boldsymbol{u}^r = \beta^r \left(\boldsymbol{v}^r -  \sum_{i=1}^{r-1} \left(\boldsymbol{k}^{r\top}\boldsymbol{k}^i\right) \boldsymbol{u}^i \right)\in \mathbb{R}^{d_v}
+\end{equation}
+```
+
+This can also be proven by induction:
+```math
+\begin{align*}
+\mathbf{H}^{r} &= \sum_{i=1}^{r} \mathbf{P}_{i+1}^r \beta^i \boldsymbol{k}^i \boldsymbol{v}^{i\top}\\
+&= \left(\mathbf{I} - \beta^r \boldsymbol{k}^r \boldsymbol{k}^{r\top}\right) \mathbf{H}^{r-1} +  \beta^r \boldsymbol{k}^r \boldsymbol{v}^{r\top}\\
+&= \sum_{i=1}^{r-1}\boldsymbol{k}^i \boldsymbol{u}^{i\top} - \beta^r \boldsymbol{k}^r \boldsymbol{k}^{r\top} \sum_{i=1}^{r-1}\boldsymbol{k}^i \boldsymbol{u}^{i\top} +\beta^r \boldsymbol{k}^r \boldsymbol{v}^{r\top}\\
+&= \sum_{i=1}^{r-1}\boldsymbol{k}^i \boldsymbol{u}^{i\top} + \boldsymbol{k}^r \left(\beta^r \boldsymbol{v}^{r\top}-\beta^r \boldsymbol{k}^{r\top} \sum_{i=1}^{r-1}\boldsymbol{k}^i \boldsymbol{u}^{i\top}\right) \\
+&= \sum_{i=1}^{r-1}\boldsymbol{k}^i \boldsymbol{u}^{i\top} + \boldsymbol{k}^r \beta^r\left(\boldsymbol{v}^{r}-\sum_{i=1}^{r-1}\left(\boldsymbol{k}^{r\top}\boldsymbol{k}^{i}\right)\boldsymbol{u}^{i} \right)^\top \\
+&=\sum_{i=1}^{r} \boldsymbol{k}^i \boldsymbol{u}^{i\top}
+\end{align*}
+```
+
+In matrix form, $\mathbf{P}$ and $\mathbf{H}$ can be written as:
+```math
+\begin{equation}
+\mathbf{P}=\mathbf{I}-\mathbf{K}^\top\mathbf{W} \in \mathbb{R}^{d_k \times d_k}, \qquad\mathbf{H}=\mathbf{K}^\top\mathbf{U} \in \mathbb{R}^{d_k\times d_v}
+\end{equation}
+```
+
+Now we can derive the matrix form of $\mathbf{W}$ and $\mathbf{U}$:
+```math
+\begin{align*}
+\mathbf{W} &= \mathrm{diag}(\beta) \mathbf{K} - \mathrm{tril}(\mathrm{diag}(\beta) \mathbf{K}\mathbf{K}^\top, -1)\mathbf{W}\\
+\left(\mathbf{I} + \mathrm{tril}(\mathrm{diag}(\beta) \mathbf{K}\mathbf{K}^\top, -1)\right) \mathbf{W} &= \mathrm{diag}(\beta) \mathbf{K}
+\end{align*}
+```
+A similar process holds for $\mathbf{U}$. We can further write $\mathbf{W}$ and $\mathbf{U}$ in matrix form:
+```math
+\begin{align*}
+\mathbf{T} &= \left(\mathbf{I} + \mathrm{tril}\left(\mathrm{diag}(\beta)\mathbf{K} \mathbf{K}^\top,-1\right)\right)^{-1}\mathrm{diag}\left(\beta\right)\in \mathbb{R}^{C \times C}\\
+\mathbf{W} &= \mathbf{T} \mathbf{K}\in \mathbb{R}^{C \times d_k}\\
+\mathbf{U} &= \mathbf{T}\mathbf{V}\in \mathbb{R}^{C \times d_v}
+\end{align*}
+```
+
+Substituting these back into the original equations yields a hardware-efficient chunkwise algorithm for DeltaNet that leverages matrix multiplications, enabling tensor core based GPU optimization:
+```math
+\begin{equation}
+\begin{aligned}
+\mathbf{S} &= \mathbf{P}\cdot\mathbf{S}^0 + \mathbf{H} \\
+&= \mathbf{S}^0 + \mathbf{K}^\top (\mathbf{U} -\mathbf{W} \mathbf{S}^0) \in \mathbb{R}^{d_k \times d_v}\\
+\mathbf{O} &= \mathbf{Q} \mathbf{S}^0 + (\mathbf{Q} \mathbf{K}^{\top} \odot \mathbf{M}) \left(\mathbf{U} - \mathbf{W} \mathbf{S}^0\right) \in \mathbb{R}^{C \times d_v}
+\end{aligned}
+\end{equation}
+```
+
+[^1]: https://arxiv.org/abs/2406.06484

fla/ops/delta_rule/__init__.py

@@ -0,0 +1,11 @@
+# -*- coding: utf-8 -*-
+
+from .chunk import chunk_delta_rule
+from .fused_chunk import fused_chunk_delta_rule
+from .fused_recurrent import fused_recurrent_delta_rule
+
+__all__ = [
+    'fused_chunk_delta_rule',
+    'fused_recurrent_delta_rule',
+    'chunk_delta_rule'
+]

fla/ops/delta_rule/parallel.py

@@ -0,0 +1,394 @@
+# -*- coding: utf-8 -*-
+# Copyright (c) 2023-2025, Songlin Yang, Yu Zhang
+
+from typing import Tuple
+
+import torch
+import triton
+import triton.language as tl
+from einops import rearrange
+
+from fla.ops.delta_rule.wy_fast import fwd_prepare_T
+from fla.utils import autocast_custom_bwd, autocast_custom_fwd, input_guard
+
+
+@triton.autotune(
+    configs=[
+        triton.Config({}, num_warps=num_warps)
+        for num_warps in [1, 2, 4]
+    ],
+    key=['BT', 'K', 'V'],
+)
+@triton.jit(do_not_specialize=['T'])
+def chunk_transform_qk_fwd_kernel(
+    q,
+    k,
+    v,
+    beta,
+    o,
+    A,
+    q_new,
+    k_new,
+    A_local,
+    scale,
+    T,
+    K: tl.constexpr,
+    V: tl.constexpr,
+    BK: tl.constexpr,
+    BV: tl.constexpr,
+    BT: tl.constexpr,
+    OUTPUT_ATTENTIONS: tl.constexpr
+):
+    i_t, i_bh = tl.program_id(0), tl.program_id(1)
+
+    p_q = tl.make_block_ptr(q + i_bh * T*K, (T, K), (K, 1), (i_t * BT, 0), (BT, BK), (1, 0))
+    p_k = tl.make_block_ptr(k + i_bh * T*K, (T, K), (K, 1), (i_t * BT, 0), (BT, BK), (1, 0))
+    p_v = tl.make_block_ptr(v + i_bh * T*V, (T, V), (V, 1), (i_t * BT, 0), (BT, BV), (1, 0))
+    b_q = (tl.load(p_q, boundary_check=(0, 1)) * scale).to(p_q.dtype.element_ty)
+    b_k = tl.load(p_k, boundary_check=(0, 1))
+    b_v = tl.load(p_v, boundary_check=(0, 1))
+
+    p_T = tl.make_block_ptr(A + i_bh * T * BT, (T, BT), (BT, 1), (i_t * BT, 0), (BT, BT), (1, 0))
+    b_T = tl.load(p_T, boundary_check=(0, 1))
+
+    o_i = tl.arange(0, BT)
+    m_t = o_i[:, None] >= o_i[None, :]
+    b_qk = tl.where(m_t, tl.dot(b_q, tl.trans(b_k), allow_tf32=False), 0).to(b_q.dtype)
+    m_t = o_i[:, None] > o_i[None, :]
+    b_kk = tl.where(m_t, tl.dot(b_k, tl.trans(b_k), allow_tf32=False), 0).to(b_k.dtype)
+
+    p_beta = tl.make_block_ptr(beta + i_bh * T, (T, ), (1, ), (i_t * BT, ), (BT, ), (0, ))
+    b_beta = tl.load(p_beta, boundary_check=(0, ))
+    b_k_beta = (b_k * b_beta[:, None]).to(b_k.dtype)
+
+    b_qkT = tl.dot(b_qk, b_T, allow_tf32=False).to(b_k.dtype)
+
+    if OUTPUT_ATTENTIONS:
+        p_a = tl.make_block_ptr(A_local + i_bh * T * BT, (T, BT), (BT, 1), (i_t * BT, 0), (BT, BT), (1, 0))
+        tl.store(p_a, b_qkT.to(p_a.dtype.element_ty), boundary_check=(0, 1))
+
+    b_kkT = tl.dot(b_kk, b_T, allow_tf32=False).to(b_k.dtype)
+    p_o = tl.make_block_ptr(o + i_bh * T*V, (T, V), (V, 1), (i_t * BT, 0), (BT, BV), (1, 0))
+    tl.store(p_o, tl.dot(b_qkT, b_v).to(p_o.dtype.element_ty), boundary_check=(0, 1))
+
+    p_q_new = tl.make_block_ptr(q_new + i_bh * T*K, (T, K), (K, 1), (i_t * BT, 0), (BT, BK), (1, 0))
+    tl.store(p_q_new, (b_q - tl.dot(b_qkT, b_k_beta, allow_tf32=False)).to(p_q_new.dtype.element_ty), boundary_check=(0, 1))
+
+    p_k_new = tl.make_block_ptr(k_new + i_bh * T*K, (T, K), (K, 1), (i_t * BT, 0), (BT, BK), (1, 0))
+    b_k_new = b_k - tl.dot(tl.trans(b_kkT), b_k_beta, allow_tf32=False)
+    tl.store(p_k_new, b_k_new.to(p_k_new.dtype.element_ty), boundary_check=(0, 1))
+
+
+def chunk_transform_qk_fwd(
+    q: torch.Tensor,
+    k: torch.Tensor,
+    v: torch.Tensor,
+    beta: torch.Tensor,
+    A: torch.Tensor,
+    scale: float,
+    chunk_size: int,
+    output_attentions: bool
+):
+    B, H, T, K = k.shape
+    BT = chunk_size
+    q_new = torch.empty_like(q)
+    k_new = torch.empty_like(k)
+    o = torch.empty_like(v)
+    grid = (triton.cdiv(T, BT), B*H)
+    V = v.shape[-1]
+    A_local = torch.empty_like(A) if output_attentions else None
+    chunk_transform_qk_fwd_kernel[grid](
+        q,
+        k,
+        v,
+        beta,
+        o,
+        A,
+        q_new,
+        k_new,
+        A_local,
+        scale=scale,
+        T=T,
+        K=K,
+        V=V,
+        BT=BT,
+        BK=triton.next_power_of_2(K),
+        BV=triton.next_power_of_2(V),
+        OUTPUT_ATTENTIONS=output_attentions
+    )
+    return q_new, k_new, o, A_local
+
+
+@triton.autotune(
+    configs=[
+        triton.Config({}, num_warps=1),
+        triton.Config({}, num_warps=2),
+    ],
+    key=['BT'],
+)
+@triton.jit(do_not_specialize=['T'])
+def save_intra_chunk_attn(
+    A,
+    A_local,
+    T,
+    BT: tl.constexpr,
+):
+    i_t, i_bh = tl.program_id(0), tl.program_id(1)
+    p_A = tl.make_block_ptr(A + i_bh * T * T, (T, T), (T, 1), (i_t * BT, i_t * BT), (BT, BT), (1, 0))
+    p_A_local = tl.make_block_ptr(A_local + i_bh * T * BT, (T, BT), (BT, 1), (i_t * BT, 0), (BT, BT), (1, 0))
+    b_A_local = tl.load(p_A_local, boundary_check=(0, 1))
+    tl.store(p_A, b_A_local.to(p_A.dtype.element_ty), boundary_check=(0, 1))
+
+
+@triton.heuristics({
+    'OUTPUT_ATTENTIONS': lambda args: args['attn'] is not None
+})
+@triton.jit(do_not_specialize=['T'])
+def parallel_delta_rule_fwd_kernel(
+    q,
+    k,
+    k2,  # original k
+    v,
+    beta,
+    o,
+    o_new,
+    attn,
+    T,
+    K: tl.constexpr,
+    V: tl.constexpr,
+    BT: tl.constexpr,
+    BS: tl.constexpr,
+    BK: tl.constexpr,
+    BV: tl.constexpr,
+    OUTPUT_ATTENTIONS: tl.constexpr
+):
+    i_t, i_bh = tl.program_id(0), tl.program_id(1)
+    p_q = tl.make_block_ptr(q + i_bh * T*K, (T, K), (K, 1), (i_t * BT, 0), (BT, BK), (1, 0))
+
+    # the Q block is kept in the shared memory throughout the whole kernel
+    # [BT, BK]
+    b_q = tl.zeros([BT, BK], dtype=tl.float32)
+    b_q += tl.load(p_q, boundary_check=(0, 1))
+
+    b_o = tl.zeros([BT, BV], dtype=tl.float32)
+    p_o = tl.make_block_ptr(o + i_bh * T*V, (T, V), (V, 1), (i_t * BT, 0), (BT, BV), (1, 0))
+    b_o += tl.load(p_o, boundary_check=(0, 1))
+
+    # As opposed to Flashattention, this kernel requires scanning the KV blocks from right to left
+    # Q block and K block have overlap.
+    # masks required
+    for offset in range((i_t + 1) * BT - 2 * BS, i_t * BT - BS, -BS):
+        p_k = tl.make_block_ptr(k + i_bh * T*K, (K, T), (1, K), (0, offset), (BK, BS), (0, 1))
+        p_k2 = tl.make_block_ptr(k2 + i_bh * T*K, (T, K), (K, 1), (offset, 0), (BS, BK), (1, 0))
+        p_v = tl.make_block_ptr(v + i_bh * T*V, (T, V), (V, 1), (offset, 0), (BS, BV), (1, 0))
+        p_beta = tl.make_block_ptr(beta + i_bh * T, (T, ), (1, ), (offset, ), (BS, ), (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))
+        # [BS]
+        b_beta = tl.load(p_beta, boundary_check=(0,))
+        # [BT, BS]
+        m_s = tl.arange(0, BT) >= (offset - i_t*BT + BS)
+        b_s = tl.dot(b_q.to(b_k.dtype), b_k, allow_tf32=False)
+        b_s = tl.where(m_s[:, None], b_s, 0)
+
+        b_o += tl.dot(b_s.to(b_v.dtype), b_v, allow_tf32=False)
+        b_k2 = (tl.load(p_k2, boundary_check=(0, 1)) * b_beta[:, None]).to(b_v.dtype)
+        b_q -= tl.dot(b_s.to(b_v.dtype), b_k2, allow_tf32=False)
+
+        if OUTPUT_ATTENTIONS:
+            p_a = tl.make_block_ptr(attn + i_bh * T * T, (T, T), (T, 1), (i_t * BT, offset), (BT, BS), (1, 0))
+            tl.store(p_a, b_s.to(p_a.dtype.element_ty), boundary_check=(0, 1))
+
+    # Q block and K block have no overlap
+    # no need for mask, thereby saving flops
+    for offset in range(i_t * BT - BS, -BS, -BS):
+        p_k = tl.make_block_ptr(k + i_bh * T*K, (K, T), (1, K), (0, offset), (BK, BS), (0, 1))
+        p_v = tl.make_block_ptr(v + i_bh * T*V, (T, V), (V, 1), (offset, 0), (BS, BV), (1, 0))
+        p_beta = tl.make_block_ptr(beta + i_bh * T, (T, ), (1, ), (offset, ), (BS, ), (0,))
+        p_k2 = tl.make_block_ptr(k2 + i_bh * T*K, (T, K), (K, 1), (offset, 0), (BS, BK), (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))
+        # [BS]
+        b_beta = tl.load(p_beta, boundary_check=(0,))
+        # [BT, BS]
+        b_s = (tl.dot(b_q.to(b_k.dtype), b_k, allow_tf32=False))
+        # [BT, BV]
+        b_o += tl.dot(b_s.to(b_v.dtype), b_v, allow_tf32=False)
+        b_k2 = (tl.load(p_k2, boundary_check=(0, 1)) * b_beta[:, None]).to(b_v.dtype)
+        b_q -= tl.dot(b_s.to(b_v.dtype), b_k2, allow_tf32=False).to(b_q.dtype)
+
+        if OUTPUT_ATTENTIONS:
+            p_a = tl.make_block_ptr(attn + i_bh * T * T, (T, T), (T, 1), (i_t * BT, offset), (BT, BS), (1, 0))
+            tl.store(p_a, b_s.to(p_a.dtype.element_ty), boundary_check=(0, 1))
+
+    p_o_new = tl.make_block_ptr(o_new + i_bh * T*V, (T, V), (V, 1), (i_t*BT, 0), (BT, BV), (1, 0))
+    tl.store(p_o_new, b_o.to(p_o.dtype.element_ty), boundary_check=(0, 1))
+
+
+class ParallelDeltaRuleFunction(torch.autograd.Function):
+
+    @staticmethod
+    @input_guard
+    @autocast_custom_fwd
+    def forward(ctx, q, k, v, beta, scale, output_attentions):
+        B, H, T, K, V = *k.shape, v.shape[-1]
+        assert q.shape[-1] <= 128, 'The maximum supported sequence length is 128.'
+        BT, BS = 128, 32
+        BK = triton.next_power_of_2(k.shape[-1])
+        BV = triton.next_power_of_2(v.shape[-1])
+        assert BT % BS == 0
+
+        A = fwd_prepare_T(k, beta, BS)
+        attn = q.new_zeros(B, H, T, T) if output_attentions else None
+        q_new, k_new, o, A_local = chunk_transform_qk_fwd(
+            q,
+            k,
+            v,
+            beta,
+            A,
+            scale,
+            BS,
+            output_attentions
+        )
+
+        num_stages = 3 if K <= 64 else 2
+        num_warps = 4
+        grid = (triton.cdiv(T, BT), B * H)
+        o_new = torch.empty_like(o)
+
+        parallel_delta_rule_fwd_kernel[grid](
+            q=q_new,
+            k=k_new,
+            k2=k,
+            v=v,
+            beta=beta,
+            o=o,
+            o_new=o_new,
+            attn=attn,
+            T=T,
+            K=K,
+            V=V,
+            BT=BT,
+            BS=BS,
+            BK=BK,
+            BV=BV,
+            num_stages=num_stages,
+            num_warps=num_warps
+        )
+
+        if output_attentions:
+            grid = (triton.cdiv(T, BS), B * H)
+            save_intra_chunk_attn[grid](
+                A=attn,
+                A_local=A_local,
+                T=T,
+                BT=BS
+            )
+        return o_new.to(q.dtype), attn
+
+    @staticmethod
+    @input_guard
+    @autocast_custom_bwd
+    def backward(ctx, do, d_attn=None):
+        raise NotImplementedError('Backward pass is not implemented. Stay tuned!')
+
+
+def parallel_delta_rule(
+    q: torch.Tensor,
+    k: torch.Tensor,
+    v: torch.Tensor,
+    beta: torch.Tensor,
+    scale: float = None,
+    output_attentions: bool = False,
+    head_first: bool = True
+) -> Tuple[torch.Tensor, torch.Tensor]:
+    r"""
+    Args:
+        q (torch.Tensor):
+            queries of shape `[B, H, T, K]` if `head_first=True` else `[B, T, H, K]`.
+        k (torch.Tensor):
+            keys of shape `[B, H, T, K]` if `head_first=True` else `[B, T, H, K]`.
+        v (torch.Tensor):
+            values of shape `[B, H, T, V]` if `head_first=True` else `[B, T, H, V]`.
+        beta (torch.Tensor):
+            betas of shape `[B, H, T]` if `head_first=True` else `[B, T, H]`.
+        scale (Optional[int]):
+            Scale factor for attention scores.
+            If not provided, it will default to `1 / sqrt(K)`. Default: `None`.
+        output_attentions (bool):
+            Whether to output the materialized attention scores of shape [B, H, T, T]. Default: `False`.
+        head_first (Optional[bool]):
+            Whether the inputs are in the head-first format.
+            Default: `True`.
+
+    Returns:
+        o (torch.Tensor):
+            Outputs of shape `[B, H, T, V]` if `head_first=True` else `[B, T, H, V]`.
+        attn (torch.Tensor):
+            Attention scores of shape `[B, H, T, T]` if `output_attentions=True` else `None`.
+    """
+    if not head_first:
+        q, k, v, beta = map(lambda x: x.transpose(1, 2), (q, k, v, beta))
+    o, attn = ParallelDeltaRuleFunction.apply(q, k, v, beta, scale, output_attentions)
+    if not head_first:
+        o = o.transpose(1, 2)
+    return o, attn
+
+
+def naive_delta_rule_parallel(q, k, v, beta, BM=128, BN=32):
+    b, h, l, d_k = q.shape
+    q = q * (d_k ** -0.5)
+    v = v * beta[..., None]
+    k_beta = k * beta[..., None]
+    # compute (I - tri(diag(beta) KK^T))^{-1}
+    q, k, v, k_beta = map(lambda x: rearrange(x, 'b h (n c) d -> b h n c d', c=BN), [q, k, v, k_beta])
+    mask = torch.triu(torch.ones(BN, BN, dtype=torch.bool, device=q.device), diagonal=0)
+    T = -(k_beta @ k.transpose(-1, -2)).masked_fill(mask, 0)
+    for i in range(1, BN):
+        T[..., i, :i] = T[..., i, :i].clone() + (T[..., i, :, None].clone() * T[..., :, :i].clone()).sum(-2)
+    T = T + torch.eye(BN, dtype=q.dtype, device=q.device)
+
+    mask2 = torch.triu(torch.ones(BN, BN, dtype=torch.bool, device=q.device), diagonal=1)
+    A_local = (q @ k.transpose(-1, -2)).masked_fill(mask2, 0) @ T
+    o_intra = A_local @ v
+
+    # apply cumprod transition matrices on k to the last position within the chunk
+    k = k - ((k @ k.transpose(-1, -2)).masked_fill(mask, 0) @ T).transpose(-1, -2) @ k_beta
+    # apply cumprod transition matrices on q to the first position within the chunk
+    q = q - A_local @ k_beta
+    o_intra = A_local @ v
+
+    A = torch.zeros(b, h, l, l, device=q.device)
+
+    q, k, v, k_beta, o_intra = map(lambda x: rearrange(x, 'b h n c d -> b h (n c) d'), [q, k, v, k_beta, o_intra])
+    o = torch.empty_like(v)
+    for i in range(0, l, BM):
+        q_i = q[:, :, i:i+BM]
+        o_i = o_intra[:, :, i:i+BM]
+        # intra block
+        for j in range(i + BM - 2 * BN, i-BN, -BN):
+            k_j = k[:, :, j:j+BN]
+            A_ij = q_i @ k_j.transpose(-1, -2)
+            mask = torch.arange(i, i+BM) >= (j + BN)
+            A_ij = A_ij.masked_fill_(~mask[:, None].to(A_ij.device), 0)
+            A[:, :, i:i+BM, j:j+BN] = A_ij
+            q_i = q_i - A_ij @ k_beta[:, :, j:j+BN]
+            o_i += A_ij @ v[:, :, j:j+BN]
+        # inter block
+        for j in range(i - BN, -BN, -BN):
+            k_j = k[:, :, j:j+BN]
+            A_ij = q_i @ k_j.transpose(-1, -2)
+            A[:, :, i:i+BM, j:j+BN] = A_ij
+            q_i = q_i - A_ij @ k_beta[:, :, j:j+BN]
+            o_i += A_ij @ v[:, :, j:j+BN]
+        o[:, :, i:i+BM] = o_i
+
+    for i in range(0, l//BN):
+        A[:, :, i*BN:i*BN+BN, i*BN:i*BN+BN] = A_local[:, :, i]
+
+    return o, A

fla/ops/gated_delta_rule/__init__.py

@@ -0,0 +1,7 @@
+from .chunk import chunk_gated_delta_rule
+from .fused_recurrent import fused_recurrent_gated_delta_rule
+
+__all__ = [
+    "chunk_gated_delta_rule",
+    "fused_recurrent_gated_delta_rule"
+]

fla/ops/generalized_delta_rule/__init__.py

@@ -0,0 +1,9 @@
+from .dplr import chunk_dplr_delta_rule, fused_recurrent_dplr_delta_rule
+from .iplr import chunk_iplr_delta_rule, fused_recurrent_iplr_delta_rule
+
+__all__ = [
+    'chunk_dplr_delta_rule',
+    'fused_recurrent_dplr_delta_rule',
+    'chunk_iplr_delta_rule',
+    'fused_recurrent_iplr_delta_rule'
+]

fla/ops/generalized_delta_rule/dplr/__init__.py

@@ -0,0 +1,7 @@
+from .chunk import chunk_dplr_delta_rule
+from .fused_recurrent import fused_recurrent_dplr_delta_rule
+
+__all__ = [
+    'chunk_dplr_delta_rule',
+    'fused_recurrent_dplr_delta_rule'
+]

fla/ops/generalized_delta_rule/dplr/chunk_A_bwd.py

@@ -0,0 +1,446 @@
+# -*- coding: utf-8 -*-
+# Copyright (c) 2023-2025, Songlin Yang, Yu Zhang
+
+from typing import Optional
+
+import torch
+import triton
+import triton.language as tl
+
+from fla.ops.utils.op import exp, gather
+from fla.utils import check_shared_mem, is_gather_supported, use_cuda_graph
+
+
+@triton.heuristics({
+    'USE_OFFSETS': lambda args: args['offsets'] is not None
+})
+@triton.autotune(
+    configs=[
+        triton.Config({}, num_warps=num_warps, num_stages=num_stages)
+        for num_warps in [2, 4, 8, 16, 32]
+        for num_stages in [2, 3, 4]
+    ],
+    key=['BK', 'NC', 'BT', 'K'],
+    use_cuda_graph=use_cuda_graph,
+)
+@triton.jit(do_not_specialize=['T'])
+def chunk_dplr_bwd_kernel_intra(
+    q,
+    k,
+    a,
+    b,
+    gi,
+    ge,
+    dAqk,
+    dAqb,
+    dAak,
+    dAab,
+    dq,
+    dk,
+    da,
+    db,
+    dqg,
+    dkg,
+    dag,
+    dbg,
+    dgk,
+    dgk_offset,
+    offsets,
+    indices,
+    scale: tl.constexpr,
+    T,
+    H: tl.constexpr,
+    K: tl.constexpr,
+    BT: tl.constexpr,
+    BC: tl.constexpr,
+    BK: tl.constexpr,
+    NC: tl.constexpr,
+    USE_OFFSETS: tl.constexpr,
+    HEAD_FIRST: tl.constexpr,
+    GATHER_SUPPORTED: tl.constexpr
+):
+    i_k, i_c, i_bh = tl.program_id(0), tl.program_id(1), tl.program_id(2)
+    i_b, i_h = i_bh // H, i_bh % H
+    i_t, i_i = i_c // NC, i_c % NC
+    if USE_OFFSETS:
+        i_n, i_t = tl.load(indices + i_t * 2).to(tl.int32), tl.load(indices + i_t * 2 + 1).to(tl.int32)
+        bos, eos = tl.load(offsets + i_n).to(tl.int32), tl.load(offsets + i_n + 1).to(tl.int32)
+    else:
+        bos, eos = i_b * T, i_b * T + T
+    T = eos - bos
+    if i_t * BT + i_i * BC >= T:
+        return
+
+    # offset calculation
+    ge += i_bh * T * K if HEAD_FIRST else (bos*H + i_h) * K
+    gi += i_bh * T * K if HEAD_FIRST else (bos*H + i_h) * K
+    q += i_bh * T * K if HEAD_FIRST else (bos*H + i_h) * K
+    a += i_bh * T * K if HEAD_FIRST else (bos*H + i_h) * K
+    b += i_bh * T * K if HEAD_FIRST else (bos*H + i_h) * K
+    k += i_bh * T * K if HEAD_FIRST else (bos*H + i_h) * K
+    dq += i_bh * T * K if HEAD_FIRST else (bos*H + i_h) * K
+    dk += i_bh * T * K if HEAD_FIRST else (bos*H + i_h) * K
+    da += i_bh * T * K if HEAD_FIRST else (bos*H + i_h) * K
+    db += i_bh * T * K if HEAD_FIRST else (bos*H + i_h) * K
+    dqg += i_bh * T * K if HEAD_FIRST else (bos*H + i_h) * K
+    dag += i_bh * T * K if HEAD_FIRST else (bos*H + i_h) * K
+    dkg += i_bh * T * K if HEAD_FIRST else (bos*H + i_h) * K
+    dbg += i_bh * T * K if HEAD_FIRST else (bos*H + i_h) * K
+    dgk += i_bh * T * K if HEAD_FIRST else (bos*H + i_h) * K
+    dgk_offset += i_bh * T * K if HEAD_FIRST else (bos*H + i_h) * K
+    dAqk += i_bh * T * BT if HEAD_FIRST else (bos*H + i_h) * BT
+    dAqb += i_bh * T * BT if HEAD_FIRST else (bos*H + i_h) * BT
+    dAak += i_bh * T * BT if HEAD_FIRST else (bos*H + i_h) * BT
+    dAab += i_bh * T * BT if HEAD_FIRST else (bos*H + i_h) * BT
+
+    stride_qk = K if HEAD_FIRST else H*K
+    stride_A = BT if HEAD_FIRST else H*BT
+
+    p_ge = tl.make_block_ptr(ge, (T, K), (stride_qk, 1), (i_t * BT + i_i * BC, i_k * BK), (BC, BK), (1, 0))
+    p_gi = tl.make_block_ptr(gi, (T, K), (stride_qk, 1), (i_t * BT + i_i * BC, i_k * BK), (BC, BK), (1, 0))
+    # [BC, BK]
+    b_ge = tl.load(p_ge, boundary_check=(0, 1))
+    b_gi = tl.load(p_gi, boundary_check=(0, 1))
+    b_dq = tl.zeros([BC, BK], dtype=tl.float32)
+    b_da = tl.zeros([BC, BK], dtype=tl.float32)
+    b_dk = tl.zeros([BC, BK], dtype=tl.float32)
+    b_db = tl.zeros([BC, BK], dtype=tl.float32)
+    # intra chunk gradient calculation
+    p_dAqk = tl.make_block_ptr(dAqk, (T, BT), (stride_A, 1), (i_t*BT + i_i*BC, i_i*BC), (BC, BC), (1, 0))
+    p_dAab = tl.make_block_ptr(dAab, (T, BT), (stride_A, 1), (i_t*BT + i_i*BC, i_i*BC), (BC, BC), (1, 0))
+    p_dAqb = tl.make_block_ptr(dAqb, (T, BT), (stride_A, 1), (i_t*BT + i_i*BC, i_i*BC), (BC, BC), (1, 0))
+    p_dAak = tl.make_block_ptr(dAak, (T, BT), (stride_A, 1), (i_t*BT + i_i*BC, i_i*BC), (BC, BC), (1, 0))
+    o_i = tl.arange(0, BC)
+    p_k = tl.make_block_ptr(k, (T, K), (stride_qk, 1), (i_t*BT + i_i*BC, i_k*BK), (BC, BK), (1, 0))
+    p_b = tl.make_block_ptr(b, (T, K), (stride_qk, 1), (i_t*BT + i_i*BC, i_k*BK), (BC, BK), (1, 0))
+    p_a = tl.make_block_ptr(a, (T, K), (stride_qk, 1), (i_t*BT + i_i*BC, i_k*BK), (BC, BK), (1, 0))
+    p_q = tl.make_block_ptr(q, (T, K), (stride_qk, 1), (i_t*BT + i_i*BC, i_k*BK), (BC, BK), (1, 0))
+    b_k = tl.load(p_k, boundary_check=(0, 1)).to(tl.float32)
+    b_b = tl.load(p_b, boundary_check=(0, 1)).to(tl.float32)
+    b_q = tl.load(p_q, boundary_check=(0, 1)).to(tl.float32)
+    b_a = tl.load(p_a, boundary_check=(0, 1)).to(tl.float32)
+    b_dAqk = tl.load(p_dAqk, boundary_check=(0, 1)).to(tl.float32)
+    b_dAab = tl.load(p_dAab, boundary_check=(0, 1)).to(tl.float32)
+    b_dAqb = tl.load(p_dAqb, boundary_check=(0, 1)).to(tl.float32)
+    b_dAak = tl.load(p_dAak, boundary_check=(0, 1)).to(tl.float32)
+
+    # inter chunk gradient calculation
+    o_k = i_k * BK + tl.arange(0, BK)
+    m_k = o_k < K
+    if i_i > 0:
+        p_gn = gi + (i_t * BT + i_i * BC - 1) * stride_qk + o_k
+        p_gn = tl.max_contiguous(tl.multiple_of(p_gn, BK), BK)
+        # [BK,]
+        b_gn = tl.load(p_gn, mask=m_k, other=0)
+        # [BK,]
+        for i_j in range(0, i_i):
+            p_kj = tl.make_block_ptr(k, (T, K), (stride_qk, 1), (i_t * BT + i_j * BC, i_k * BK), (BC, BK), (1, 0))
+            p_bj = tl.make_block_ptr(b, (T, K), (stride_qk, 1), (i_t * BT + i_j * BC, i_k * BK), (BC, BK), (1, 0))
+            p_gkj = tl.make_block_ptr(gi, (T, K), (stride_qk, 1), (i_t * BT + i_j * BC, i_k * BK), (BC, BK), (1, 0))
+            p_dAqikj = tl.make_block_ptr(dAqk, (T, BT), (stride_A, 1), (i_t * BT + i_i * BC, i_j * BC), (BC, BC), (1, 0))
+            p_dAaibj = tl.make_block_ptr(dAab, (T, BT), (stride_A, 1), (i_t * BT + i_i * BC, i_j * BC), (BC, BC), (1, 0))
+            p_dAqibj = tl.make_block_ptr(dAqb, (T, BT), (stride_A, 1), (i_t * BT + i_i * BC, i_j * BC), (BC, BC), (1, 0))
+            p_dAaikj = tl.make_block_ptr(dAak, (T, BT), (stride_A, 1), (i_t * BT + i_i * BC, i_j * BC), (BC, BC), (1, 0))
+            # [BC, BK]
+            b_kj = tl.load(p_kj, boundary_check=(0, 1))
+            b_bj = tl.load(p_bj, boundary_check=(0, 1))
+            b_gkj = tl.load(p_gkj, boundary_check=(0, 1))
+            tmp = exp(b_gn[None, :] - b_gkj)
+            b_kjg = b_kj * tmp
+            b_bjg = b_bj * tmp
+            # [BC, BC]
+            b_dAqikj = tl.load(p_dAqikj, boundary_check=(0, 1))
+            b_dAaibj = tl.load(p_dAaibj, boundary_check=(0, 1))
+            b_dAqibj = tl.load(p_dAqibj, boundary_check=(0, 1))
+            b_dAaikj = tl.load(p_dAaikj, boundary_check=(0, 1))
+            # [BC, BK]
+            b_dq += tl.dot(b_dAqikj, b_kjg)
+            b_dq += tl.dot(b_dAqibj, b_bjg)
+            # [BC, BC]
+            b_da += tl.dot(b_dAaibj, b_bjg)
+            b_da += tl.dot(b_dAaikj, b_kjg)
+        b_dq *= exp(b_gi - b_gn[None, :])
+        b_da *= exp(b_ge - b_gn[None, :])
+
+    NC = min(NC, tl.cdiv(T - i_t * BT, BC))
+    if i_i < NC - 1:
+        p_gn = gi + (min(i_t * BT + i_i * BC + BC, T) - 1)*stride_qk + o_k
+        p_gn = tl.max_contiguous(tl.multiple_of(p_gn, BK), BK)
+        # [BK,]
+        b_gn = tl.load(p_gn, mask=m_k, other=0)
+        for i_j in range(i_i + 1, NC):
+            m_j = (i_t * BT + i_j * BC + tl.arange(0, BC)) < T
+            p_qj = tl.make_block_ptr(q, (T, K), (stride_qk, 1), (i_t * BT + i_j * BC, i_k * BK), (BC, BK), (1, 0))
+            p_aj = tl.make_block_ptr(a, (T, K), (stride_qk, 1), (i_t * BT + i_j * BC, i_k * BK), (BC, BK), (1, 0))
+            p_gij = tl.make_block_ptr(gi, (T, K), (stride_qk, 1), (i_t * BT + i_j * BC, i_k * BK), (BC, BK), (1, 0))
+            p_gej = tl.make_block_ptr(ge, (T, K), (stride_qk, 1), (i_t * BT + i_j * BC, i_k * BK), (BC, BK), (1, 0))
+            p_dAqjki = tl.make_block_ptr(dAqk, (BT, T), (1, stride_A), (i_i*BC, i_t*BT + i_j*BC), (BC, BC), (0, 1))
+            p_dAajbi = tl.make_block_ptr(dAab, (BT, T), (1, stride_A), (i_i*BC, i_t*BT + i_j*BC), (BC, BC), (0, 1))
+            p_dAqjbi = tl.make_block_ptr(dAqb, (BT, T), (1, stride_A), (i_i*BC, i_t*BT + i_j*BC), (BC, BC), (0, 1))
+            p_dAajki = tl.make_block_ptr(dAak, (BT, T), (1, stride_A), (i_i*BC, i_t*BT + i_j*BC), (BC, BC), (0, 1))
+            b_qj = tl.load(p_qj, boundary_check=(0, 1))
+            b_aj = tl.load(p_aj, boundary_check=(0, 1))
+            b_gij = tl.load(p_gij, boundary_check=(0, 1))
+            b_gej = tl.load(p_gej, boundary_check=(0, 1))
+            b_gij = tl.where(m_j[:, None] & m_k, b_gij, float('-inf'))
+            b_gej = tl.where(m_j[:, None] & m_k, b_gej, float('-inf'))
+            b_qjg = b_qj * exp(b_gij - b_gn[None, :])
+            b_ajg = b_aj * exp(b_gej - b_gn[None, :])
+            # [BC, BC]
+            b_dAqjki = tl.load(p_dAqjki, boundary_check=(0, 1))
+            b_dAajbi = tl.load(p_dAajbi, boundary_check=(0, 1))
+            b_dAqjbi = tl.load(p_dAqjbi, boundary_check=(0, 1))
+            b_dAajki = tl.load(p_dAajki, boundary_check=(0, 1))
+            b_dk += tl.dot(b_dAqjki, b_qjg)
+            b_dk += tl.dot(b_dAajki, b_ajg)
+            b_db += tl.dot(b_dAqjbi, b_qjg)
+            b_db += tl.dot(b_dAajbi, b_ajg)
+        tmp = exp(b_gn[None, :] - b_gi)
+        b_dk *= tmp
+        b_db *= tmp
+
+    # intra chunk gradient calculation
+    for j in range(0, min(BC, T - i_t * BT - i_i * BC)):
+        # trick to index the block
+        if GATHER_SUPPORTED:
+            row_idx = tl.full([1, BK], j, dtype=tl.int16)
+            col_idx = tl.full([BC, 1], j, dtype=tl.int16)
+            row_idx_bc = tl.full([1, BC], j, dtype=tl.int16)
+            # [1, BK]
+            b_kj = gather(b_k, row_idx, axis=0)
+            b_bj = gather(b_b, row_idx, axis=0)
+            b_gij = gather(b_gi, row_idx, axis=0)
+            b_gej = gather(b_ge, row_idx, axis=0)
+            b_qj = gather(b_q, row_idx, axis=0)
+            b_aj = gather(b_a, row_idx, axis=0)
+            # [BC, 1]
+            b_dAqk_j = gather(b_dAqk, col_idx, axis=1)
+            b_dAab_j = gather(b_dAab, col_idx, axis=1)
+            b_dAqb_j = gather(b_dAqb, col_idx, axis=1)
+            b_dAak_j = gather(b_dAak, col_idx, axis=1)
+            # [1, BC] -> [BC, 1]
+            b_dA_qk_j = tl.sum(gather(b_dAqk, row_idx_bc, axis=0), 0)[:, None]
+            b_dA_qk_j = tl.sum(gather(b_dAqk, row_idx_bc, axis=0), 0)[:, None]
+            b_dA_ab_j = tl.sum(gather(b_dAab, row_idx_bc, axis=0), 0)[:, None]
+            b_dA_qb_j = tl.sum(gather(b_dAqb, row_idx_bc, axis=0), 0)[:, None]
+            b_dA_ak_j = tl.sum(gather(b_dAak, row_idx_bc, axis=0), 0)[:, None]
+        else:
+            mask_idx = tl.arange(0, BC) == j
+            b_kj = tl.sum(tl.where(mask_idx[:, None], b_k, 0), 0)[None, :]
+            b_bj = tl.sum(tl.where(mask_idx[:, None], b_b, 0), 0)[None, :]
+            b_gij = tl.sum(tl.where(mask_idx[:, None], b_gi, 0), 0)[None, :]
+            b_gej = tl.sum(tl.where(mask_idx[:, None], b_ge, 0), 0)[None, :]
+            b_dAqk_j = tl.sum(tl.where(mask_idx[None, :], b_dAqk, 0), 1)[:, None]
+            b_dAab_j = tl.sum(tl.where(mask_idx[None, :], b_dAab, 0), 1)[:, None]
+            b_dAqb_j = tl.sum(tl.where(mask_idx[None, :], b_dAqb, 0), 1)[:, None]
+            b_dAak_j = tl.sum(tl.where(mask_idx[None, :], b_dAak, 0), 1)[:, None]
+            b_dA_qk_j = tl.sum(tl.where(mask_idx[:, None], b_dAqk, 0), 0)[:, None]
+            b_dA_ab_j = tl.sum(tl.where(mask_idx[:, None], b_dAab, 0), 0)[:, None]
+            b_dA_qb_j = tl.sum(tl.where(mask_idx[:, None], b_dAqb, 0), 0)[:, None]
+            b_dA_ak_j = tl.sum(tl.where(mask_idx[:, None], b_dAak, 0), 0)[:, None]
+            # [1, BK] b_qj, b_aj
+            b_qj = tl.sum(tl.where(mask_idx[:, None], b_q, 0), 0)[None, :]
+            b_aj = tl.sum(tl.where(mask_idx[:, None], b_a, 0), 0)[None, :]
+        # tl.static_print(b_kj)
+        m_e = o_i[:, None] > j
+        m_i = o_i[:, None] >= j
+        tmp1 = exp(b_gi - b_gij)
+        tmp2 = exp(b_ge - b_gij)
+        b_dq += tl.where(m_i, b_dAqk_j * b_kj * tmp1, 0.)
+        b_dq += tl.where(m_i, b_dAqb_j * b_bj * tmp1, 0.)
+        b_da += tl.where(m_e, b_dAab_j * b_bj * tmp2, 0.)
+        b_da += tl.where(m_e, b_dAak_j * b_kj * tmp2, 0.)
+
+        m_i = o_i[:, None] <= j
+        m_e = o_i[:, None] < j
+        tmp1 = exp(b_gij - b_gi)
+        tmp2 = exp(b_gej - b_gi)
+        b_dk += tl.where(m_i, b_dA_qk_j * b_qj * tmp1, 0.)
+        b_dk += tl.where(m_e, b_dA_ak_j * b_aj * tmp2, 0.)
+        b_db += tl.where(m_i, b_dA_qb_j * b_qj * tmp1, 0.)
+        b_db += tl.where(m_e, b_dA_ab_j * b_aj * tmp2, 0.)
+    # post processing
+    p_dq = tl.make_block_ptr(dq, (T, K), (stride_qk, 1), (i_t * BT + i_i * BC, i_k * BK), (BC, BK), (1, 0))
+    p_dk = tl.make_block_ptr(dk, (T, K), (stride_qk, 1), (i_t * BT + i_i * BC, i_k * BK), (BC, BK), (1, 0))
+    p_da = tl.make_block_ptr(da, (T, K), (stride_qk, 1), (i_t * BT + i_i * BC, i_k * BK), (BC, BK), (1, 0))
+    p_db = tl.make_block_ptr(db, (T, K), (stride_qk, 1), (i_t * BT + i_i * BC, i_k * BK), (BC, BK), (1, 0))
+    p_dgk = tl.make_block_ptr(dgk, (T, K), (stride_qk, 1), (i_t * BT + i_i * BC, i_k * BK), (BC, BK), (1, 0))
+    p_dgk_offset = tl.make_block_ptr(dgk_offset, (T, K), (stride_qk, 1), (i_t * BT + i_i * BC, i_k * BK), (BC, BK), (1, 0))
+    p_dqg = tl.make_block_ptr(dqg, (T, K), (stride_qk, 1), (i_t * BT + i_i * BC, i_k * BK), (BC, BK), (1, 0))
+    p_dkg = tl.make_block_ptr(dkg, (T, K), (stride_qk, 1), (i_t * BT + i_i * BC, i_k * BK), (BC, BK), (1, 0))
+    p_dag = tl.make_block_ptr(dag, (T, K), (stride_qk, 1), (i_t * BT + i_i * BC, i_k * BK), (BC, BK), (1, 0))
+    p_dbg = tl.make_block_ptr(dbg, (T, K), (stride_qk, 1), (i_t * BT + i_i * BC, i_k * BK), (BC, BK), (1, 0))
+    p_gn = gi + (min(i_t * BT + BT, T) - 1)*stride_qk + o_k
+    p_gn = tl.max_contiguous(tl.multiple_of(p_gn, BK), BK)
+    b_gn = tl.load(p_gn, mask=m_k, other=0)
+    b_da += tl.load(p_dag, boundary_check=(0, 1)) * exp(b_ge)
+    b_dq += tl.load(p_dqg, boundary_check=(0, 1)) * exp(b_gi) * scale
+    tmp = exp(b_gn[None, :] - b_gi)
+    b_dk += tl.load(p_dkg, boundary_check=(0, 1)) * tmp
+    b_db += tl.load(p_dbg, boundary_check=(0, 1)) * tmp
+    tl.store(p_dq, (b_dq).to(p_dq.dtype.element_ty), boundary_check=(0, 1))
+    tl.store(p_dk, b_dk.to(p_dk.dtype.element_ty), boundary_check=(0, 1))
+    tl.store(p_da, b_da.to(p_da.dtype.element_ty), boundary_check=(0, 1))
+    tl.store(p_db, b_db.to(p_db.dtype.element_ty), boundary_check=(0, 1))
+    b_dgk = b_dq * b_q + b_da * b_a - b_dk * b_k - b_db * b_b
+    b_dgk_offset = b_da * b_a
+    tl.store(p_dgk, b_dgk.to(p_dgk.dtype.element_ty), boundary_check=(0, 1))
+    tl.store(p_dgk_offset, b_dgk_offset.to(p_dgk_offset.dtype.element_ty), boundary_check=(0, 1))
+
+
+@triton.heuristics({
+    'USE_OFFSETS': lambda args: args['offsets'] is not None,
+})
+@triton.autotune(
+    configs=[
+        triton.Config({'BK': BK}, num_warps=num_warps, num_stages=num_stages)
+        for num_warps in [2, 4, 8, 16, 32]
+        for num_stages in [2, 3, 4]
+        for BK in [32, 64]
+    ],
+    key=['BK', 'BT', 'K'],
+    use_cuda_graph=use_cuda_graph,
+)
+@triton.jit(do_not_specialize=['T'])
+def chunk_dplr_bwd_dgk_kernel(
+    dgk,
+    dgk_offset,
+    dgk_last,
+    dgk_output,
+    offsets,
+    indices,
+    T,
+    H: tl.constexpr,
+    K: tl.constexpr,
+    BT: tl.constexpr,
+    BK: tl.constexpr,
+    USE_OFFSETS: tl.constexpr,
+    HEAD_FIRST: tl.constexpr,
+):
+    i_t, i_k, i_bh = tl.program_id(0), tl.program_id(1), tl.program_id(2)
+    i_b, i_h = i_bh // H, i_bh % H
+    if USE_OFFSETS:
+        i_tg = i_t
+        i_n, i_t = tl.load(indices + i_t * 2).to(tl.int32), tl.load(indices + i_t * 2 + 1).to(tl.int32)
+        bos, eos = tl.load(offsets + i_n).to(tl.int32), tl.load(offsets + i_n + 1).to(tl.int32)
+        T = eos - bos
+        NT = tl.cdiv(T, BT)
+    else:
+        NT = tl.cdiv(T, BT)
+        i_tg = i_b * NT + i_t
+        bos, eos = i_b * T, i_b * T + T
+    T = eos - bos
+    stride_qk = K if HEAD_FIRST else H * K
+    dgk += i_bh * T * K if HEAD_FIRST else (bos * H + i_h) * K
+    dgk_offset += i_bh * T * K if HEAD_FIRST else (bos * H + i_h) * K
+    dgk_last += ((i_bh * NT + i_t) * K) if HEAD_FIRST else (i_tg * H + i_h) * K
+    dgk_output += i_bh * T * K if HEAD_FIRST else (bos * H + i_h) * K
+    p_dgk_last = dgk_last + tl.arange(0, BK) + i_k * BK
+    m_k = tl.arange(0, BK) + i_k * BK < K
+    b_dgk_last = tl.load(p_dgk_last, mask=m_k, other=0)
+    p_dgk_offset = tl.make_block_ptr(dgk_offset, (T, K), (stride_qk, 1), (i_t * BT, i_k * BK), (BT, BK), (1, 0))
+    p_dgk = tl.make_block_ptr(dgk, (T, K), (stride_qk, 1), (i_t * BT, i_k * BK), (BT, BK), (1, 0))
+    b_dgk = tl.load(p_dgk, boundary_check=(0, 1))
+    b_dgk_offset = tl.load(p_dgk_offset, boundary_check=(0, 1))
+    # m_inv_cumsum = (tl.arange(0, BT)[:, None] <= tl.arange(0, BT)[None, :]).to(tl.float32)
+    # b_dgk_cumsum = tl.dot(m_inv_cumsum, b_dgk, allow_tf32=False)
+    b_dgk_cumsum = tl.cumsum(b_dgk, 0, reverse=True)
+    b_dgk_cumsum += b_dgk_last[None, :]
+    b_dgk_cumsum -= b_dgk_offset
+    p_dgk_output = tl.make_block_ptr(dgk_output, (T, K), (stride_qk, 1), (i_t * BT, i_k * BK), (BT, BK), (1, 0))
+    tl.store(p_dgk_output, b_dgk_cumsum.to(p_dgk_output.dtype.element_ty), boundary_check=(0, 1))
+
+
+def chunk_dplr_bwd_dqk_intra(
+    q: torch.Tensor,
+    k: torch.Tensor,
+    a: torch.Tensor,
+    b: torch.Tensor,
+    gi: torch.Tensor,
+    ge: torch.Tensor,
+    dAqk: torch.Tensor,
+    dAqb: torch.Tensor,
+    dAak: torch.Tensor,
+    dAab: torch.Tensor,
+    dqg: torch.Tensor,
+    dkg: torch.Tensor,
+    dag: torch.Tensor,
+    dbg: torch.Tensor,
+    dgk_last: torch.Tensor,
+    offsets: Optional[torch.LongTensor] = None,
+    indices: Optional[torch.LongTensor] = None,
+    head_first: bool = True,
+    scale: float = 1.0,
+    chunk_size: int = 64,
+):
+    if head_first:
+        B, H, T, K = q.shape
+    else:
+        B, T, H, K = q.shape
+    BT = min(chunk_size, max(16, triton.next_power_of_2(T)))
+    BC = min(16, BT)
+    BK = min(64, triton.next_power_of_2(K)) if check_shared_mem() else min(32, triton.next_power_of_2(K))
+    NT = triton.cdiv(T, BT) if offsets is None else len(indices)
+    NC = triton.cdiv(BT, BC)
+    NK = triton.cdiv(K, BK)
+
+    dq = torch.empty_like(q)
+    dk = torch.empty_like(k)
+    da = torch.empty_like(a)
+    db = torch.empty_like(b)
+    dgk = torch.empty_like(gi, dtype=torch.float)
+    dgk_offset = torch.empty_like(gi, dtype=torch.float)
+
+    grid = (NK, NT * NC, B * H)
+    chunk_dplr_bwd_kernel_intra[grid](
+        q=q,
+        k=k,
+        a=a,
+        b=b,
+        gi=gi,
+        ge=ge,
+        dAqk=dAqk,
+        dAqb=dAqb,
+        dAak=dAak,
+        dAab=dAab,
+        dq=dq,
+        dk=dk,
+        dgk=dgk,
+        dgk_offset=dgk_offset,
+        dqg=dqg,
+        dkg=dkg,
+        dag=dag,
+        dbg=dbg,
+        da=da,
+        db=db,
+        offsets=offsets,
+        indices=indices,
+        scale=scale,
+        T=T,
+        H=H,
+        K=K,
+        BT=BT,
+        BC=BC,
+        BK=BK,
+        NC=NC,
+        HEAD_FIRST=head_first,
+        GATHER_SUPPORTED=is_gather_supported
+    )
+
+    def grid2(meta): return (NT, triton.cdiv(K, meta['BK']), B * H)
+    dgk_output = torch.empty_like(dgk)
+
+    chunk_dplr_bwd_dgk_kernel[grid2](
+        dgk=dgk,
+        dgk_offset=dgk_offset,
+        dgk_last=dgk_last,
+        dgk_output=dgk_output,
+        offsets=offsets,
+        indices=indices,
+        T=T,
+        H=H,
+        K=K,
+        BT=BT,
+        HEAD_FIRST=head_first
+    )
+    return dq, dk, da, db, dgk_output

fla/ops/generalized_delta_rule/dplr/chunk_A_fwd.py

@@ -0,0 +1,324 @@
+# -*- coding: utf-8 -*-
+# Copyright (c) 2023-2025, Songlin Yang, Yu Zhang
+
+from typing import Optional
+
+import torch
+import triton
+import triton.language as tl
+
+from fla.ops.utils.op import exp, gather
+from fla.utils import is_gather_supported, use_cuda_graph
+
+
+@triton.heuristics({
+    'USE_OFFSETS': lambda args: args['offsets'] is not None
+})
+@triton.autotune(
+    configs=[
+        triton.Config({'BK': BK}, num_warps=num_warps, num_stages=num_stages)
+        for BK in [32, 64]
+        for num_warps in [2, 4, 8, 16]
+        for num_stages in [2, 3, 4]
+    ],
+    key=['BC', 'K'],
+    use_cuda_graph=use_cuda_graph,
+)
+@triton.jit(do_not_specialize=['T'])
+def chunk_dplr_fwd_A_kernel_intra_sub_inter(
+    q,
+    k,
+    a,
+    b,
+    gi,  # cumsum
+    ge,  # before cumsum
+    Aqk,
+    Aqb,
+    Aab,
+    Aak,
+    offsets,
+    indices,
+    scale: tl.constexpr,
+    T,
+    H: tl.constexpr,
+    K: tl.constexpr,
+    BT: tl.constexpr,
+    BC: tl.constexpr,
+    BK: tl.constexpr,
+    NC: tl.constexpr,
+    USE_OFFSETS: tl.constexpr,
+    HEAD_FIRST: tl.constexpr,
+):
+    i_t, i_c, i_bh = tl.program_id(0), tl.program_id(1), tl.program_id(2)
+    i_b, i_h = i_bh // H, i_bh % H
+    i_i, i_j = i_c // NC, i_c % NC
+    if USE_OFFSETS:
+        i_n, i_t = tl.load(indices + i_t * 2).to(tl.int32), tl.load(indices + i_t * 2 + 1).to(tl.int32)
+        bos, eos = tl.load(offsets + i_n).to(tl.int32), tl.load(offsets + i_n + 1).to(tl.int32)
+        T = eos - bos
+    else:
+        bos, eos = i_b * T, i_b * T + T
+
+    if i_t * BT + i_i * BC >= T:
+        return
+    if i_i <= i_j:
+        return
+
+    b_Aqk = tl.zeros([BC, BC], dtype=tl.float32)
+    b_Aqb = tl.zeros([BC, BC], dtype=tl.float32)
+    b_Aab = tl.zeros([BC, BC], dtype=tl.float32)
+    b_Aak = tl.zeros([BC, BC], dtype=tl.float32)
+    for i_k in range(tl.cdiv(K, BK)):
+        o_k = i_k * BK + tl.arange(0, BK)
+        m_k = o_k < K
+
+        if HEAD_FIRST:
+            p_q = tl.make_block_ptr(q + i_bh * T*K, (T, K), (K, 1), (i_t * BT + i_i * BC, i_k * BK), (BC, BK), (1, 0))
+            p_a = tl.make_block_ptr(a + i_bh * T*K, (T, K), (K, 1), (i_t * BT + i_i * BC, i_k * BK), (BC, BK), (1, 0))
+            p_gq_i = tl.make_block_ptr(gi + i_bh * T*K, (T, K), (K, 1), (i_t * BT + i_i * BC, i_k * BK), (BC, BK), (1, 0))
+            p_gq_e = tl.make_block_ptr(ge + i_bh * T*K, (T, K), (K, 1), (i_t * BT + i_i * BC, i_k * BK), (BC, BK), (1, 0))
+            p_k = tl.make_block_ptr(k + i_bh * T*K, (K, T), (1, K), (i_k * BK, i_t * BT + i_j * BC), (BK, BC), (0, 1))
+            p_b = tl.make_block_ptr(b + i_bh * T*K, (K, T), (1, K), (i_k * BK, i_t * BT + i_j * BC), (BK, BC), (0, 1))
+            p_gk = tl.make_block_ptr(gi + i_bh * T*K, (K, T), (1, K), (i_k * BK, i_t * BT + i_j * BC), (BK, BC), (0, 1))
+            p_gn = tl.max_contiguous(tl.multiple_of(gi + (i_bh * T + i_t * BT + i_i * BC - 1) * K + o_k, BK), BK)
+        else:
+            p_q = tl.make_block_ptr(q + (bos*H+i_h)*K, (T, K), (H*K, 1), (i_t * BT + i_i * BC, i_k * BK), (BC, BK), (1, 0))
+            p_a = tl.make_block_ptr(a + (bos*H+i_h)*K, (T, K), (H*K, 1), (i_t * BT + i_i * BC, i_k * BK), (BC, BK), (1, 0))
+            p_gq_i = tl.make_block_ptr(gi + (bos*H+i_h)*K, (T, K), (H*K, 1), (i_t * BT + i_i * BC, i_k * BK), (BC, BK), (1, 0))
+            p_gq_e = tl.make_block_ptr(ge + (bos*H+i_h)*K, (T, K), (H*K, 1), (i_t * BT + i_i * BC, i_k * BK), (BC, BK), (1, 0))
+            p_k = tl.make_block_ptr(k + (bos*H+i_h)*K, (K, T), (1, H*K), (i_k * BK, i_t * BT + i_j * BC), (BK, BC), (0, 1))
+            p_b = tl.make_block_ptr(b + (bos*H+i_h)*K, (K, T), (1, H*K), (i_k * BK, i_t * BT + i_j * BC), (BK, BC), (0, 1))
+            p_gk = tl.make_block_ptr(gi + (bos*H+i_h)*K, (K, T), (1, H*K), (i_k * BK, i_t * BT + i_j * BC), (BK, BC), (0, 1))
+            p_gn = gi + (bos + i_t * BT + i_i * BC - 1) * H*K + i_h * K + o_k
+        # [BK,]
+        b_gn = tl.load(p_gn, mask=m_k, other=0).to(tl.float32)
+        # [BC, BK]
+        b_q = tl.load(p_q, boundary_check=(0, 1))
+        b_a = tl.load(p_a, boundary_check=(0, 1))
+        b_gq_i = tl.load(p_gq_i, boundary_check=(0, 1))
+        b_gq_e = tl.load(p_gq_e, boundary_check=(0, 1))
+        b_ag = b_a * exp(b_gq_e - b_gn[None, :])
+        b_qg = b_q * exp(b_gq_i - b_gn[None, :]) * scale
+        # [BK, BC]
+        b_k = tl.load(p_k, boundary_check=(0, 1))
+        b_b = tl.load(p_b, boundary_check=(0, 1))
+        b_gk = tl.load(p_gk, boundary_check=(0, 1)).to(tl.float32)
+        tmp = exp(b_gn[:, None] - b_gk)
+        b_kg = b_k * tmp
+        b_bg = b_b * tmp
+        # [BC, BC] using tf32 to improve precision here.
+        b_Aab += tl.dot(b_ag, b_bg)
+        b_Aak += tl.dot(b_ag, b_kg)
+        b_Aqk += tl.dot(b_qg, b_kg)
+        b_Aqb += tl.dot(b_qg, b_bg)
+
+    if HEAD_FIRST:
+        p_Aqk = tl.make_block_ptr(Aqk + i_bh*T*BT, (T, BT), (BT, 1), (i_t * BT + i_i * BC, i_j * BC), (BC, BC), (1, 0))
+        p_Aqb = tl.make_block_ptr(Aqb + i_bh*T*BT, (T, BT), (BT, 1), (i_t * BT + i_i * BC, i_j * BC), (BC, BC), (1, 0))
+        p_Aab = tl.make_block_ptr(Aab + i_bh*T*BT, (T, BT), (BT, 1), (i_t * BT + i_i * BC, i_j * BC), (BC, BC), (1, 0))
+        p_Aak = tl.make_block_ptr(Aak + i_bh*T*BT, (T, BT), (BT, 1), (i_t * BT + i_i * BC, i_j * BC), (BC, BC), (1, 0))
+    else:
+        p_Aqk = tl.make_block_ptr(Aqk + (bos*H+i_h)*BT, (T, BT), (H*BT, 1), (i_t * BT + i_i * BC, i_j * BC), (BC, BC), (1, 0))
+        p_Aqb = tl.make_block_ptr(Aqb + (bos*H+i_h)*BT, (T, BT), (H*BT, 1), (i_t * BT + i_i * BC, i_j * BC), (BC, BC), (1, 0))
+        p_Aab = tl.make_block_ptr(Aab + (bos*H+i_h)*BT, (T, BT), (H*BT, 1), (i_t * BT + i_i * BC, i_j * BC), (BC, BC), (1, 0))
+        p_Aak = tl.make_block_ptr(Aak + (bos*H+i_h)*BT, (T, BT), (H*BT, 1), (i_t * BT + i_i * BC, i_j * BC), (BC, BC), (1, 0))
+    tl.store(p_Aqk, b_Aqk.to(Aqk.dtype.element_ty, fp_downcast_rounding="rtne"), boundary_check=(0, 1))
+    tl.store(p_Aqb, b_Aqb.to(Aqb.dtype.element_ty, fp_downcast_rounding="rtne"), boundary_check=(0, 1))
+    tl.store(p_Aab, b_Aab.to(Aab.dtype.element_ty, fp_downcast_rounding="rtne"), boundary_check=(0, 1))
+    tl.store(p_Aak, b_Aak.to(Aak.dtype.element_ty, fp_downcast_rounding="rtne"), boundary_check=(0, 1))
+
+
+@triton.heuristics({
+    'USE_OFFSETS': lambda args: args['offsets'] is not None
+})
+@triton.autotune(
+    configs=[
+        triton.Config({}, num_warps=num_warps, num_stages=num_stages)
+        for num_warps in [2, 4, 8, 16, 32]
+        for num_stages in [2, 3, 4]
+    ],
+    key=['BK', 'BT'],
+    use_cuda_graph=use_cuda_graph,
+)
+@triton.jit(do_not_specialize=['T'])
+def chunk_dplr_fwd_A_kernel_intra_sub_intra(
+    q,
+    k,
+    a,
+    b,
+    gi,
+    ge,
+    qg,
+    kg,
+    ag,
+    bg,
+    Aqk,
+    Aqb,
+    Aab,
+    Aak,
+    offsets,
+    indices,
+    scale: tl.constexpr,
+    T,
+    H: tl.constexpr,
+    K: tl.constexpr,
+    BT: tl.constexpr,
+    BC: tl.constexpr,
+    BK: tl.constexpr,
+    NC: tl.constexpr,
+    USE_OFFSETS: tl.constexpr,
+    HEAD_FIRST: tl.constexpr,
+    GATHER_SUPPORTED: tl.constexpr
+):
+    i_t, i_i, i_bh = tl.program_id(0), tl.program_id(1), tl.program_id(2)
+    i_b, i_h = i_bh // H, i_bh % H
+    i_j = i_i
+    if USE_OFFSETS:
+        i_n, i_t = tl.load(indices + i_t * 2).to(tl.int32), tl.load(indices + i_t * 2 + 1).to(tl.int32)
+        bos, eos = tl.load(offsets + i_n).to(tl.int32), tl.load(offsets + i_n + 1).to(tl.int32)
+        T = eos - bos
+    else:
+        bos, eos = i_b * T, i_b * T + T
+
+    if i_t * BT + i_i * BC >= T:
+        return
+
+    o_i = tl.arange(0, BC)
+    o_k = tl.arange(0, BK)
+    m_k = o_k < K
+    m_A = (i_t * BT + i_i * BC + tl.arange(0, BC)) < T
+    last_idx = min((i_t+1) * BT, T) - 1
+    if HEAD_FIRST:
+        o_A = i_bh * T*BT + (i_t * BT + i_i * BC + tl.arange(0, BC)) * BT + i_j * BC
+        p_q = tl.make_block_ptr(q + i_bh * T*K, (T, K), (K, 1), (i_t * BT + i_i * BC, 0), (BC, BK), (1, 0))
+        p_k = tl.make_block_ptr(k + i_bh * T*K, (T, K), (K, 1), (i_t * BT + i_i * BC, 0), (BC, BK), (1, 0))
+        p_a = tl.make_block_ptr(a + i_bh * T*K, (T, K), (K, 1), (i_t * BT + i_i * BC, 0), (BC, BK), (1, 0))
+        p_b = tl.make_block_ptr(b + i_bh * T*K, (T, K), (K, 1), (i_t * BT + i_i * BC, 0), (BC, BK), (1, 0))
+        p_gi = tl.make_block_ptr(gi + i_bh * T*K, (T, K), (K, 1), (i_t * BT + i_i * BC, 0), (BC, BK), (1, 0))
+        p_ge = tl.make_block_ptr(ge + i_bh * T*K, (T, K), (K, 1), (i_t * BT + i_i * BC, 0), (BC, BK), (1, 0))
+        p_g_last = gi + i_bh * T*K + last_idx * K + tl.arange(0, BK)
+        b_g_last = tl.load(p_g_last, mask=m_k, other=0)
+
+        p_qg = tl.make_block_ptr(qg + i_bh * T*K, (T, K), (K, 1), (i_t * BT + i_i * BC, 0), (BC, BK), (1, 0))
+        p_kg = tl.make_block_ptr(kg + i_bh * T*K, (T, K), (K, 1), (i_t * BT + i_i * BC, 0), (BC, BK), (1, 0))
+        p_ag = tl.make_block_ptr(ag + i_bh * T*K, (T, K), (K, 1), (i_t * BT + i_i * BC, 0), (BC, BK), (1, 0))
+        p_bg = tl.make_block_ptr(bg + i_bh * T*K, (T, K), (K, 1), (i_t * BT + i_i * BC, 0), (BC, BK), (1, 0))
+    else:
+        o_A = (bos + i_t * BT + i_i * BC + tl.arange(0, BC)) * H*BT + i_h * BT + i_j * BC
+        p_q = tl.make_block_ptr(q + (bos * H + i_h) * K, (T, K), (H*K, 1), (i_t * BT + i_i * BC, 0), (BC, BK), (1, 0))
+        p_k = tl.make_block_ptr(k + (bos * H + i_h) * K, (T, K), (H*K, 1), (i_t * BT + i_i * BC, 0), (BC, BK), (1, 0))
+        p_a = tl.make_block_ptr(a + (bos * H + i_h) * K, (T, K), (H*K, 1), (i_t * BT + i_i * BC, 0), (BC, BK), (1, 0))
+        p_b = tl.make_block_ptr(b + (bos * H + i_h) * K, (T, K), (H*K, 1), (i_t * BT + i_i * BC, 0), (BC, BK), (1, 0))
+        p_gi = tl.make_block_ptr(gi + (bos * H + i_h) * K, (T, K), (H*K, 1), (i_t * BT + i_i * BC, 0), (BC, BK), (1, 0))
+        p_ge = tl.make_block_ptr(ge + (bos * H + i_h) * K, (T, K), (H*K, 1), (i_t * BT + i_i * BC, 0), (BC, BK), (1, 0))
+        p_g_last = gi + (bos * H + i_h) * K + last_idx * H * K + tl.arange(0, BK)
+        b_g_last = tl.load(p_g_last, mask=m_k, other=0)
+        p_qg = tl.make_block_ptr(qg + (bos * H + i_h) * K, (T, K), (H*K, 1), (i_t * BT + i_i * BC, 0), (BC, BK), (1, 0))
+        p_kg = tl.make_block_ptr(kg + (bos * H + i_h) * K, (T, K), (H*K, 1), (i_t * BT + i_i * BC, 0), (BC, BK), (1, 0))
+        p_ag = tl.make_block_ptr(ag + (bos * H + i_h) * K, (T, K), (H*K, 1), (i_t * BT + i_i * BC, 0), (BC, BK), (1, 0))
+        p_bg = tl.make_block_ptr(bg + (bos * H + i_h) * K, (T, K), (H*K, 1), (i_t * BT + i_i * BC, 0), (BC, BK), (1, 0))
+
+    b_q = tl.load(p_q, boundary_check=(0, 1))
+    b_q = b_q * scale
+    b_k = tl.load(p_k, boundary_check=(0, 1))
+    b_a = tl.load(p_a, boundary_check=(0, 1))
+    b_b = tl.load(p_b, boundary_check=(0, 1))
+    b_gi = tl.load(p_gi, boundary_check=(0, 1)).to(tl.float32)
+    b_ge = tl.load(p_ge, boundary_check=(0, 1)).to(tl.float32)
+
+    # deal with decay term.
+    g_exp = exp(b_gi)
+    g_exp_inv = exp(-b_gi + b_g_last[None, :])
+    b_qg = b_q * g_exp
+    b_kg = b_k * g_exp_inv
+    b_bg = b_b * g_exp_inv
+    b_ag = b_a * exp(b_ge)
+    tl.store(p_qg, b_qg.to(p_qg.dtype.element_ty, fp_downcast_rounding="rtne"), boundary_check=(0, 1))
+    tl.store(p_bg, b_bg.to(p_bg.dtype.element_ty, fp_downcast_rounding="rtne"), boundary_check=(0, 1))
+    tl.store(p_ag, b_ag.to(p_ag.dtype.element_ty, fp_downcast_rounding="rtne"), boundary_check=(0, 1))
+    tl.store(p_kg, b_kg.to(p_kg.dtype.element_ty, fp_downcast_rounding="rtne"), boundary_check=(0, 1))
+    # tl.debug_barrier()
+
+    b_q = b_q.to(b_k.dtype)
+    # inner attn
+    for j in range(0, min(BC, T - i_t * BT - i_i * BC)):
+        # a trick to index the j-th row of b_k, b_g, b_b
+        if GATHER_SUPPORTED:
+            row_idx = tl.full([1, BK], j, dtype=tl.int16)
+            # [1, BK]
+            b_k_j = gather(b_k, row_idx, axis=0)
+            b_gk_j = gather(b_gi, row_idx, axis=0)
+            b_b_j = gather(b_b, row_idx, axis=0)
+        else:
+            mask = tl.arange(0, BC) == j
+            b_k_j = tl.sum(tl.where(mask[:, None], b_k, 0), 0)[None, :]
+            b_gk_j = tl.sum(tl.where(mask[:, None], b_gi, 0), 0)[None, :]
+            b_b_j = tl.sum(tl.where(mask[:, None], b_b, 0), 0)[None, :]
+        mask = tl.arange(0, BC) == j
+        tmp = exp(b_gi - b_gk_j)
+        b_A_qk = tl.sum(b_q * b_k_j * tmp, 1)
+        b_A_qk = tl.where(o_i >= j, b_A_qk, 0.)
+        b_A_qb = tl.sum(b_q * b_b_j * tmp, 1)
+        b_A_qb = tl.where(o_i >= j, b_A_qb, 0.)
+        tmp2 = exp(b_ge - b_gk_j)
+        b_A_ak = tl.sum(b_a * b_k_j * tmp2, 1)
+        b_A_ak = tl.where(o_i > j, b_A_ak, 0.)
+        b_A_ab = tl.sum(b_a * b_b_j * tmp2, 1)
+        b_A_ab = tl.where(o_i > j, b_A_ab, 0.)
+        tl.store(Aqk + o_A + j, b_A_qk.to(dtype=Aqk.dtype.element_ty, fp_downcast_rounding="rtne"), mask=m_A)
+        tl.store(Aqb + o_A + j, b_A_qb.to(dtype=Aqb.dtype.element_ty, fp_downcast_rounding="rtne"), mask=m_A)
+        tl.store(Aab + o_A + j, b_A_ab.to(dtype=Aqb.dtype.element_ty, fp_downcast_rounding="rtne"), mask=m_A)
+        tl.store(Aak + o_A + j, b_A_ak.to(dtype=Aqk.dtype.element_ty, fp_downcast_rounding="rtne"), mask=m_A)
+
+
+def chunk_fwd_intra_dplr_fn(
+    q: torch.Tensor,
+    k: torch.Tensor,
+    a: torch.Tensor,
+    b: torch.Tensor,
+    gi: torch.Tensor,
+    ge: torch.Tensor,
+    scale: float,
+    chunk_size: int,
+    offsets: Optional[torch.LongTensor] = None,
+    indices: Optional[torch.LongTensor] = None,
+    head_first: bool = True,
+):
+    if head_first:
+        B, H, T, K = k.shape
+    else:
+        B, T, H, K = k.shape
+    BT = min(chunk_size, max(16, triton.next_power_of_2(T)))
+    NT = triton.cdiv(T, BT) if offsets is None else len(indices)
+    BC = min(16, BT)
+    NC = triton.cdiv(BT, BC)
+
+    Aqk = q.new_empty(B, *((H, T) if head_first else (T, H)), BT, dtype=q.dtype)
+    Aqb = q.new_empty(B, *((H, T) if head_first else (T, H)), BT, dtype=q.dtype)
+    # involving matrix inverse and it'd be better to use float here.
+    Aab = q.new_empty(B, *((H, T) if head_first else (T, H)), BT, dtype=torch.float)
+    Aak = q.new_empty(B, *((H, T) if head_first else (T, H)), BT, dtype=torch.float)
+    grid = (NT, NC * NC, B * H)
+
+    chunk_dplr_fwd_A_kernel_intra_sub_inter[grid](
+        q=q, k=k, a=a, b=b, gi=gi, ge=ge, Aqk=Aqk, Aqb=Aqb, Aab=Aab, Aak=Aak,
+        offsets=offsets, indices=indices,
+        scale=scale,
+        T=T, H=H, K=K, BT=BT, BC=BC, NC=NC,
+        HEAD_FIRST=head_first
+    )
+    grid = (NT, NC, B * H)
+    BK = triton.next_power_of_2(K)
+    qg = torch.empty_like(q)
+    kg = torch.empty_like(k, dtype=q.dtype)
+    ag = torch.empty_like(a, dtype=q.dtype)
+    bg = torch.empty_like(b, dtype=q.dtype)
+    chunk_dplr_fwd_A_kernel_intra_sub_intra[grid](
+        q=q, k=k, a=a, b=b, gi=gi, ge=ge, Aqk=Aqk, Aqb=Aqb, Aab=Aab, Aak=Aak,
+        qg=qg, kg=kg, ag=ag, bg=bg,
+        offsets=offsets, indices=indices,
+        scale=scale,
+        T=T, H=H, K=K, BT=BT, BC=BC, BK=BK, HEAD_FIRST=head_first, NC=NC,
+        GATHER_SUPPORTED=is_gather_supported
+    )
+    return Aab, Aqk, Aak, Aqb, qg, kg, ag, bg

fla/ops/generalized_delta_rule/dplr/chunk_h_bwd.py

@@ -0,0 +1,196 @@
+# -*- coding: utf-8 -*-
+# Copyright (c) 2023-2025, Songlin Yang, Yu Zhang
+
+from typing import Optional, Tuple
+
+import torch
+import triton
+import triton.language as tl
+
+from fla.ops.common.utils import prepare_chunk_offsets
+from fla.ops.utils.op import exp
+from fla.utils import check_shared_mem, use_cuda_graph
+
+
+@triton.heuristics({
+    'USE_FINAL_STATE_GRADIENT': lambda args: args['dht'] is not None,
+    'USE_INITIAL_STATE': lambda args: args['dh0'] is not None,
+    'USE_OFFSETS': lambda args: args['offsets'] is not None,
+})
+@triton.autotune(
+    configs=[
+        triton.Config({}, num_warps=num_warps, num_stages=num_stages)
+        for num_warps in [2, 4, 8, 16, 32]
+        for num_stages in [2, 3, 4]
+    ],
+    key=['BT', 'BK', 'BV', "V"],
+    use_cuda_graph=use_cuda_graph,
+)
+@triton.jit(do_not_specialize=['T'])
+def chunk_dplr_bwd_kernel_dhu(
+    qg,
+    bg,
+    w,
+    gk,
+    dht,
+    dh0,
+    do,
+    dh,
+    dv,
+    dv2,
+    offsets,
+    chunk_offsets,
+    T,
+    H: tl.constexpr,
+    K: tl.constexpr,
+    V: tl.constexpr,
+    BT: tl.constexpr,
+    BC: tl.constexpr,
+    BK: tl.constexpr,
+    BV: tl.constexpr,
+    USE_FINAL_STATE_GRADIENT: tl.constexpr,
+    USE_INITIAL_STATE: tl.constexpr,
+    USE_OFFSETS: tl.constexpr,
+    HEAD_FIRST: tl.constexpr
+):
+    i_k, i_v, i_nh = tl.program_id(0), tl.program_id(1), tl.program_id(2)
+    i_n, i_h = i_nh // H, i_nh % H
+    if USE_OFFSETS:
+        bos, eos = tl.load(offsets + i_n).to(tl.int32), tl.load(offsets + i_n + 1).to(tl.int32)
+        T = eos - bos
+        NT = tl.cdiv(T, BT)
+        boh = tl.load(chunk_offsets + i_n).to(tl.int32)
+    else:
+        bos, eos = i_n * T, i_n * T + T
+        NT = tl.cdiv(T, BT)
+        boh = i_n * NT
+
+    # [BK, BV]
+    b_dh = tl.zeros([BK, BV], dtype=tl.float32)
+    if USE_FINAL_STATE_GRADIENT:
+        p_dht = tl.make_block_ptr(dht + i_nh * K*V, (K, V), (V, 1), (i_k * BK, i_v * BV), (BK, BV), (1, 0))
+        b_dh += tl.load(p_dht, boundary_check=(0, 1))
+
+    mask_k = tl.arange(0, BK) < K
+    for i_t in range(NT - 1, -1, -1):
+        if HEAD_FIRST:
+            p_dh = tl.make_block_ptr(dh + (i_nh * NT + i_t) * K*V, (K, V), (V, 1), (i_k * BK, i_v * BV), (BK, BV), (1, 0))
+        else:
+            p_dh = tl.make_block_ptr(dh + ((boh+i_t) * H + i_h) * K*V, (K, V), (V, 1), (i_k * BK, i_v * BV), (BK, BV), (1, 0))
+        tl.store(p_dh, b_dh.to(p_dh.dtype.element_ty), boundary_check=(0, 1))
+        b_dh_tmp = tl.zeros([BK, BV], dtype=tl.float32)
+        for i_c in range(tl.cdiv(BT, BC) - 1, -1, -1):
+            if HEAD_FIRST:
+                p_qg = tl.make_block_ptr(qg + i_nh * T*K, (K, T), (1, K), (i_k * BK, i_t * BT + i_c * BC), (BK, BC), (0, 1))
+                p_bg = tl.make_block_ptr(bg + i_nh * T*K, (T, K), (K, 1), (i_t * BT + i_c * BC, i_k * BK), (BC, BK), (1, 0))
+                p_w = tl.make_block_ptr(w + i_nh * T*K, (K, T), (1, K), (i_k * BK, i_t * BT + i_c * BC), (BK, BC), (0, 1))
+                p_dv = tl.make_block_ptr(dv + i_nh * T*V, (T, V), (V, 1), (i_t * BT + i_c * BC, i_v * BV), (BC, BV), (1, 0))
+                p_do = tl.make_block_ptr(do + i_nh * T*V, (T, V), (V, 1), (i_t * BT + i_c * BC, i_v * BV), (BC, BV), (1, 0))
+                p_dv2 = tl.make_block_ptr(dv2 + i_nh * T*V, (T, V), (V, 1), (i_t * BT + i_c * BC, i_v * BV), (BC, BV), (1, 0))
+            else:
+                p_qg = tl.make_block_ptr(qg+(bos*H+i_h)*K, (K, T), (1, H*K), (i_k * BK, i_t * BT + i_c * BC), (BK, BC), (0, 1))
+                p_bg = tl.make_block_ptr(bg+(bos*H+i_h)*K, (T, K), (H*K, 1), (i_t * BT + i_c * BC, i_k * BK), (BC, BK), (1, 0))
+                p_w = tl.make_block_ptr(w+(bos*H+i_h)*K, (K, T), (1, H*K), (i_k * BK, i_t * BT + i_c * BC), (BK, BC), (0, 1))
+                p_dv = tl.make_block_ptr(dv+(bos*H+i_h)*V, (T, V), (H*V, 1), (i_t*BT + i_c * BC, i_v * BV), (BC, BV), (1, 0))
+                p_do = tl.make_block_ptr(do+(bos*H+i_h)*V, (T, V), (H*V, 1), (i_t*BT + i_c * BC, i_v * BV), (BC, BV), (1, 0))
+                p_dv2 = tl.make_block_ptr(dv2+(bos*H+i_h)*V, (T, V), (H*V, 1), (i_t*BT + i_c * BC, i_v * BV), (BC, BV), (1, 0))
+            # [BK, BT]
+            b_qg = tl.load(p_qg, boundary_check=(0, 1))
+            # [BT, BK]
+            b_bg = tl.load(p_bg, boundary_check=(0, 1))
+            b_w = tl.load(p_w, boundary_check=(0, 1))
+            # [BT, V]
+            b_do = tl.load(p_do, boundary_check=(0, 1))
+            b_dv = tl.load(p_dv, boundary_check=(0, 1))
+            b_dv2 = b_dv + tl.dot(b_bg, b_dh.to(b_bg.dtype))
+            tl.store(p_dv2, b_dv2.to(p_dv.dtype.element_ty), boundary_check=(0, 1))
+            # [BK, BV]
+            b_dh_tmp += tl.dot(b_qg, b_do.to(b_qg.dtype))
+            b_dh_tmp += tl.dot(b_w, b_dv2.to(b_qg.dtype))
+        last_idx = min((i_t + 1) * BT, T) - 1
+        if HEAD_FIRST:
+            bg_last = tl.load(gk + (i_nh * T + last_idx) * K + tl.arange(0, BK), mask=mask_k)
+        else:
+            bg_last = tl.load(gk + ((bos + last_idx) * H + i_h) * K + tl.arange(0, BK), mask=mask_k)
+        b_dh *= exp(bg_last)[:, None]
+        b_dh += b_dh_tmp
+
+    if USE_INITIAL_STATE:
+        p_dh0 = tl.make_block_ptr(dh0 + i_nh * K*V, (K, V), (V, 1), (i_k * BK, i_v * BV), (BK, BV), (1, 0))
+        tl.store(p_dh0, b_dh.to(p_dh0.dtype.element_ty), boundary_check=(0, 1))
+
+
+def chunk_dplr_bwd_dhu(
+    qg: torch.Tensor,
+    bg: torch.Tensor,
+    w: torch.Tensor,
+    gk: torch.Tensor,
+    h0: torch.Tensor,
+    dht: Optional[torch.Tensor],
+    do: torch.Tensor,
+    dv: torch.Tensor,
+    offsets: Optional[torch.LongTensor] = None,
+    indices: Optional[torch.LongTensor] = None,
+    head_first: bool = True,
+    chunk_size: int = 64
+) -> Tuple[torch.Tensor, torch.Tensor, torch.Tensor]:
+    if head_first:
+        B, H, T, K, V = *qg.shape, do.shape[-1]
+    else:
+        B, T, H, K, V = *qg.shape, do.shape[-1]
+    BT = min(chunk_size, max(triton.next_power_of_2(T), 16))
+    BK = triton.next_power_of_2(K)
+    assert BK <= 256, "current kernel does not support head dimension being larger than 256."
+    # H100
+    if check_shared_mem('hopper', qg.device.index):
+        BV = 64
+        BC = 64 if K <= 128 else 32
+    elif check_shared_mem('ampere', qg.device.index):  # A100
+        BV = 32
+        BC = 32
+    else:  # Etc: 4090
+        BV = 16
+        BC = 16
+
+    # N: the actual number of sequences in the batch with either equal or variable lengths
+    if offsets is None:
+        N, NT, chunk_offsets = B, triton.cdiv(T, BT), None
+    else:
+        N, NT, chunk_offsets = len(offsets) - 1, len(indices), prepare_chunk_offsets(offsets, BT)
+
+    BC = min(BT, BC)
+    NK, NV = triton.cdiv(K, BK), triton.cdiv(V, BV)
+    assert NK == 1, 'NK > 1 is not supported because it involves time-consuming synchronization'
+
+    if head_first:
+        dh = qg.new_empty(B, H, NT, K, V)
+    else:
+        dh = qg.new_empty(B, NT, H, K, V)
+    dh0 = torch.empty_like(h0, dtype=torch.float32) if h0 is not None else None
+    dv2 = torch.zeros_like(dv)
+
+    grid = (NK, NV, N * H)
+    chunk_dplr_bwd_kernel_dhu[grid](
+        qg=qg,
+        bg=bg,
+        w=w,
+        gk=gk,
+        dht=dht,
+        dh0=dh0,
+        do=do,
+        dh=dh,
+        dv=dv,
+        dv2=dv2,
+        offsets=offsets,
+        chunk_offsets=chunk_offsets,
+        T=T,
+        H=H,
+        K=K,
+        V=V,
+        BT=BT,
+        BC=BC,
+        BK=BK,
+        BV=BV,
+        HEAD_FIRST=head_first
+    )
+    return dh, dh0, dv2

fla/ops/generalized_delta_rule/dplr/chunk_h_fwd.py

@@ -0,0 +1,197 @@
+# -*- coding: utf-8 -*-
+# Copyright (c) 2023-2025, Songlin Yang, Yu Zhang
+
+from typing import Optional, Tuple
+
+import torch
+import triton
+import triton.language as tl
+
+from fla.ops.common.utils import prepare_chunk_offsets
+from fla.ops.utils.op import exp
+from fla.utils import check_shared_mem, use_cuda_graph
+
+
+@triton.heuristics({
+    'USE_INITIAL_STATE': lambda args: args['h0'] is not None,
+    'STORE_FINAL_STATE': lambda args: args['ht'] is not None,
+    'USE_OFFSETS': lambda args: args['offsets'] is not None,
+})
+@triton.autotune(
+    configs=[
+        triton.Config({}, num_warps=num_warps, num_stages=num_stages)
+        for num_warps in [2, 4, 8, 16, 32]
+        for num_stages in [2, 3, 4]
+    ],
+    key=['BT', 'BK', 'BV'],
+    use_cuda_graph=use_cuda_graph,
+)
+@triton.jit(do_not_specialize=['T'])
+def chunk_dplr_fwd_kernel_h(
+    kg,
+    v,
+    w,
+    bg,
+    u,
+    v_new,
+    gk,
+    h,
+    h0,
+    ht,
+    offsets,
+    chunk_offsets,
+    T,
+    H: tl.constexpr,
+    K: tl.constexpr,
+    V: tl.constexpr,
+    BT: tl.constexpr,
+    BC: tl.constexpr,
+    BK: tl.constexpr,
+    BV: tl.constexpr,
+    NT: tl.constexpr,
+    USE_INITIAL_STATE: tl.constexpr,
+    STORE_FINAL_STATE: tl.constexpr,
+    USE_OFFSETS: tl.constexpr,
+    HEAD_FIRST: tl.constexpr,
+):
+    i_k, i_v, i_nh = tl.program_id(0), tl.program_id(1), tl.program_id(2)
+    i_n, i_h = i_nh // H, i_nh % H
+    if USE_OFFSETS:
+        bos, eos = tl.load(offsets + i_n).to(tl.int32), tl.load(offsets + i_n + 1).to(tl.int32)
+        T = eos - bos
+        NT = tl.cdiv(T, BT)
+        boh = tl.load(chunk_offsets + i_n).to(tl.int32)
+    else:
+        bos, eos = i_n * T, i_n * T + T
+        NT = tl.cdiv(T, BT)
+        boh = i_n * NT
+
+    # [BK, BV]
+    b_h = tl.zeros([BK, BV], dtype=tl.float32)
+    if USE_INITIAL_STATE:
+        p_h0 = tl.make_block_ptr(h0 + i_nh * K*V, (K, V), (V, 1), (i_k * BK, i_v * BV), (BK, BV), (1, 0))
+        b_h = tl.load(p_h0, boundary_check=(0, 1)).to(tl.float32)
+
+    for i_t in range(NT):
+        if HEAD_FIRST:
+            p_h = tl.make_block_ptr(h + (i_nh * NT + i_t) * K*V, (K, V), (V, 1), (i_k * BK, i_v * BV), (BK, BV), (1, 0))
+        else:
+            p_h = tl.make_block_ptr(h + ((boh + i_t) * H + i_h) * K*V, (K, V), (V, 1), (i_k * BK, i_v * BV), (BK, BV), (1, 0))
+        tl.store(p_h, b_h.to(p_h.dtype.element_ty), boundary_check=(0, 1))
+
+        b_hc = tl.zeros([BK, BV], dtype=tl.float32)
+        # since we need to make all DK in the SRAM. we face serve SRAM memory burden. By subchunking we allievate such burden
+        for i_c in range(tl.cdiv(min(BT, T - i_t * BT), BC)):
+            if HEAD_FIRST:
+                p_kg = tl.make_block_ptr(kg + i_nh * T*K, (K, T), (1, K), (i_k * BK, i_t * BT + i_c * BC), (BK, BC), (0, 1))
+                p_bg = tl.make_block_ptr(bg + i_nh * T*K, (K, T), (1, K), (i_k * BK, i_t * BT + i_c * BC), (BK, BC), (0, 1))
+                p_w = tl.make_block_ptr(w + i_nh * T*K, (T, K), (K, 1), (i_t * BT + i_c * BC, i_k * BK), (BC, BK), (1, 0))
+                p_v = tl.make_block_ptr(v + i_nh * T*V, (T, V), (V, 1), (i_t * BT + i_c * BC, i_v * BV), (BC, BV), (1, 0))
+                p_u = tl.make_block_ptr(u + i_nh * T*V, (T, V), (V, 1), (i_t * BT + i_c * BC, i_v * BV), (BC, BV), (1, 0))
+                p_v_new = tl.make_block_ptr(v_new+i_nh*T*V, (T, V), (V, 1), (i_t * BT + i_c * BC, i_v * BV), (BC, BV), (1, 0))
+            else:
+                p_kg = tl.make_block_ptr(kg+(bos*H+i_h)*K, (K, T), (1, H*K), (i_k * BK, i_t * BT + i_c * BC), (BK, BC), (0, 1))
+                p_bg = tl.make_block_ptr(bg+(bos*H+i_h)*K, (K, T), (1, H*K), (i_k * BK, i_t * BT + i_c * BC), (BK, BC), (0, 1))
+                p_w = tl.make_block_ptr(w+(bos*H+i_h)*K, (T, K), (H*K, 1), (i_t * BT + i_c * BC, i_k * BK), (BC, BK), (1, 0))
+                p_v = tl.make_block_ptr(v+(bos*H+i_h)*V, (T, V), (H*V, 1), (i_t * BT + i_c * BC, i_v * BV), (BC, BV), (1, 0))
+                p_u = tl.make_block_ptr(u+(bos*H+i_h)*V, (T, V), (H*V, 1), (i_t * BT + i_c * BC, i_v * BV), (BC, BV), (1, 0))
+                p_v_new = tl.make_block_ptr(v_new+(bos*H+i_h)*V, (T, V), (H*V, 1), (i_t*BT+i_c*BC, i_v * BV), (BC, BV), (1, 0))
+            # [BK, BC]
+            b_kg = tl.load(p_kg, boundary_check=(0, 1))
+            b_v = tl.load(p_v, boundary_check=(0, 1))
+            b_w = tl.load(p_w, boundary_check=(0, 1))
+            b_bg = tl.load(p_bg, boundary_check=(0, 1))
+            b_v2 = tl.dot(b_w, b_h.to(b_w.dtype)) + tl.load(p_u, boundary_check=(0, 1))
+            b_hc += tl.dot(b_kg, b_v)
+            b_hc += tl.dot(b_bg.to(b_hc.dtype), b_v2)
+            tl.store(p_v_new, b_v2.to(p_v_new.dtype.element_ty), boundary_check=(0, 1))
+
+        last_idx = min((i_t + 1) * BT, T) - 1
+        if HEAD_FIRST:
+            b_g_last = tl.load(gk + i_nh * T * K + last_idx * K + tl.arange(0, BK), mask=tl.arange(0, BK) < K).to(tl.float32)
+        else:
+            b_g_last = tl.load(gk + (bos + last_idx) * H * K + i_h * K +
+                               tl.arange(0, BK), mask=tl.arange(0, BK) < K).to(tl.float32)
+        b_h *= exp(b_g_last[:, None])
+        b_h += b_hc
+
+    if STORE_FINAL_STATE:
+        p_ht = tl.make_block_ptr(ht + i_nh * K*V, (K, V), (V, 1), (i_k * BK, i_v * BV), (BK, BV), (1, 0))
+        tl.store(p_ht, b_h.to(p_ht.dtype.element_ty, fp_downcast_rounding="rtne"), boundary_check=(0, 1))
+
+
+def chunk_dplr_fwd_h(
+    kg: torch.Tensor,
+    v: torch.Tensor,
+    w: torch.Tensor,
+    u: torch.Tensor,
+    bg: torch.Tensor,
+    gk: torch.Tensor,
+    initial_state: Optional[torch.Tensor] = None,
+    output_final_state: bool = False,
+    offsets: Optional[torch.LongTensor] = None,
+    indices: Optional[torch.LongTensor] = None,
+    head_first: bool = True,
+    chunk_size: int = 64
+) -> Tuple[torch.Tensor, torch.Tensor]:
+    if head_first:
+        B, H, T, K, V = *kg.shape, u.shape[-1]
+    else:
+        B, T, H, K, V = *kg.shape, u.shape[-1]
+    BT = min(chunk_size, max(triton.next_power_of_2(T), 16))
+    # N: the actual number of sequences in the batch with either equal or variable lengths
+    if offsets is None:
+        N, NT, chunk_offsets = B, triton.cdiv(T, BT), None
+    else:
+        N, NT, chunk_offsets = len(offsets) - 1, len(indices), prepare_chunk_offsets(offsets, BT)
+    BK = triton.next_power_of_2(K)
+    assert BK <= 256, "current kernel does not support head dimension larger than 256."
+    # H100 can have larger block size
+
+    if check_shared_mem('hopper', kg.device.index):
+        BV = 64
+        BC = 64 if K <= 128 else 32
+    elif check_shared_mem('ampere', kg.device.index):  # A100
+        BV = 32
+        BC = 32
+    else:
+        BV = 16
+        BC = 16
+
+    BC = min(BT, BC)
+    NK = triton.cdiv(K, BK)
+    NV = triton.cdiv(V, BV)
+    assert NK == 1, 'NK > 1 is not supported because it involves time-consuming synchronization'
+
+    if head_first:
+        h = kg.new_empty(B, H, NT, K, V)
+    else:
+        h = kg.new_empty(B, NT, H, K, V)
+    final_state = kg.new_empty(N, H, K, V, dtype=torch.float32) if output_final_state else None
+    v_new = torch.empty_like(u)
+    grid = (NK, NV, N * H)
+    chunk_dplr_fwd_kernel_h[grid](
+        kg=kg,
+        v=v,
+        w=w,
+        bg=bg,
+        u=u,
+        v_new=v_new,
+        h=h,
+        gk=gk,
+        h0=initial_state,
+        ht=final_state,
+        offsets=offsets,
+        chunk_offsets=chunk_offsets,
+        T=T,
+        H=H,
+        K=K,
+        V=V,
+        BT=BT,
+        BC=BC,
+        BK=BK,
+        BV=BV,
+        NT=NT,
+        HEAD_FIRST=head_first
+    )
+    return h, v_new, final_state

fla/ops/generalized_delta_rule/dplr/fused_recurrent.py

@@ -0,0 +1,292 @@
+# -*- coding: utf-8 -*-
+# Copyright (c) 2023-2025, Songlin Yang, Yu Zhang
+
+from typing import Optional, Tuple
+
+import torch
+import triton
+import triton.language as tl
+
+from fla.ops.utils.op import exp
+from fla.utils import autocast_custom_bwd, autocast_custom_fwd, input_guard, use_cuda_graph
+
+
+@triton.heuristics({
+    'USE_INITIAL_STATE': lambda args: args['h0'] is not None,
+    'STORE_FINAL_STATE': lambda args: args['ht'] is not None,
+    'USE_OFFSETS': lambda args: args['offsets'] is not None
+})
+@triton.autotune(
+    configs=[
+        triton.Config({'BV': BV}, num_warps=num_warps, num_stages=num_stages)
+        for BV in [16, 32, 64]
+        for num_warps in [2, 4, 8, 16]
+        for num_stages in [2, 3, 4]
+    ],
+    key=['BK'],
+    use_cuda_graph=use_cuda_graph,
+)
+@triton.jit(do_not_specialize=['T'])
+def fused_recurrent_dplr_delta_rule_fwd_kernel(
+    q,
+    k,
+    v,
+    a,
+    b,
+    gk,
+    o,
+    h0,
+    ht,
+    offsets,
+    scale,
+    T,
+    B: tl.constexpr,
+    H: tl.constexpr,
+    K: tl.constexpr,
+    V: tl.constexpr,
+    BK: tl.constexpr,
+    BV: tl.constexpr,
+    REVERSE: tl.constexpr,
+    USE_INITIAL_STATE: tl.constexpr,
+    STORE_FINAL_STATE: tl.constexpr,
+    USE_OFFSETS: tl.constexpr,
+    HEAD_FIRST: tl.constexpr
+):
+    i_v, i_nh = tl.program_id(0).to(tl.int64), tl.program_id(1).to(tl.int64)
+    i_n, i_h = i_nh // H, i_nh % H
+
+    if USE_OFFSETS:
+        bos, eos = tl.load(offsets + i_n).to(tl.int64), tl.load(offsets + i_n + 1).to(tl.int64)
+        T = eos - bos
+    else:
+        bos, eos = i_n * T, i_n * T + T
+
+    o_k = tl.arange(0, BK)
+    o_v = i_v * BV + tl.arange(0, BV)
+    if HEAD_FIRST:
+        p_q = q + i_nh * T*K + ((T-1) * K if REVERSE else 0) + o_k
+        p_k = k + i_nh * T*K + ((T-1) * K if REVERSE else 0) + o_k
+        p_a = a + i_nh * T*K + ((T-1) * K if REVERSE else 0) + o_k
+        p_b = b + i_nh * T*K + ((T-1) * K if REVERSE else 0) + o_k
+        p_gk = gk + i_nh * T*K + ((T-1) * K if REVERSE else 0) + o_k
+        p_v = v + i_nh * T*V + ((T-1) * V if REVERSE else 0) + o_v
+        p_o = o + i_nh * T*V + ((T-1) * V if REVERSE else 0) + o_v
+
+    else:
+        p_q = q + (bos + ((T-1) if REVERSE else 0)) * H*K + i_h * K + o_k
+        p_k = k + (bos + ((T-1) if REVERSE else 0)) * H*K + i_h * K + o_k
+        p_a = a + (bos + ((T-1) if REVERSE else 0)) * H*K + i_h * K + o_k
+        p_b = b + (bos + ((T-1) if REVERSE else 0)) * H*K + i_h * K + o_k
+        p_gk = gk + (bos + ((T-1) if REVERSE else 0)) * H*K + i_h * K + o_k
+        p_v = v + (bos + ((T-1) if REVERSE else 0)) * H*V + i_h * V + o_v
+        p_o = o + (bos + ((T-1) if REVERSE else 0)) * H*V + i_h * V + o_v
+
+    mask_k = o_k < K
+    mask_v = o_v < V
+    mask_h = mask_k[None, :] & mask_v[:, None]
+    b_h = tl.zeros([BV, BK], dtype=tl.float32)
+
+    if USE_INITIAL_STATE:
+        p_h0 = h0 + i_nh * K*V + o_k[None, :] * V + o_v[:, None]
+        b_h += tl.load(p_h0, mask=mask_h, other=0).to(tl.float32)
+
+    for _ in range(0, T):
+        b_q = tl.load(p_q, mask=mask_k, other=0).to(tl.float32) * scale
+        b_k = tl.load(p_k, mask=mask_k, other=0).to(tl.float32)
+        b_a = tl.load(p_a, mask=mask_k, other=0).to(tl.float32)
+        b_b = tl.load(p_b, mask=mask_k, other=0).to(tl.float32)
+        b_gk = tl.load(p_gk, mask=mask_k, other=0).to(tl.float32)
+        b_v = tl.load(p_v, mask=mask_v, other=0).to(tl.float32)
+
+        tmp = tl.sum(b_h * b_a[None, :], axis=1)
+        b_h = exp(b_gk)[None, :] * b_h + (tmp[:, None] * b_b[None, :] + b_k[None, :] * b_v[:, None])
+        b_o = tl.sum(b_h * b_q[None, :], axis=1)
+
+        tl.store(p_o, b_o.to(p_o.dtype.element_ty), mask=mask_v)
+        p_q += (-1 if REVERSE else 1) * (1 if HEAD_FIRST else H) * K
+        p_k += (-1 if REVERSE else 1) * (1 if HEAD_FIRST else H) * K
+        p_a += (-1 if REVERSE else 1) * (1 if HEAD_FIRST else H) * K
+        p_b += (-1 if REVERSE else 1) * (1 if HEAD_FIRST else H) * K
+        p_gk += (-1 if REVERSE else 1) * (1 if HEAD_FIRST else H) * K
+        p_v += (-1 if REVERSE else 1) * (1 if HEAD_FIRST else H) * V
+        p_o += (-1 if REVERSE else 1) * (1 if HEAD_FIRST else H) * V
+
+    if STORE_FINAL_STATE:
+        p_ht = ht + i_nh * K*V + o_k[None, :] * V + o_v[:, None]
+        tl.store(p_ht, b_h.to(p_ht.dtype.element_ty), mask=mask_h)
+
+
+def fused_recurrent_dplr_delta_rule_fwd(
+    q: torch.Tensor,
+    k: torch.Tensor,
+    v: torch.Tensor,
+    a: torch.Tensor,
+    b: torch.Tensor,
+    gk: torch.Tensor,
+    scale: Optional[float] = 1.0,
+    initial_state: Optional[torch.Tensor] = None,
+    output_final_state: bool = False,
+    reverse: bool = False,
+    offsets: Optional[torch.LongTensor] = None,
+    head_first: bool = True
+):
+    if head_first:
+        B, H, T, K, V = *k.shape, v.shape[-1]
+    else:
+        B, T, H, K, V = *k.shape, v.shape[-1]
+    N = B if offsets is None else len(offsets) - 1
+    BK = triton.next_power_of_2(K)
+
+    h0 = initial_state
+    if output_final_state:
+        ht = q.new_empty(N, H, K, V, dtype=torch.float32)
+    else:
+        ht = None
+    o = torch.empty_like(v)
+
+    def grid(meta): return (triton.cdiv(V, meta['BV']), N * H)
+    fused_recurrent_dplr_delta_rule_fwd_kernel[grid](
+        q,
+        k,
+        v,
+        a,
+        b,
+        gk,
+        o,
+        h0,
+        ht,
+        offsets,
+        scale,
+        T=T,
+        B=B,
+        H=H,
+        K=K,
+        V=V,
+        BK=BK,
+        REVERSE=reverse,
+        HEAD_FIRST=head_first
+    )
+    return o, ht
+
+
+class FusedRecurrentDPLRDeltaRuleFunction(torch.autograd.Function):
+
+    @staticmethod
+    @input_guard
+    @autocast_custom_fwd
+    def forward(
+        ctx,
+        q: torch.Tensor,
+        k: torch.Tensor,
+        v: torch.Tensor,
+        a: torch.Tensor,
+        b: torch.Tensor,
+        gk: torch.Tensor,
+        scale: Optional[float] = 1.0,
+        initial_state: Optional[torch.Tensor] = None,
+        output_final_state: bool = False,
+        reverse: bool = False,
+        offsets: Optional[torch.LongTensor] = None,
+        head_first: bool = False
+    ):
+        o, ht = fused_recurrent_dplr_delta_rule_fwd(
+            q=q,
+            k=k,
+            v=v,
+            a=a,
+            b=b,
+            gk=gk,
+            scale=scale,
+            initial_state=initial_state,
+            output_final_state=output_final_state,
+            reverse=reverse,
+            offsets=offsets,
+            head_first=head_first
+        )
+        return o, ht
+
+    @staticmethod
+    @input_guard
+    @autocast_custom_bwd
+    def backward(ctx, do, dht):
+        raise NotImplementedError(
+            "Backward pass for fused_recurrent_dplr_delta_rule is not implemented and will not be supported. "
+            "This kernel is only for inference. "
+            "For training, please use `chunk_dplr_delta_rule`."
+        )
+
+
+def fused_recurrent_dplr_delta_rule(
+    q: torch.Tensor,
+    k: torch.Tensor,
+    v: torch.Tensor,
+    a: torch.Tensor,
+    b: torch.Tensor,
+    gk: torch.Tensor,
+    scale: Optional[float] = 1.0,
+    initial_state: Optional[torch.Tensor] = None,
+    output_final_state: bool = False,
+    reverse: bool = False,
+    cu_seqlens: Optional[torch.Tensor] = None,
+    head_first: bool = False
+) -> Tuple[torch.Tensor, torch.Tensor]:
+    r"""
+    This function computes the recurrence S_t = S_t @ (I + a_t b_t^T) + v_t k_t^T in a recurrent manner.
+
+    Args:
+        q (torch.Tensor):
+            queries of shape `[B, H, T, K]`
+        k (torch.Tensor):
+            keys of shape `[B, H, T, K]`
+        v (torch.Tensor):
+            values of shape `[B, H, T, V]`
+        a (torch.Tensor):
+            as of shape `[B, H, T, K]`
+        b (torch.Tensor):
+             bs of shape `[B, H, T, K]`
+        gk (torch.Tensor):
+            gk of shape `[B, H, T, K]`
+        scale (Optional[int]):
+            Scale factor for the RetNet attention scores.
+            If None, it will default to `1 / sqrt(K)`. Default: `1.0`.
+        initial_state (Optional[torch.Tensor]):
+            Initial state of shape `[B, H, K, V]`. Default: `None`.
+        output_final_state (Optional[bool]):
+            Whether to output the final state of shape `[B, H, K, V]`. Default: `False`.
+        reverse (Optional[bool]):
+            If `True`, process the state passing in reverse order. Default: `False`.
+        cu_seqlens (Optional[torch.Tensor]):
+            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, which is not supported for variable-length inputs.
+            Default: `False`.
+    """
+    if cu_seqlens is not None:
+        if 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 head_first:
+            raise RuntimeError("Sequences with variable lengths are not supported for head-first mode")
+        if initial_state is not None and initial_state.shape[0] != len(cu_seqlens) - 1:
+            raise ValueError(f"The number of initial states is expected to be equal to the number of input sequences, "
+                             f"i.e., {len(cu_seqlens) - 1} rather than {initial_state.shape[0]}.")
+    if scale is None:
+        scale = q.shape[-1] ** -0.5
+    else:
+        assert scale > 0, "scale must be positive"
+    o, final_state = FusedRecurrentDPLRDeltaRuleFunction.apply(
+        q,
+        k,
+        v,
+        a,
+        b,
+        gk,
+        scale,
+        initial_state,
+        output_final_state,
+        reverse,
+        cu_seqlens,
+        head_first
+    )
+    return o, final_state

fla/ops/generalized_delta_rule/dplr/naive.py

@@ -0,0 +1,96 @@
+# -*- coding: utf-8 -*-
+
+import torch
+from einops import rearrange
+
+# S_t = S_t @ (I + alpha_t beta_t^T) + v_t k_t^T
+# q, k, alpha, beta [B, H, L, D_K]
+# v [B, H, L, D_V]
+
+
+def dplr_recurrence(q, k, v, alpha, beta, gk, initial_state=None, output_final_state=True):
+    orig_dtype = q.dtype
+    b, h, l, d_k = q.shape
+    q, k, v, beta, gk = map(lambda x: x.float(), [q, k, v, beta, gk])
+    d_v = v.shape[-1]
+    o = torch.zeros_like(v)
+    S = torch.zeros(b, h, d_k, d_v).to(v)
+    q = q * (d_k ** -0.5)
+
+    if initial_state is not None:
+        S += initial_state
+
+    for i in range(l):
+        _k = k[:, :, i]
+        _q = q[:, :, i]
+        _v = v[:, :, i]
+        _alpha = alpha[:, :, i].clone()
+        _beta = beta[:, :, i].clone()
+        _kv = _k[..., None] * _v[..., None, :] + (S.clone() * _alpha[..., None]).sum(-2, keepdim=True) * _beta[..., None]
+        S = S.clone() * gk[:, :, i].exp()[..., None] + _kv
+        o[:, :, i] = torch.einsum('bhd,bhdm->bhm', _q, S)
+    S = None if output_final_state is False else S
+    return o.to(orig_dtype), S
+
+
+def dplr_chunkwise(q, k, v, alpha, beta, gk, initial_state=None, output_final_state=True, chunk_size=32):
+    b, h, l, d_k = q.shape
+    d_v = v.shape[-1]
+    q = q * (d_k ** -0.5)
+    v = v
+    assert l % chunk_size == 0
+
+    S = k.new_zeros(b, h, d_k, d_v).to(q)
+    if initial_state is not None:
+        S += initial_state
+
+    # note that diagonal is masked.
+    mask = torch.triu(torch.ones(chunk_size, chunk_size, dtype=torch.bool, device=q.device), diagonal=0)
+    q, k, v, alpha, beta, gk = map(lambda x: rearrange(x, 'b h (n c) d -> b h n c d',
+                                   c=chunk_size).float(), [q, k, v, alpha, beta, gk])
+
+    gk_cumsum = gk.cumsum(-2)
+
+    # v2 = (alpha @ k.transpose(-1, -2)).masked_fill_(mask, 0) @ v
+    A_ab = torch.zeros(b, h, l // chunk_size, chunk_size, chunk_size).to(q.device)
+    A_qk = torch.zeros(b, h, l // chunk_size, chunk_size, chunk_size).to(q.device)
+    A_ak = torch.zeros(b, h, l // chunk_size, chunk_size, chunk_size).to(q.device)
+    A_qb = torch.zeros(b, h, l // chunk_size, chunk_size, chunk_size).to(q.device)
+
+    for i in range(chunk_size):
+        alpha_i = alpha[:, :, :, i, None]
+        q_i = q[:, :, :, i, None]
+        gk_i = gk_cumsum[:, :, :, i, None]
+        mask = (torch.arange(chunk_size) <= i).to(q.device)
+        attn_i = (gk_i - gk_cumsum).masked_fill(~mask.unsqueeze(-1), float('-inf')).exp()
+        A_qk[:, :, :, i, :] = (q_i * k * attn_i).sum(-1).clone()
+        A_qb[:, :, :, i, :] = (q_i * beta * attn_i).sum(-1).clone()
+        mask = (torch.arange(chunk_size) < i).to(q.device)
+        # shift by one.
+        attn_i = (gk_i - gk[:, :, :, i, None] - gk_cumsum).masked_fill(~mask.unsqueeze(-1), float('-inf')).exp()
+        A_ab[:, :, :, i, :] = (alpha_i * beta * attn_i).sum(-1).clone()
+        A_ak[:, :, :, i, :] = (alpha_i * k * attn_i).sum(-1).clone()
+
+    A_ab = A_ab
+    for i in range(1, chunk_size):
+        A_ab[..., i, :i] = A_ab[..., i, :i].clone() + (A_ab[..., i, :, None].clone() * A_ab[..., :, :i].clone()).sum(-2)
+
+    A_ab = A_ab + torch.eye(chunk_size, dtype=torch.float, device=q.device)
+    u = A_ab @ (A_ak @ v)
+    w = A_ab @ ((gk_cumsum-gk).exp() * alpha)
+
+    o = torch.zeros_like(v)
+    mask = torch.triu(torch.ones(chunk_size, chunk_size, dtype=torch.bool, device=q.device), diagonal=1)
+    for i in range(0, l // chunk_size):
+        q_i, k_i, v_i, u_i, w_i, beta_i = q[:, :, i], k[:, :, i], v[:, :, i], u[:, :, i], w[:, :, i], beta[:, :, i]
+        v2_i = u_i + w_i @ S
+
+        o_1 = A_qk[:, :, i] @ v_i
+        o_2 = A_qb[:, :, i] @ v2_i
+        o_3 = (q_i * gk_cumsum[:, :, i].exp()) @ S
+        o[:, :, i] = o_1 + o_2 + o_3
+        decay = (gk_cumsum[:, :, i, -1, None] - gk_cumsum[:, :, i]).exp()
+        S = S*gk_cumsum[:, :, i, -1, :, None].exp() + (k_i * decay).transpose(-1, -2) @ v_i + \
+            (beta_i * decay).transpose(-1, -2) @ v2_i
+    S = None if output_final_state is False else S
+    return rearrange(o, 'b h n c d -> b h (n c) d'), S

fla/ops/generalized_delta_rule/iplr/wy_fast.py

@@ -0,0 +1,338 @@
+
+# -*- coding: utf-8 -*-
+# Copyright (c) 2023-2025, Songlin Yang, Yu Zhang
+
+from typing import Optional, Tuple
+
+import torch
+import triton
+import triton.language as tl
+
+from fla.utils import check_shared_mem, is_nvidia_hopper
+
+NUM_WARPS = [2, 4] if is_nvidia_hopper else [2, 4, 8]
+
+
+@triton.heuristics({
+    'USE_OFFSETS': lambda args: args['offsets'] is not None
+})
+@triton.autotune(
+    configs=[
+        triton.Config({}, num_warps=num_warps)
+        for num_warps in [1, 2, 4, 8, 16]
+    ],
+    key=['BK']
+)
+@triton.jit(do_not_specialize=['T'])
+def fwd_prepare_wy_repr_kernel_chunk32(
+    a,
+    b,
+    A,
+    offsets,
+    indices,
+    T,
+    H: tl.constexpr,
+    K: tl.constexpr,
+    BT: tl.constexpr,
+    BK: tl.constexpr,
+    BC: tl.constexpr,  # dummy placeholder
+    USE_OFFSETS: tl.constexpr,
+    HEAD_FIRST: tl.constexpr,
+):
+    i_t, i_bh = tl.program_id(0), tl.program_id(1)
+    i_b, i_h = i_bh // H, i_bh % H
+    if USE_OFFSETS:
+        i_n, i_t = tl.load(indices + i_t * 2).to(tl.int32), tl.load(indices + i_t * 2 + 1).to(tl.int32)
+        bos, eos = tl.load(offsets + i_n).to(tl.int32), tl.load(offsets + i_n + 1).to(tl.int32)
+        T = eos - bos
+    else:
+        bos, eos = i_b * T, i_b * T + T
+
+    b_A = tl.zeros([BT, BT], dtype=tl.float32)
+    for i_k in range(tl.cdiv(K, BK)):
+        if HEAD_FIRST:
+            p_a = tl.make_block_ptr(a + i_bh * T*K, (T, K), (K, 1), (i_t * BT, i_k * BK), (BT, BK), (1, 0))
+            p_b = tl.make_block_ptr(b + i_bh * T*K, (K, T), (1, K), (i_k * BK, i_t * BT), (BK, BT), (0, 1))
+        else:
+            p_a = tl.make_block_ptr(a + (bos * H + i_h) * K, (T, K), (H*K, 1), (i_t * BT, i_k * BK), (BT, BK), (1, 0))
+            p_b = tl.make_block_ptr(b + (bos * H + i_h) * K, (K, T), (1, K*H), (i_k * BK, i_t * BT), (BK, BT), (0, 1))
+        b_a = tl.load(p_a, boundary_check=(0, 1))
+        b_b = tl.load(p_b, boundary_check=(0, 1))
+        b_A += tl.dot(b_a, b_b)
+
+    b_A = tl.where(tl.arange(0, BT)[:, None] > tl.arange(0, BT)[None, :], b_A, 0)
+    for i in range(1, BT):
+        mask = tl.arange(0, BT) == i
+        b_a = tl.sum(tl.where(mask[:, None], b_A, 0), 0)
+        b_a = b_a + tl.sum(b_a[:, None] * b_A, 0) * (tl.arange(0, BT) < i)
+        b_A = tl.where(mask[:, None], b_a, b_A)
+    b_A += tl.arange(0, BT)[:, None] == tl.arange(0, BT)[None, :]
+
+    if HEAD_FIRST:
+        p_A = tl.make_block_ptr(A + i_bh * T * BT, (T, BT), (BT, 1), (i_t * BT, 0), (BT, BT), (1, 0))
+    else:
+        p_A = tl.make_block_ptr(A + (bos*H + i_h) * BT, (T, BT), (H*BT, 1), (i_t * BT, 0), (BT, BT), (1, 0))
+    tl.store(p_A, b_A.to(p_A.dtype.element_ty), boundary_check=(0, 1))
+
+
+@triton.heuristics({
+    'USE_OFFSETS': lambda args: args['offsets'] is not None
+})
+@triton.autotune(
+    configs=[
+        triton.Config({}, num_warps=num_warps)
+        for num_warps in [1, 2, 4, 8, 16]
+    ],
+    key=['BK']
+)
+@triton.jit(do_not_specialize=['T'])
+def fwd_prepare_wy_repr_kernel_chunk64(
+    a,
+    b,
+    A,
+    offsets,
+    indices,
+    T,
+    H: tl.constexpr,
+    K: tl.constexpr,
+    BT: tl.constexpr,
+    BK: tl.constexpr,
+    BC: tl.constexpr,
+    USE_OFFSETS: tl.constexpr,
+    HEAD_FIRST: tl.constexpr
+):
+    i_t, i_bh = tl.program_id(0), tl.program_id(1)
+    i_b, i_h = i_bh // H, i_bh % H
+    if USE_OFFSETS:
+        i_n, i_t = tl.load(indices + i_t * 2).to(tl.int32), tl.load(indices + i_t * 2 + 1).to(tl.int32)
+        bos, eos = tl.load(offsets + i_n).to(tl.int32), tl.load(offsets + i_n + 1).to(tl.int32)
+        T = eos - bos
+    else:
+        bos, eos = i_b * T, i_b * T + T
+
+    b_A = tl.zeros([BC, BC], dtype=tl.float32)
+    b_A2 = tl.zeros([BC, BC], dtype=tl.float32)
+    b_A3 = tl.zeros([BC, BC], dtype=tl.float32)
+
+    for i_k in range(tl.cdiv(K, BK)):
+        if HEAD_FIRST:
+            p_a1 = tl.make_block_ptr(a + i_bh * T*K, (T, K), (K, 1), (i_t * BT, i_k * BK), (BC, BK), (1, 0))
+            p_a2 = tl.make_block_ptr(a + i_bh * T*K, (T, K), (K, 1), (i_t * BT + BC, i_k * BK), (BC, BK), (1, 0))
+            p_b1 = tl.make_block_ptr(b + i_bh * T*K, (K, T), (1, K), (i_k * BK, i_t * BT), (BK, BC), (0, 1))
+            p_b2 = tl.make_block_ptr(b + i_bh * T*K, (K, T), (1, K), (i_k * BK, i_t * BT + BC), (BK, BC), (0, 1))
+        else:
+            p_a1 = tl.make_block_ptr(a + (bos * H + i_h) * K, (T, K), (H*K, 1), (i_t * BT, i_k * BK), (BC, BK), (1, 0))
+            p_a2 = tl.make_block_ptr(a + (bos * H + i_h) * K, (T, K), (H*K, 1), (i_t * BT + BC, i_k * BK), (BC, BK), (1, 0))
+            p_b1 = tl.make_block_ptr(b + (bos * H + i_h) * K, (K, T), (1, K*H), (i_k * BK, i_t * BT), (BK, BC), (0, 1))
+            p_b2 = tl.make_block_ptr(b + (bos * H + i_h) * K, (K, T), (1, K*H), (i_k * BK, i_t * BT + BC), (BK, BC), (0, 1))
+        b_a1 = tl.load(p_a1, boundary_check=(0, 1))
+        b_a2 = tl.load(p_a2, boundary_check=(0, 1))
+        b_b1 = tl.load(p_b1, boundary_check=(0, 1))
+        b_b2 = tl.load(p_b2, boundary_check=(0, 1))
+        b_A += tl.dot(b_a1, b_b1, allow_tf32=False)
+        b_A2 += tl.dot(b_a2, b_b2, allow_tf32=False)
+        b_A3 += tl.dot(b_a2, b_b1, allow_tf32=False)
+
+    b_A = tl.where(tl.arange(0, BC)[:, None] > tl.arange(0, BC)[None, :], b_A, 0)
+    b_A2 = tl.where(tl.arange(0, BC)[:, None] > tl.arange(0, BC)[None, :], b_A2, 0)
+
+    for i in range(1, BC):
+        mask = tl.arange(0, BC) == i
+        b_a = tl.sum(tl.where(mask[:, None], b_A, 0), 0)
+        b_a2 = tl.sum(tl.where(mask[:, None], b_A2, 0), 0)
+        b_a = b_a + tl.sum(b_a[:, None] * b_A, 0) * (tl.arange(0, BC) < i)
+        b_a2 = b_a2 + tl.sum(b_a2[:, None] * b_A2, 0) * (tl.arange(0, BC) < i)
+        b_A = tl.where(mask[:, None], b_a, b_A)
+        b_A2 = tl.where(mask[:, None], b_a2, b_A2)
+
+    # blockwise computation of lower triangular matrix's inverse
+    # i.e., [A11, 0; A21, A22]^-1 = [A11^-1, 0; -A22^-1 A21 A11^-1, A22^-1]
+    b_A += tl.arange(0, BC)[:, None] == tl.arange(0, BC)[None, :]
+    b_A2 += tl.arange(0, BC)[:, None] == tl.arange(0, BC)[None, :]
+    b_A3 = tl.dot(tl.dot(b_A2, b_A3, allow_tf32=False), b_A, allow_tf32=False)
+
+    if HEAD_FIRST:
+        p_A1 = tl.make_block_ptr(A + i_bh * T * BT, (T, BT), (BT, 1), (i_t * BT, 0), (BC, BC), (1, 0))
+        p_A2 = tl.make_block_ptr(A + i_bh * T * BT, (T, BT), (BT, 1), (i_t * BT + BC, BC), (BC, BC), (1, 0))
+        p_A3 = tl.make_block_ptr(A + i_bh * T * BT, (T, BT), (BT, 1), (i_t * BT + BC, 0), (BC, BC), (1, 0))
+        p_A4 = tl.make_block_ptr(A + i_bh * T * BT, (T, BT), (BT, 1), (i_t * BT, BC), (BC, BC), (1, 0))
+    else:
+        p_A1 = tl.make_block_ptr(A + (bos*H + i_h) * BT, (T, BT), (H*BT, 1), (i_t * BT, 0), (BC, BC), (1, 0))
+        p_A2 = tl.make_block_ptr(A + (bos*H + i_h) * BT, (T, BT), (H*BT, 1), (i_t * BT + BC, BC), (BC, BC), (1, 0))
+        p_A3 = tl.make_block_ptr(A + (bos*H + i_h) * BT, (T, BT), (H*BT, 1), (i_t * BT + BC, 0), (BC, BC), (1, 0))
+        p_A4 = tl.make_block_ptr(A + (bos*H + i_h) * BT, (T, BT), (H*BT, 1), (i_t * BT, BC), (BC, BC), (1, 0))
+    tl.store(p_A1, b_A.to(p_A1.dtype.element_ty), boundary_check=(0, 1))
+    tl.store(p_A2, b_A2.to(p_A2.dtype.element_ty), boundary_check=(0, 1))
+    tl.store(p_A3, b_A3.to(p_A3.dtype.element_ty), boundary_check=(0, 1))
+    # causal mask
+    tl.store(p_A4, tl.zeros([BC, BC], dtype=tl.float32).to(p_A4.dtype.element_ty), boundary_check=(0, 1))
+
+
+@triton.heuristics({
+    'USE_OFFSETS': lambda args: args['offsets'] is not None
+})
+@triton.autotune(
+    configs=[
+        triton.Config({}, num_warps=num_warps)
+        for num_warps in NUM_WARPS
+    ],
+    key=['BT', 'BK', 'BV']
+)
+@triton.jit(do_not_specialize=['T'])
+def fwd_wu_kernel(
+    w,
+    u,
+    a,
+    k,
+    v,
+    A,
+    offsets,
+    indices,
+    T,
+    H: tl.constexpr,
+    K: tl.constexpr,
+    V: tl.constexpr,
+    BT: tl.constexpr,
+    BK: tl.constexpr,
+    BV: tl.constexpr,
+    USE_OFFSETS: tl.constexpr,
+    HEAD_FIRST: tl.constexpr
+):
+    i_t, i_bh = tl.program_id(0), tl.program_id(1)
+    i_b, i_h = i_bh // H, i_bh % H
+    if USE_OFFSETS:
+        i_n, i_t = tl.load(indices + i_t * 2).to(tl.int32), tl.load(indices + i_t * 2 + 1).to(tl.int32)
+        bos, eos = tl.load(offsets + i_n).to(tl.int32), tl.load(offsets + i_n + 1).to(tl.int32)
+        T = eos - bos
+    else:
+        bos, eos = i_b * T, i_b * T + T
+
+    if HEAD_FIRST:
+        p_A = tl.make_block_ptr(A + i_bh * T * BT, (T, BT), (BT, 1), (i_t * BT, 0), (BT, BT), (1, 0))
+    else:
+        p_A = tl.make_block_ptr(A + (bos*H + i_h) * BT, (T, BT), (H*BT, 1), (i_t * BT, 0), (BT, BT), (1, 0))
+
+    b_A = tl.load(p_A, boundary_check=(0, 1))
+    b_Aak = tl.zeros([BT, BT], dtype=tl.float32)
+
+    for i_k in range(tl.cdiv(K, BK)):
+        if HEAD_FIRST:
+            p_k = tl.make_block_ptr(k + i_bh * T*K, (T, K), (K, 1), (i_t * BT, i_k * BK), (BT, BK), (1, 0))
+            p_a = tl.make_block_ptr(a + i_bh * T*K, (T, K), (K, 1), (i_t * BT, i_k * BK), (BT, BK), (1, 0))
+            p_w = tl.make_block_ptr(w + i_bh * T*K, (T, K), (K, 1), (i_t * BT, i_k * BK), (BT, BK), (1, 0))
+        else:
+            p_k = tl.make_block_ptr(k + (bos * H + i_h) * K, (T, K), (H*K, 1), (i_t * BT, i_k * BK), (BT, BK), (1, 0))
+            p_a = tl.make_block_ptr(a + (bos * H + i_h) * K, (T, K), (H*K, 1), (i_t * BT, i_k * BK), (BT, BK), (1, 0))
+            p_w = tl.make_block_ptr(w + (bos * H + i_h) * K, (T, K), (H*K, 1), (i_t * BT, i_k * BK), (BT, BK), (1, 0))
+        b_k = tl.load(p_k, boundary_check=(0, 1))
+        b_a = tl.load(p_a, boundary_check=(0, 1))
+        b_w = tl.dot(b_A, b_a)
+        b_Aak += tl.dot(b_a, tl.trans(b_k))
+        tl.store(p_w, b_w.to(p_w.dtype.element_ty), boundary_check=(0, 1))
+
+    b_Aak = tl.where(tl.arange(0, BT)[:, None] > tl.arange(0, BT)[None, :], b_Aak, 0)
+    b_Aak = b_Aak.to(k.dtype.element_ty)
+
+    for i_v in range(tl.cdiv(V, BV)):
+        if HEAD_FIRST:
+            p_v = tl.make_block_ptr(v + i_bh * T*V, (T, V), (V, 1), (i_t * BT, i_v * BV), (BT, BV), (1, 0))
+            p_u = tl.make_block_ptr(u + i_bh * T*V, (T, V), (V, 1), (i_t * BT, i_v * BV), (BT, BV), (1, 0))
+        else:
+            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_u = tl.make_block_ptr(u + (bos*H + i_h) * V, (T, V), (H*V, 1), (i_t * BT, i_v * BV), (BT, BV), (1, 0))
+        b_v = tl.load(p_v, boundary_check=(0, 1))
+        b_v = tl.dot(b_Aak, b_v).to(v.dtype.element_ty)
+        b_u = tl.dot(b_A, b_v)
+        tl.store(p_u, b_u.to(p_u.dtype.element_ty), boundary_check=(0, 1))
+
+
+def fwd_prepare_wy_repr(
+    a: torch.Tensor,
+    b: torch.Tensor,
+    v: torch.Tensor,
+    k: torch.Tensor,
+    offsets: Optional[torch.LongTensor],
+    indices: Optional[torch.LongTensor],
+    head_first: bool = True,
+    chunk_size: int = 64
+) -> Tuple[torch.Tensor, torch.Tensor, torch.Tensor]:
+    if head_first:
+        B, H, T, K = a.shape
+    else:
+        B, T, H, K = a.shape
+    BT = min(chunk_size, max(triton.next_power_of_2(T), 16))
+    NT = triton.cdiv(T, BT) if offsets is None else len(indices)
+    BC = min(BT, 32)
+    BK = min(triton.next_power_of_2(K), 64)
+
+    A = torch.empty(B, *((H, T) if head_first else (T, H)), BT, device=a.device, dtype=a.dtype)
+    fwd_fn = fwd_prepare_wy_repr_kernel_chunk64 if BT == 64 else fwd_prepare_wy_repr_kernel_chunk32
+
+    fwd_fn[(NT, B * H)](
+        a=a,
+        b=b,
+        A=A,
+        offsets=offsets,
+        indices=indices,
+        T=T,
+        H=H,
+        K=K,
+        BT=BT,
+        BK=BK,
+        BC=BC,
+        HEAD_FIRST=head_first
+    )
+    w, u = fwd_wu(
+        a=a,
+        v=v,
+        k=k,
+        A=A,
+        offsets=offsets,
+        indices=indices,
+        head_first=head_first,
+        chunk_size=chunk_size
+    )
+    return w, u, A
+
+
+def fwd_wu(
+    a: torch.Tensor,
+    v: torch.Tensor,
+    k: torch.Tensor,
+    A: torch.Tensor,
+    offsets: Optional[torch.LongTensor],
+    indices: Optional[torch.LongTensor],
+    head_first: bool,
+    chunk_size: int
+) -> Tuple[torch.Tensor, torch.Tensor]:
+    if head_first:
+        B, H, T, K, V = *a.shape, v.shape[-1]
+    else:
+        B, T, H, K, V = *a.shape, v.shape[-1]
+    BT = min(chunk_size, max(triton.next_power_of_2(T), 16))
+    NT = triton.cdiv(T, BT) if offsets is None else len(indices)
+    CONST_TILING = 64 if check_shared_mem() else 32
+    BK = min(triton.next_power_of_2(K), CONST_TILING)
+    BV = min(triton.next_power_of_2(V), CONST_TILING)
+
+    u = torch.empty_like(v)
+    w = torch.empty_like(a)
+    fwd_wu_kernel[(NT, B*H)](
+        a=a,
+        v=v,
+        w=w,
+        u=u,
+        A=A,
+        k=k,
+        offsets=offsets,
+        indices=indices,
+        T=T,
+        H=H,
+        K=K,
+        V=V,
+        BT=BT,
+        BK=BK,
+        BV=BV,
+        HEAD_FIRST=head_first
+    )
+    return w, u

fla/ops/gsa/__init__.py

@@ -0,0 +1,9 @@
+# -*- coding: utf-8 -*-
+
+from .chunk import chunk_gsa
+from .fused_recurrent import fused_recurrent_gsa
+
+__all__ = [
+    'chunk_gsa',
+    'fused_recurrent_gsa'
+]

fla/ops/gsa/naive.py

@@ -0,0 +1,68 @@
+# -*- coding: utf-8 -*-
+
+from typing import Optional
+
+import torch
+from einops import repeat
+
+
+def naive_recurrent_gsa(
+    q: torch.Tensor,
+    k: torch.Tensor,
+    v: torch.Tensor,
+    s: torch.Tensor,
+    g: Optional[torch.Tensor] = None,
+    scale: Optional[int] = None,
+    initial_state: Optional[torch.Tensor] = None,
+    output_final_state: Optional[bool] = False
+) -> torch.Tensor:
+    dtype = q.dtype
+
+    NG = q.shape[1]//k.shape[1]
+    # [batch_size, n_heads, seq_len, n_slots]
+    if g is None:
+        z = s.float().logcumsumexp(2)
+        g = torch.cat((z[:, :, :1], z[:, :, :-1]), 2) - z
+        s = torch.exp(s - z)
+    q, k, v, s, g = map(lambda x: x.float(), (q, k, v, s, g))
+    k, v, s, g = map(lambda x: repeat(x, 'b h t d -> b (h g) t d', g=NG), (k, v, s, g))
+    if initial_state is not None:
+        initial_state = tuple(map(lambda x: repeat(x, 'b h k v -> b (h g) k v', g=NG), initial_state))
+
+    B, H, T, K, V, M = *q.shape, v.shape[-1], s.shape[-1]
+
+    hk = torch.zeros(B, H, K, M, dtype=torch.float, device=q.device)
+    ok = torch.zeros_like(s)
+
+    if scale is None:
+        scale = q.shape[-1] ** -0.5
+
+    final_state = None
+    if initial_state is not None:
+        hk += initial_state[0]
+
+    for i in range(T):
+        q_i = q[:, :, i] * scale
+        k_i = k[:, :, i]
+        v_i = s[:, :, i]
+        g_i = g[:, :, i].exp()
+        hk = hk * g_i[..., None, :] + k_i[..., None] * v_i[..., None, :]
+        ok[:, :, i] = (q_i[..., None] * hk).sum(-2)
+
+    qv = ok.softmax(-1)
+    hv = torch.zeros(B, H, M, V, dtype=torch.float, device=q.device)
+    ov = torch.zeros_like(v)
+    if initial_state is not None:
+        hv += initial_state[1]
+
+    for i in range(T):
+        q_i = qv[:, :, i]
+        k_i = s[:, :, i]
+        v_i = v[:, :, i]
+        g_i = g[:, :, i].exp()
+        hv = hv * g_i[..., :, None] + k_i[..., None] * v_i[..., None, :]
+        ov[:, :, i] = (q_i[..., None] * hv).sum(-2)
+
+    if output_final_state:
+        final_state = (hk.view(B, -1, NG, K, M)[:, :, 0], hv.view(B, -1, NG, M, V)[:, :, 0])
+    return ov.to(dtype), final_state

fla/ops/hgrn/chunk.py

@@ -0,0 +1,282 @@
+# -*- coding: utf-8 -*-
+# Copyright (c) 2023-2025, Songlin Yang, Yu Zhang
+
+# this function implements the chunkwise form of HGRN, inspired by
+# [Volodymyr Kyrylov in his blog post](https://proger.github.io/posts/scan/chunk.html)
+# also refer to the `accelerated-scan` lib: https://github.com/proger/accelerated-scan
+
+# from tests on H800, with B, D = 16, 128, we see that the chunk can be greatly faster than the recurrent:
+#
+# Performance:
+#    seq_len     chunk  recurrent  chunk_bwd  recurrent_bwd
+# 0    128.0  0.039360   0.061056   0.312160       0.205008
+# 1    256.0  0.045824   0.123712   0.308784       0.297696
+# 2    512.0  0.058688   0.241952   0.310720       0.626528
+# 3   1024.0  0.088288   0.476992   0.313184       1.333152
+# 4   2048.0  0.169472   0.943264   0.452464       2.724864
+# 5   4096.0  0.329920   1.886144   0.881600       5.551520
+# 6   8192.0  0.647872   3.755040   1.740496      11.117184
+# 7  16384.0  1.272064   7.520576   3.446608      22.362528
+
+from typing import Tuple
+
+import torch
+import triton
+import triton.language as tl
+
+from fla.ops.utils.op import exp
+from fla.utils import input_guard
+
+
+@triton.autotune(
+    configs=[
+        triton.Config({'BD': 32}, num_warps=1),
+        triton.Config({'BD': 32}, num_warps=2),
+        triton.Config({'BD': 32}, num_warps=4),
+        triton.Config({'BD': 32}, num_warps=8),
+        triton.Config({'BD': 64}, num_warps=1),
+        triton.Config({'BD': 64}, num_warps=2),
+        triton.Config({'BD': 64}, num_warps=4),
+        triton.Config({'BD': 64}, num_warps=8),
+        triton.Config({'BD': 128}, num_warps=1),
+        triton.Config({'BD': 128}, num_warps=2),
+        triton.Config({'BD': 128}, num_warps=4),
+        triton.Config({'BD': 128}, num_warps=8),
+    ],
+    key=['D']
+)
+@triton.jit(do_not_specialize=['T'])
+def chunk_hgrn_fwd_kernel_h(
+    x,
+    g,
+    gc,
+    o,
+    h0,
+    T,
+    D: tl.constexpr,
+    BT: tl.constexpr,
+    BD: tl.constexpr,
+    USE_INITIAL_STATE: tl.constexpr
+):
+    i_d, i_t, i_b = tl.program_id(0), tl.program_id(1), tl.program_id(2)
+    o_d = i_d * BD + tl.arange(0, BD)
+    mask = o_d < D
+
+    p_x = x + i_b * T * D + i_t * BT * D + o_d
+    p_g = g + i_b * T * D + i_t * BT * D + o_d
+    p_gc = gc + i_b * T * D + i_t * BT * D + o_d
+    p_o = o + i_b * T * D + i_t * BT * D + o_d
+
+    b_h = tl.zeros([BD], dtype=tl.float32)
+    b_gc = tl.zeros([BD], dtype=tl.float32)
+    if USE_INITIAL_STATE:
+        if i_t == 0:
+            b_h += tl.load(h0 + i_b * D + o_d, mask=mask, other=0).to(tl.float32)
+    for i in range(0, BT):
+        mask_t = mask & ((i_t * BT + i) < T)
+        b_x = tl.load(p_x, mask=mask_t, other=0).to(tl.float32)
+        b_g = tl.load(p_g, mask=mask_t, other=0).to(tl.float32)
+        b_h = exp(b_g) * b_h + b_x
+        b_gc = b_gc + b_g
+        tl.store(p_gc, b_gc.to(p_o.dtype.element_ty), mask=mask_t)
+        tl.store(p_o, b_h.to(p_o.dtype.element_ty), mask=mask_t)
+
+        p_x += D
+        p_g += D
+        p_gc += D
+        p_o += D
+
+
+@triton.jit(do_not_specialize=['T'])
+def chunk_hgrn_fwd_kernel_o(
+    gc,
+    o,
+    s_b,
+    s_t,
+    s_d,
+    T,
+    D: tl.constexpr,
+    BT: tl.constexpr,
+    BD: tl.constexpr
+):
+    i_d, i_b = tl.program_id(0), tl.program_id(1)
+    o_d = i_d * BD + tl.arange(0, BD)
+    mask = o_d < D
+
+    for i_t in range(1, tl.cdiv(T, BT)):
+        p_gc = tl.make_block_ptr(gc + i_b * s_b, (T, D), (s_t, s_d), (i_t * BT, i_d * BD), (BT, BD), (1, 0))
+        p_o = tl.make_block_ptr(o + i_b * s_b, (T, D), (s_t, s_d), (i_t * BT, i_d * BD), (BT, BD), (1, 0))
+
+        # [BD,]
+        b_h0 = tl.load(o + i_b * T * D + i_t * BT * D - D + o_d, mask=mask, other=0).to(tl.float32)
+        # [BT, BD]
+        b_gc = tl.load(p_gc, boundary_check=(0, 1)).to(tl.float32)
+        b_o = tl.load(p_o, boundary_check=(0, 1)).to(tl.float32)
+        b_o = b_o + exp(b_gc) * b_h0[None, :]
+        tl.store(p_o, b_o.to(p_o.dtype.element_ty), boundary_check=(0, 1))
+
+
+@triton.autotune(
+    configs=[
+        triton.Config({'BD': BD}, num_warps=num_warps)
+        for BD in [32, 64, 128]
+        for num_warps in [1, 2, 4, 8]
+    ],
+    key=['D']
+)
+@triton.jit(do_not_specialize=['T'])
+def chunk_hgrn_bwd_kernel_h(
+    g,
+    gc,
+    dx,
+    do,
+    T,
+    D: tl.constexpr,
+    BT: tl.constexpr,
+    BD: tl.constexpr
+):
+    i_d, i_t, i_b = tl.program_id(0), tl.program_id(1), tl.program_id(2)
+    o_d = i_d * BD + tl.arange(0, BD)
+    mask = o_d < D
+    BC = min(BT, T - i_t * BT)
+    NT = tl.num_programs(1)
+
+    p_g = g + (i_b * T + i_t * BT + BC - 1) * D + o_d
+    p_gc = gc + (i_b * T + i_t * BT + BC - 1) * D + o_d
+    p_dx = dx + (i_b * T + i_t * BT + BC - 1) * D + o_d
+    p_do = do + (i_b * T + i_t * BT + BC - 1) * D + o_d
+
+    if i_t == NT - 1:
+        b_gc = tl.zeros([BD], dtype=tl.float32)
+    else:
+        b_gc = tl.load(g + (i_b * T + i_t * BT + BT) * D + o_d, mask=mask, other=0).to(tl.float32)
+    b_dh = tl.zeros([BD], dtype=tl.float32)
+    for _ in range(BC - 1, -1, -1):
+        tl.store(p_gc, b_gc.to(p_gc.dtype.element_ty), mask=mask)
+
+        b_g = tl.load(p_g, mask=mask, other=0).to(tl.float32)
+        b_do = tl.load(p_do, mask=mask, other=0).to(tl.float32)
+
+        b_gc = b_gc + b_g
+        b_dh = b_dh + b_do
+        b_dx = b_dh
+        b_dh = b_dh * exp(b_g)
+
+        tl.store(p_dx, b_dx.to(p_dx.dtype.element_ty), mask=mask)
+
+        p_g -= D
+        p_gc -= D
+        p_dx -= D
+        p_do -= D
+
+
+@triton.jit(do_not_specialize=['T'])
+def chunk_hgrn_bwd_kernel_o(
+    g,
+    gc,
+    o,
+    dx,
+    dg,
+    s_b,
+    s_t,
+    s_d,
+    T,
+    D: tl.constexpr,
+    BT: tl.constexpr,
+    BD: tl.constexpr
+):
+    i_d, i_b = tl.program_id(0), tl.program_id(1)
+    o_d = i_d * BD + tl.arange(0, BD)
+    mask = o_d < D
+
+    for i_t in range(tl.cdiv(T, BT) - 1, -1, -1):
+        p_g = tl.make_block_ptr(g + i_b * s_b, (T, D), (s_t, s_d), (i_t * BT, i_d * BD), (BT, BD), (1, 0))
+        p_gc = tl.make_block_ptr(gc + i_b * s_b, (T, D), (s_t, s_d), (i_t * BT, i_d * BD), (BT, BD), (1, 0))
+        p_o = tl.make_block_ptr(o + i_b * s_b, (T, D), (s_t, s_d), (i_t * BT - 1, i_d * BD), (BT, BD), (1, 0))
+        p_dx = tl.make_block_ptr(dx + i_b * s_b, (T, D), (s_t, s_d), (i_t * BT, i_d * BD), (BT, BD), (1, 0))
+        p_dg = tl.make_block_ptr(dg + i_b * s_b, (T, D), (s_t, s_d), (i_t * BT, i_d * BD), (BT, BD), (1, 0))
+
+        # [BD,]
+        mask_t = mask & ((i_t + 1) * BT < T)
+        b_ht = tl.load(dx + i_b * T * D + (i_t + 1) * BT * D + o_d, mask=mask_t, other=0).to(tl.float32)
+        # [BT, BD]
+        b_g = tl.load(p_g, boundary_check=(0, 1)).to(tl.float32)
+        b_gc = tl.load(p_gc, boundary_check=(0, 1)).to(tl.float32)
+        b_o = tl.load(p_o, boundary_check=(0, 1)).to(tl.float32)
+        b_dx = tl.load(p_dx, boundary_check=(0, 1)).to(tl.float32)
+
+        b_dx = b_dx + exp(b_gc) * b_ht[None, :]
+        b_dg = b_o * b_dx * exp(b_g)
+        tl.store(p_dx, b_dx.to(p_dx.dtype.element_ty), boundary_check=(0, 1))
+        tl.store(p_dg, b_dg.to(p_dg.dtype.element_ty), boundary_check=(0, 1))
+
+
+class ChunkHGRNFunction(torch.autograd.Function):
+
+    @staticmethod
+    @input_guard
+    def forward(ctx, x, g, initial_state=None, output_final_state=False):
+        B, T, D = x.shape
+        BT, BD = 128, min(64, triton.next_power_of_2(D))
+        num_warps = 8 if BD == 64 else 4
+
+        gc = torch.empty_like(g, dtype=torch.float)
+        o = torch.empty_like(x, dtype=torch.float)
+        def grid(meta): return (triton.cdiv(D, meta['BD']), triton.cdiv(T, meta['BT']), B)
+        chunk_hgrn_fwd_kernel_h[grid](
+            x, g, gc, o, initial_state,
+            T=T, D=D, BT=BT,
+            USE_INITIAL_STATE=initial_state is not None
+        )
+        def grid(meta): return (triton.cdiv(D, meta['BD']), B)
+        chunk_hgrn_fwd_kernel_o[grid](
+            gc, o,
+            o.stride(-3), o.stride(-2), o.stride(-1),
+            T=T, D=D, BT=BT, BD=BD,
+            num_warps=num_warps
+        )
+        final_state = None
+        if output_final_state:
+            final_state = o[:, -1].clone()
+        o = o.to(x.dtype)
+        ctx.save_for_backward(g, o, initial_state)
+        return o, final_state
+
+    @staticmethod
+    @input_guard
+    def backward(ctx, do, dht=None):
+        g, o, initial_state = ctx.saved_tensors
+        B, T, D = do.shape
+        BT, BD = 128, min(64, triton.next_power_of_2(D))
+        num_warps = 8 if BD == 64 else 4
+
+        gc = torch.empty_like(g, dtype=torch.float)
+        dx = torch.empty_like(o, dtype=torch.float)
+        def grid(meta): return (triton.cdiv(D, meta['BD']), triton.cdiv(T, meta['BT']), B)
+        chunk_hgrn_bwd_kernel_h[grid](
+            g, gc, dx, do,
+            T=T, D=D, BT=BT
+        )
+
+        dg = torch.empty_like(g, dtype=torch.float)
+        def grid(meta): return (triton.cdiv(D, meta['BD']), B)
+        chunk_hgrn_bwd_kernel_o[grid](
+            g, gc, o, dx, dg,
+            o.stride(-3), o.stride(-2), o.stride(-1),
+            T=T, D=D, BT=BT, BD=BD,
+            num_warps=num_warps
+        )
+        if initial_state is not None:
+            dg[:, 0] = (initial_state * dx[:, 0] * g[:, 0].float().exp()).to(dg.dtype)
+
+        return dx.to(o.dtype), dg, None, None
+
+
+@torch.compiler.disable
+def chunk_hgrn(
+    x: torch.Tensor,
+    g: torch.Tensor,
+    initial_state: torch.Tensor = None,
+    output_final_state: bool = False
+) -> Tuple[torch.Tensor, torch.Tensor]:
+    return ChunkHGRNFunction.apply(x, g, initial_state, output_final_state)

fla/ops/hgrn/fused_recurrent.py

@@ -0,0 +1,308 @@
+# -*- coding: utf-8 -*-
+# Copyright (c) 2023-2025, Songlin Yang, Yu Zhang
+
+from typing import Optional, Tuple
+
+import torch
+import triton
+import triton.language as tl
+
+from fla.ops.utils.op import exp
+from fla.utils import input_guard
+
+
+@triton.heuristics({
+    'USE_INITIAL_STATE': lambda args: args['h0'] is not None,
+    'STORE_FINAL_STATE': lambda args: args['ht'] is not None,
+    'USE_OFFSETS': lambda args: args['offsets'] is not None
+})
+@triton.autotune(
+    configs=[
+        triton.Config({'BD': BD}, num_warps=num_warps)
+        for BD in [32, 64, 128]
+        for num_warps in [1, 2, 4, 8]
+    ],
+    key=['D']
+)
+@triton.jit(do_not_specialize=['T'])
+def fused_recurrent_hgrn_fwd_kernel(
+    x,
+    g,
+    o,
+    h0,
+    ht,
+    offsets,
+    T,
+    D: tl.constexpr,
+    BD: tl.constexpr,
+    USE_INITIAL_STATE: tl.constexpr,
+    STORE_FINAL_STATE: tl.constexpr,
+    USE_OFFSETS: tl.constexpr
+):
+    i_d, i_n = tl.program_id(0), tl.program_id(1)
+    if USE_OFFSETS:
+        bos, eos = tl.load(offsets + i_n).to(tl.int64), tl.load(offsets + i_n + 1).to(tl.int64)
+        T = eos - bos
+    else:
+        bos, eos = i_n * T, i_n * T + T
+
+    o_d = i_d * BD + tl.arange(0, BD)
+    mask = o_d < D
+
+    p_x = x + bos * D + o_d
+    p_g = g + bos * D + o_d
+    p_o = o + bos * D + o_d
+
+    b_h = tl.zeros([BD], dtype=tl.float32)
+    if USE_INITIAL_STATE:
+        p_h0 = h0 + i_n * D + o_d
+        b_h += tl.load(p_h0, mask=mask, other=0).to(tl.float32)
+    for _ in range(0, T):
+        b_x = tl.load(p_x, mask=mask, other=0).to(tl.float32)
+        b_g = tl.load(p_g, mask=mask, other=0).to(tl.float32)
+        b_h = exp(b_g) * b_h + b_x
+        tl.store(p_o, b_h.to(p_o.dtype.element_ty), mask=mask)
+
+        p_x += D
+        p_g += D
+        p_o += D
+
+    if STORE_FINAL_STATE:
+        p_ht = ht + i_n * D + o_d
+        tl.store(p_ht, b_h.to(p_ht.dtype.element_ty), mask=mask)
+
+
+@triton.heuristics({
+    'USE_INITIAL_STATE': lambda args: args['h0'] is not None,
+    'USE_FINAL_STATE_GRADIENT': lambda args: args['dht'] is not None,
+    'USE_OFFSETS': lambda args: args['offsets'] is not None
+})
+@triton.autotune(
+    configs=[
+        triton.Config({'BD': BD}, num_warps=num_warps)
+        for BD in [32, 64, 128]
+        for num_warps in [1, 2, 4, 8]
+    ],
+    key=['D']
+)
+@triton.jit(do_not_specialize=['T'])
+def fused_recurrent_hgrn_bwd_kernel(
+    g,
+    o,
+    h0,
+    dx,
+    dg,
+    do,
+    dht,
+    dh0,
+    offsets,
+    T,
+    D: tl.constexpr,
+    BD: tl.constexpr,
+    USE_INITIAL_STATE: tl.constexpr,
+    USE_FINAL_STATE_GRADIENT: tl.constexpr,
+    USE_OFFSETS: tl.constexpr
+):
+    i_d, i_n = tl.program_id(0), tl.program_id(1)
+    if USE_OFFSETS:
+        bos, eos = tl.load(offsets + i_n).to(tl.int64), tl.load(offsets + i_n + 1).to(tl.int64)
+        T = eos - bos
+    else:
+        bos, eos = i_n * T, i_n * T + T
+
+    o_d = i_d * BD + tl.arange(0, BD)
+    mask = o_d < D
+
+    p_g = g + (bos + T - 1) * D + o_d
+    p_o = o + (bos + T - 2) * D + o_d
+    p_dx = dx + (bos + T - 1) * D + o_d
+    p_dg = dg + (bos + T - 1) * D + o_d
+    p_do = do + (bos + T - 1) * D + o_d
+
+    b_dh = tl.zeros([BD], dtype=tl.float32)
+    if USE_FINAL_STATE_GRADIENT:
+        p_dht = dht + i_n * D + o_d
+        b_dh += tl.load(p_dht, mask=mask, other=0).to(tl.float32)
+
+    for i in range(T - 1, -1, -1):
+        b_g = tl.load(p_g, mask=mask, other=0).to(tl.float32)
+        b_do = tl.load(p_do, mask=mask, other=0).to(tl.float32)
+        if i > 0:
+            b_o = tl.load(p_o, mask=mask, other=0).to(tl.float32)
+        elif USE_INITIAL_STATE:
+            b_o = tl.load(h0 + i_n * D + o_d, mask=mask, other=0).to(tl.float32)
+        else:
+            b_o = tl.zeros([BD], dtype=tl.float32)
+
+        b_dh = b_dh + b_do
+        b_dx = b_dh
+        b_dh = b_dh * exp(b_g)
+        b_dg = b_dh * b_o
+        tl.store(p_dx, b_dx.to(p_dx.dtype.element_ty), mask=mask)
+        tl.store(p_dg, b_dg.to(p_dg.dtype.element_ty), mask=mask)
+
+        p_g -= D
+        p_o -= D
+        p_dx -= D
+        p_dg -= D
+        p_do -= D
+
+    if USE_INITIAL_STATE:
+        p_dh0 = dh0 + i_n * D + o_d
+        tl.store(p_dh0, b_dh.to(p_dh0.dtype.element_ty), mask=mask)
+
+
+def fused_recurrent_hgrn_fwd(
+    x: torch.Tensor,
+    g: torch.Tensor,
+    initial_state: torch.Tensor = None,
+    output_final_state: bool = False,
+    offsets: Optional[torch.LongTensor] = None,
+) -> Tuple[torch.Tensor, torch.Tensor]:
+    B, T, D = x.shape
+    N = B if offsets is None else len(offsets) - 1
+
+    o = torch.empty_like(x)
+    final_state = x.new_empty(N, D) if output_final_state else None
+
+    def grid(meta): return (triton.cdiv(D, meta['BD']), N)
+    fused_recurrent_hgrn_fwd_kernel[grid](
+        x=x,
+        g=g,
+        o=o,
+        h0=initial_state,
+        ht=final_state,
+        offsets=offsets,
+        T=T,
+        D=D
+    )
+    return o, final_state
+
+
+def fused_recurrent_hgrn_bwd(
+    g: torch.Tensor,
+    o: torch.Tensor,
+    do: torch.Tensor,
+    dht: torch.Tensor = None,
+    initial_state: torch.Tensor = None,
+    offsets: Optional[torch.LongTensor] = None
+) -> Tuple[torch.Tensor, torch.Tensor]:
+    B, T, D = do.shape
+    N = B if offsets is None else len(offsets) - 1
+
+    dx = torch.empty_like(o, dtype=torch.float)
+    dg = torch.empty_like(g, dtype=torch.float)
+    dh0 = torch.empty_like(initial_state, dtype=torch.float) if initial_state is not None else None
+    def grid(meta): return (triton.cdiv(D, meta['BD']), N)
+    fused_recurrent_hgrn_bwd_kernel[grid](
+        g=g,
+        o=o,
+        h0=initial_state,
+        dx=dx,
+        dg=dg,
+        do=do,
+        dht=dht,
+        dh0=dh0,
+        offsets=offsets,
+        T=T,
+        D=D
+    )
+    return dx, dg, dh0
+
+
+class FusedRecurrentHGRNFunction(torch.autograd.Function):
+
+    @staticmethod
+    @input_guard
+    def forward(
+        ctx,
+        x: torch.Tensor,
+        g: torch.Tensor,
+        initial_state: torch.Tensor = None,
+        output_final_state: bool = False,
+        offsets: Optional[torch.LongTensor] = None
+    ):
+        o, ht = fused_recurrent_hgrn_fwd(
+            x=x,
+            g=g,
+            initial_state=initial_state,
+            output_final_state=output_final_state,
+            offsets=offsets
+        )
+        ctx.save_for_backward(g, o, initial_state)
+        ctx.offsets = offsets
+        return o, ht
+
+    @staticmethod
+    @input_guard
+    def backward(ctx, do, dht=None):
+        g, o, initial_state = ctx.saved_tensors
+        offsets = ctx.offsets
+
+        dx, dg, dh0 = fused_recurrent_hgrn_bwd(
+            g=g,
+            o=o,
+            do=do,
+            dht=dht,
+            initial_state=initial_state,
+            offsets=offsets
+        )
+        return dx, dg, dh0, None, None
+
+
+@torch.compiler.disable
+def fused_recurrent_hgrn(
+    x: torch.Tensor,
+    g: torch.Tensor,
+    initial_state: torch.Tensor = None,
+    output_final_state: bool = False,
+    cu_seqlens: Optional[torch.LongTensor] = None,
+) -> Tuple[torch.Tensor, torch.Tensor]:
+    r"""
+    Args:
+        x (torch.Tensor):
+            inputs of shape `[B, T, D].
+        g (torch.Tensor):
+            Forget gates of shape `[B, T, D]`.
+        initial_state (Optional[torch.Tensor]):
+            Initial state of shape `[N, D]` for `N` input sequences.
+            For equal-length input sequences, `N` equals the batch size `B`.
+            Default: `None`.
+        output_final_state (Optional[bool]):
+            Whether to output the final state of shape `[N, D]`. Default: `False`.
+        cu_seqlens (torch.LongTensor):
+            Cumulative sequence lengths of shape `[N+1]` used for variable-length training,
+            consistent with the FlashAttention API.
+
+    Returns:
+        o (torch.Tensor):
+            Outputs of shape `[B, T, D]`.
+        final_state (torch.Tensor):
+            Final state of shape `[N, D]` if `output_final_state=True` else `None`.
+
+    Examples::
+        >>> import torch
+        >>> import torch.nn.functional as F
+        >>> from einops import rearrange
+        >>> from fla.ops.hgrn import fused_recurrent_hgrn
+        # inputs with equal lengths
+        >>> B, T, D = 4, 2048, 512
+        >>> x = torch.randn(B, T, D, device='cuda')
+        >>> g = F.logsigmoid(torch.randn(B, T, D, device='cuda'))
+        >>> h0 = torch.randn(B, D, device='cuda')
+        >>> o, ht = fused_recurrent_hgrn(x, g, initial_state=h0, output_final_state=True)
+        # for variable-length inputs, the batch size `B` is expected to be 1 and `cu_seqlens` is required
+        >>> x, g = map(lambda x: rearrange(x, 'b t d -> 1 (b t) d'), (x, g))
+        # for a batch with 4 sequences, `cu_seqlens` with 5 start/end positions are expected
+        >>> cu_seqlens = x.new_tensor([0, 2048, 4096, 6144, 8192], dtype=torch.long)
+        >>> o_var, ht_var = fused_recurrent_hgrn(x, g, initial_state=h0, output_final_state=True, cu_seqlens=cu_seqlens)
+        >>> assert o.allclose(o_var.view(o.shape))
+        >>> assert ht.allclose(ht_var)
+    """
+    return FusedRecurrentHGRNFunction.apply(
+        x,
+        g,
+        initial_state,
+        output_final_state,
+        cu_seqlens
+    )

fla/ops/linear_attn/fused_recurrent.py

@@ -0,0 +1,251 @@
+# -*- coding: utf-8 -*-
+# Copyright (c) 2024, Songlin Yang, Yu Zhang
+
+from typing import Optional, Tuple
+
+import torch
+import triton
+import triton.language as tl
+
+from fla.ops.linear_attn.utils import normalize_output
+from fla.utils import input_guard
+
+
+@triton.jit
+def fused_recurrent_linear_attn_fwd_kernel(
+    q,  # query [B, H, L, K]
+    k,  # key [B, H, L, V]
+    v,  # value [B, H, L, V]
+    o,  # output [B, H, L, V]
+    h0,
+    ht,  # final hidden state [B, H, K, V]
+
+    s_k_h,  # stride size: L * K
+    s_v_h,  # stride size: L * V
+
+    scale,
+    B,  # batch size
+    H,  # H
+    T,  # T
+    K: tl.constexpr,  # K
+    V: tl.constexpr,  # V
+    BK: tl.constexpr,  # BLOCK SIZE along the K dimension
+    BV: tl.constexpr,  # BLOCK SIZE along the V dimension
+    USE_INITIAL_STATE: tl.constexpr,  # whether to use initial state
+    STORE_FINAL_STATE: tl.constexpr,  # whether to store final state
+):
+    # indices
+    i_v, i_k, i_bh = tl.program_id(0), tl.program_id(1), tl.program_id(2)
+
+    p_q = q + i_bh * s_k_h + i_k * BK + tl.arange(0, BK)
+    p_k = k + i_bh * s_k_h + i_k * BK + tl.arange(0, BK)
+    p_v = v + i_bh * s_v_h + i_v * BV + tl.arange(0, BV)
+    p_o = o + (i_bh + i_k * B * H) * s_v_h + i_v * BV + tl.arange(0, BV)
+
+    mask_bk = (i_k * BK + tl.arange(0, BK)) < K
+    mask_bv = (i_v * BV + tl.arange(0, BV)) < V
+    mask_kv = mask_bk[None, :] & mask_bv[:, None]
+
+    b_h = tl.zeros([BV, BK], dtype=tl.float32)
+
+    if USE_INITIAL_STATE:
+        p_h0 = h0 + i_bh * K * V + (i_k * BK + tl.arange(0, BK)[None, :]) * V + (i_v * BV + tl.arange(0, BV)[:, None])
+        b_h += tl.load(p_h0, mask=mask_kv, other=0).to(tl.float32)
+
+    for _ in range(0, T):
+        b_k = tl.load(p_k, mask=mask_bk, other=0).to(tl.float32)
+        b_v = tl.load(p_v, mask=mask_bv, other=0).to(tl.float32)
+        b_q = tl.load(p_q, mask=mask_bk, other=0).to(tl.float32) * scale
+
+        b_h += b_k[None, :] * b_v[:, None]
+        b_o = b_h * b_q[None, :]
+        b_o = tl.sum(b_o, axis=1)
+        tl.store(p_o, b_o.to(p_o.dtype.element_ty), mask=mask_bv)
+
+        p_q += K
+        p_k += K
+        p_o += V
+        p_v += V
+
+    if STORE_FINAL_STATE:
+        p_ht = ht + i_bh * K * V + (i_k * BK + tl.arange(0, BK)[None, :]) * V + (i_v * BV + tl.arange(0, BV)[:, None])
+        tl.store(p_ht, b_h.to(p_ht.dtype.element_ty), mask=mask_kv)
+
+
+# Similar to Algorithm1 of https://arxiv.org/abs/2006.16236
+@triton.jit
+def fused_recurrent_linear_attn_bwd_kernel(
+    q,  # query [B, H, L, K]
+    k,  # key [B, H, L, V]
+    v,  # value [B, H, L, V]
+
+    do,  # gradient of output [B, H, L, V]
+    dq,  # gradient of query [NV, B, H, L, K]
+    dk,  # gradient of key [NV, B, H, L, K]
+    dv,  # gradient of value [NK, B, H, L, V]
+    h0,  # initial hidden state initialization [B, H, K, V]
+
+    s_k_h,  # stride size: L * K
+    s_v_h,  # stride size: L * V
+    scale,  # K ** -0.5
+
+    B,  # B
+    H,  # H
+    T,  # T
+    K: tl.constexpr,  # K
+    V: tl.constexpr,  # V
+    BK: tl.constexpr,  # BLOCK SIZE along the K dimension
+    BV: tl.constexpr,  # BLOCK SIZE along the V dimension
+    USE_INITIAL_STATE: tl.constexpr,  # whether to use initial state
+):
+    i_v, i_k, i_bh = tl.program_id(0), tl.program_id(1), tl.program_id(2)
+
+    p_q = q + i_bh * s_k_h + i_k * BK + tl.arange(0, BK)
+    p_k = k + i_bh * s_k_h + i_k * BK + tl.arange(0, BK)
+    p_v = v + i_bh * s_v_h + i_v * BV + tl.arange(0, BV)
+    p_do = do + i_bh * s_v_h + i_v * BV + tl.arange(0, BV)
+
+    p_dq = dq + (i_bh + i_v * B * H) * s_k_h + i_k * BK + tl.arange(0, BK)
+    mask_bk = i_k * BK + tl.arange(0, BK) < K
+    mask_bv = i_v * BV + tl.arange(0, BV) < V
+
+    b_h = tl.zeros([BK, BV], dtype=tl.float32)
+
+    if USE_INITIAL_STATE:
+        mask_kv = mask_bk[:, None] & mask_bv[None, :]
+        p_h0 = h0 + i_bh * K * V + (i_k * BK + tl.arange(0, BK)[:, None]) * V + (i_v * BV + tl.arange(0, BV)[None, :])
+        b_h += tl.load(p_h0, mask=mask_kv, other=0).to(tl.float32)
+
+    for _ in range(0, T):
+        b_k = tl.load(p_k, mask=mask_bk, other=0).to(tl.float32)
+        b_v = tl.load(p_v, mask=mask_bv, other=0).to(tl.float32)
+        b_do = tl.load(p_do, mask=mask_bv, other=0).to(tl.float32)
+
+        b_h += b_k[:, None] * b_v[None, :]
+        _d_q = b_h * b_do[None, :]
+        d_q = tl.sum(_d_q, axis=1) * scale
+        tl.store(p_dq, d_q.to(p_dq.dtype.element_ty), mask=mask_bk)
+
+        p_k += K
+        p_do += V
+        p_v += V
+        p_dq += K
+
+    # sync threads
+    tl.debug_barrier()
+
+    p_q = q + i_bh * s_k_h + i_k * BK + tl.arange(0, BK) + (T - 1) * K
+    p_k = k + i_bh * s_k_h + i_k * BK + tl.arange(0, BK) + (T - 1) * K
+    p_do = do + i_bh * s_v_h + i_v * BV + tl.arange(0, BV) + (T - 1) * V
+    p_v = v + i_bh * s_v_h + i_v * BV + tl.arange(0, BV) + (T - 1) * V
+    p_dk = dk + (i_bh + i_v * B * H) * s_k_h + i_k * BK + tl.arange(0, BK) + (T - 1) * K
+    p_dv = dv + (i_bh + i_k * B * H) * s_v_h + i_v * BV + tl.arange(0, BV) + (T - 1) * V
+    d_h = tl.zeros([BK, BV], dtype=tl.float32)
+
+    for _ in range(T):
+        b_do = tl.load(p_do, mask=mask_bv, other=0).to(tl.float32)
+        b_q = tl.load(p_q, mask=mask_bk, other=0).to(tl.float32) * scale
+        b_k = tl.load(p_k, mask=mask_bk, other=0).to(tl.float32)
+        b_v = tl.load(p_v, mask=mask_bv, other=0).to(tl.float32)
+        d_h += b_q[:, None] * b_do[None, :]
+        d_k = tl.sum(d_h * b_v[None, :], axis=1)
+        d_v = tl.sum(d_h * b_k[:, None], axis=0)
+
+        tl.store(p_dk, d_k.to(p_dk.dtype.element_ty), mask=mask_bk)
+        tl.store(p_dv, d_v.to(p_dv.dtype.element_ty), mask=mask_bv)
+
+        p_do -= V
+        p_q -= K
+        p_k -= K
+        p_v -= V
+        p_dk -= K
+        p_dv -= V
+
+
+class FusedRecurrentLinearAttentionFunction(torch.autograd.Function):
+
+    @staticmethod
+    @input_guard
+    def forward(ctx, q, k, v, scale, initial_state=None, output_final_state=False):
+        B, H, T, K = q.shape
+        V = v.shape[-1]
+
+        BK, BV = min(K, 32), min(V, 32)
+        NK, NV = triton.cdiv(K, BK), triton.cdiv(V, BV)
+        num_warps = 1
+        num_stages = 1
+
+        o = q.new_empty(NK, B, H, T, V)
+        final_state = q.new_empty(B, H, K, V) if output_final_state else None
+
+        grid = (NV, NK, B * H)
+        fused_recurrent_linear_attn_fwd_kernel[grid](
+            q, k, v, o, initial_state, final_state,
+            q.stride(1),
+            v.stride(1), scale,
+            B=B, H=H, T=T, K=K, V=V, BK=BK, BV=BV,
+            USE_INITIAL_STATE=initial_state is not None,
+            STORE_FINAL_STATE=final_state is not None,
+            num_warps=num_warps,
+            num_stages=num_stages
+        )
+
+        o = o.sum(0)
+        ctx.save_for_backward(q, k, v, initial_state)
+        ctx.scale = scale
+        return o, final_state
+
+    @staticmethod
+    @input_guard
+    def backward(ctx, do, dht=None):
+        q, k, v, initial_state = ctx.saved_tensors
+        B, H, T, K = q.shape
+        V = v.shape[-1]
+        scale = ctx.scale
+
+        BK, BV = min(K, 32), min(V, 32)
+        NK, NV = triton.cdiv(K, BK), triton.cdiv(V, BV)
+        num_warps = 1
+        num_stages = 1
+
+        dq = q.new_empty(NV, B, H, T, K)
+        dk = q.new_empty(NV, B, H, T, K)
+        dv = q.new_empty(NK, B, H, T, V)
+        grid = (NV, NK, B * H)
+
+        fused_recurrent_linear_attn_bwd_kernel[grid](
+            q, k, v, do, dq, dk, dv, initial_state,
+            q.stride(1),
+            v.stride(1),
+            scale,
+            B=B, H=H, T=T, K=K, V=V, BK=BK, BV=BV,
+            USE_INITIAL_STATE=initial_state is not None,
+            num_warps=num_warps,
+            num_stages=num_stages
+        )
+        dq = dq.sum(0)
+        dk = dk.sum(0)
+        dv = dv.sum(0)
+        return dq, dk, dv, None, None, None
+
+
+def fused_recurrent_linear_attn(
+    q: torch.Tensor,
+    k: torch.Tensor,
+    v: torch.Tensor,
+    scale: Optional[float] = None,
+    initial_state: torch.Tensor = None,
+    output_final_state: bool = False,
+    normalize: bool = False,
+    head_first: bool = True
+) -> Tuple[torch.Tensor, torch.Tensor]:
+    if scale is None:
+        scale = q.shape[-1] ** -0.5
+    if not head_first:
+        q, k, v = map(lambda x: x.transpose(1, 2), (q, k, v))
+    o, final_state = FusedRecurrentLinearAttentionFunction.apply(q, k, v, scale, initial_state, output_final_state)
+    if normalize:
+        o = normalize_output(q * scale, k, o)
+    if not head_first:
+        o = o.transpose(1, 2)
+    return o, final_state

fla/ops/rebased/parallel.py

@@ -0,0 +1,466 @@
+
+# -*- coding: utf-8 -*-
+# Copyright (c) 2023-2025, Songlin Yang, Yu Zhang
+
+import torch
+import triton
+import triton.language as tl
+
+from fla.utils import autocast_custom_bwd, autocast_custom_fwd, input_guard
+
+# Rebased: Linear Transformers with Learnable Kernel Functions are Better In-Context Models
+# https://github.com/corl-team/rebased/blob/main/flash_linear_attention/fla/ops/triton/rebased_fast/parallel.py
+
+
+@triton.jit(do_not_specialize=['T'])
+def parallel_rebased_fwd_kernel(
+    q,
+    k,
+    v,
+    o,
+    z,
+    scale,
+    T,
+    B: tl.constexpr,
+    H: tl.constexpr,
+    K: tl.constexpr,
+    V: tl.constexpr,
+    BTL: tl.constexpr,
+    BTS: tl.constexpr,
+    BK: tl.constexpr,
+    BV: tl.constexpr,
+):
+    # i_c: chunk index. used for sequence parallelism
+    i_kv, i_c, i_bh = tl.program_id(0), tl.program_id(1), tl.program_id(2)
+    NV = tl.cdiv(V, BV)
+    i_k = i_kv // (NV)
+    i_v = i_kv % (NV)
+
+    p_q = tl.make_block_ptr(q + i_bh * T*K, (T, K), (K, 1), (i_c*BTL, i_k*BK), (BTL, BK), (1, 0))
+    p_k = tl.make_block_ptr(k + i_bh * T*K, (K, T), (1, K), (i_k*BK, 0), (BK, BTS), (0, 1))
+    p_v = tl.make_block_ptr(v + i_bh * T*V, (T, V), (V, 1), (0, i_v*BV), (BTS, BV), (1, 0))
+
+    # [BQ, BD] block Q, in the shared memory throughout the whole kernel
+    b_q = tl.load(p_q, boundary_check=(0, 1))
+    b_q = (b_q * scale).to(b_q.dtype)
+    b_o = tl.zeros([BTL, BV], dtype=tl.float32)
+    b_z = tl.zeros([BTL], dtype=tl.float32)
+
+    # Q block and K block have no overlap
+    # no need for mask, thereby saving flops
+    for _ in range(0, i_c*BTL, BTS):
+        # [BK, BTS]
+        b_k = tl.load(p_k, boundary_check=(0, 1))
+
+        # [BTS, BV]
+        b_v = tl.load(p_v, boundary_check=(0, 1))
+        # [BTL, BTS]
+        b_s = tl.dot(b_q, (b_k), allow_tf32=False)
+        b_s = b_s * b_s
+        b_z += tl.sum(b_s, axis=1)
+
+        # [BQ, BD]
+        b_o = b_o + tl.dot(b_s.to(b_v.dtype), b_v, allow_tf32=False)
+        p_k = tl.advance(p_k, (0, BTS))
+        p_v = tl.advance(p_v, (BTS, 0))
+
+    # # rescale interchunk output
+    tl.debug_barrier()
+    o_q = tl.arange(0, BTL)
+    # # sync threads, easy for compiler to optimize
+    # tl.debug_barrier()
+
+    o_k = tl.arange(0, BTS)
+    p_k = tl.make_block_ptr(k + i_bh * T*K, (K, T), (1, K), (i_k*BK, i_c*BTL), (BK, BTS), (0, 1))
+    p_v = tl.make_block_ptr(v + i_bh * T*V, (T, V), (V, 1), (i_c*BTL, i_v*BV), (BTS, BV), (1, 0))
+    # Q block and K block have overlap. masks required
+    for _ in range(i_c*BTL, (i_c + 1) * BTL, BTS):
+        # [BK, BTS]
+        b_k = tl.load(p_k, boundary_check=(0, 1))
+        # [BTS, BV]
+        b_v = tl.load(p_v, boundary_check=(0, 1))
+        # [BTL, BTS]
+        m_s = o_q[:, None] >= o_k[None, :]
+        b_s = tl.dot(b_q, b_k, allow_tf32=False)
+        b_s = b_s * b_s
+        b_s = tl.where(m_s, b_s, 0)
+        b_z += tl.sum(b_s, axis=1)
+        # [BTL, BV]
+        b_o += tl.dot(b_s.to(b_q.dtype), b_v, allow_tf32=False)
+        p_k = tl.advance(p_k, (0, BTS))
+        p_v = tl.advance(p_v, (BTS, 0))
+        o_k += BTS
+
+    p_o = tl.make_block_ptr(o + (i_bh + B * H * i_k) * T*V, (T, V), (V, 1), (i_c*BTL, i_v*BV), (BTL, BV), (1, 0))
+    p_z = z + (i_bh + B * H * i_k) * T + i_c*BTL + tl.arange(0, BTL)
+    tl.store(p_o, b_o.to(p_o.dtype.element_ty), boundary_check=(0, 1))
+    tl.store(p_z, b_z.to(p_z.dtype.element_ty), mask=((i_c*BTL + tl.arange(0, BTL)) < T))
+
+
+@triton.jit(do_not_specialize=['T'])
+def _parallel_rebased_bwd_dq(
+    i_bh,
+    i_c,
+    i_k,
+    i_v,
+    i_h,
+    q,
+    k,
+    v,
+    do,
+    dz,
+    dq,
+    scale,
+    T,
+    B: tl.constexpr,
+    H: tl.constexpr,
+    K: tl.constexpr,
+    V: tl.constexpr,
+    BTL: tl.constexpr,
+    BTS: tl.constexpr,
+    BK: tl.constexpr,
+    BV: tl.constexpr
+):
+    p_do = tl.make_block_ptr(do + i_bh * T*V, (T, V), (V, 1), (i_c*BTL, i_v*BV), (BTL, BV), (1, 0))
+    p_q = tl.make_block_ptr(q + (i_bh) * T*K, (T, K), (K, 1), (i_c*BTL, i_k*BK), (BTL, BK), (1, 0))
+    b_q = tl.load(p_q, boundary_check=(0, 1))
+    b_do = tl.load(p_do, boundary_check=(0, 1)).to(b_q.dtype)
+    b_q = (b_q * scale).to(b_q.dtype)
+    b_dq = tl.zeros([BTL, BK], dtype=tl.float32)
+    p_k = tl.make_block_ptr(k + i_bh * T*K, (T, K), (K, 1), (0, i_k*BK), (BTS, BK), (1, 0))
+    p_v = tl.make_block_ptr(v + i_bh * T*V, (V, T), (1, V), (i_v*BV, 0), (BV, BTS), (0, 1))
+    p_dz = dz + i_bh * T + i_c*BTL + tl.arange(0, BTL)
+    b_dz = tl.load(p_dz, mask=(i_c*BTL + tl.arange(0, BTL)) < T)
+
+    for _ in range(0, i_c*BTL, BTS):
+        # [BTS, BK]
+        b_k = tl.load(p_k, boundary_check=(0, 1))
+        # [BV, BTS]
+        b_v = tl.load(p_v, boundary_check=(0, 1))
+        # [BTL, BTS]
+        b_ds = tl.dot(b_do, b_v, allow_tf32=False)
+        if i_v == 0:
+            b_ds += b_dz[:, None]
+        else:
+            b_ds = b_ds
+        b_s = tl.dot(b_q, tl.trans(b_k), allow_tf32=False)
+        # [BQ, BD]
+        b_dq += tl.dot((2 * b_ds * b_s).to(b_v.dtype), b_k, allow_tf32=False)
+        p_k = tl.advance(p_k, (BTS, 0))
+        p_v = tl.advance(p_v, (0, BTS))
+
+    b_dq *= scale
+    o_q = tl.arange(0, BTL)
+    o_k = tl.arange(0, BTS)
+    p_k = tl.make_block_ptr(k + i_bh * T*K, (T, K), (K, 1), (i_c*BTL, i_k*BK), (BTS, BK), (1, 0))
+    p_v = tl.make_block_ptr(v + i_bh * T*V, (V, T), (1, V), (i_v*BV, i_c*BTL), (BV, BTS), (0, 1))
+    # Q block and K block have overlap. masks required
+    for _ in range(i_c*BTL, (i_c + 1) * BTL, BTS):
+        # [BTS, BK]
+        b_k = tl.load(p_k, boundary_check=(0, 1))
+        # [BV, BTS]
+        b_v = tl.load(p_v, boundary_check=(0, 1))
+        # [BTL, BTS]
+        m_s = o_q[:, None] >= o_k[None, :]
+        b_ds = tl.dot(b_do, b_v, allow_tf32=False)
+        if i_v == 0:
+            b_ds += b_dz[:, None]
+        else:
+            b_ds = b_ds
+        b_ds = tl.where(m_s, b_ds, 0) * scale
+        b_s = tl.dot(b_q, tl.trans(b_k), allow_tf32=False)
+        b_s = tl.where(m_s, b_s, 0)
+        # [BTL, BK]
+        b_dq += tl.dot((2 * b_ds * b_s).to(b_k.dtype),
+                       b_k, allow_tf32=False)
+        p_k = tl.advance(p_k, (BTS, 0))
+        p_v = tl.advance(p_v, (0, BTS))
+        o_k += BTS
+    p_dq = tl.make_block_ptr(dq + (i_bh + B * H * i_v) * T*K, (T, K), (K, 1), (i_c*BTL, i_k*BK), (BTL, BK), (1, 0))
+    tl.store(p_dq, b_dq.to(p_dq.dtype.element_ty), boundary_check=(0, 1))
+    return
+
+
+@triton.jit(do_not_specialize=['T'])
+def _parallel_rebased_bwd_dkv(
+    i_bh,
+    i_c,
+    i_k,
+    i_v,
+    i_h,
+    q,
+    k,
+    v,
+    do,
+    dz,
+    dk,
+    dv,
+    scale,
+    T,
+    B: tl.constexpr,
+    H: tl.constexpr,
+    K: tl.constexpr,
+    V: tl.constexpr,
+    BTL: tl.constexpr,
+    BTS: tl.constexpr,
+    BK: tl.constexpr,
+    BV: tl.constexpr,
+):
+    # compute dk dv
+    p_k = tl.make_block_ptr(k + i_bh * T*K, (T, K), (K, 1), (i_c*BTL, i_k*BK), (BTL, BK), (1, 0))
+    p_v = tl.make_block_ptr(v + i_bh * T*V, (T, V), (V, 1), (i_c*BTL, i_v*BV), (BTL, BV), (1, 0))
+    b_k, b_v = tl.load(p_k, boundary_check=(0, 1)), tl.load(p_v, boundary_check=(0, 1))
+    b_dk, b_dv = tl.zeros([BTL, BK], dtype=tl.float32), tl.zeros(
+        [BTL, BV], dtype=tl.float32)
+
+    for i in range((tl.cdiv(T, BTS) * BTS)-BTS, (i_c + 1) * BTL - BTS, -BTS):
+        p_q = tl.make_block_ptr(q + i_bh * T*K, (K, T), (1, K), (i_k*BK, i), (BK, BTS), (0, 1))
+        p_do = tl.make_block_ptr(do + i_bh * T*V, (V, T), (1, V), (i_v*BV, i), (BV, BTS), (0, 1))
+        p_dz = dz + i_bh * T + i + tl.arange(0, BTS)
+        # [BK, BTS]
+        b_q = tl.load(p_q, boundary_check=(0, 1))
+        # [BV, BTS]
+        b_do = tl.load(p_do, boundary_check=(0, 1)).to(b_q.dtype)
+        b_dz = tl.load(p_dz, mask=(i + tl.arange(0, BTS)) < T)
+        # [BTL, BTS]
+        b_s = tl.dot(b_k.to(b_q.dtype), b_q, allow_tf32=False) * scale
+        b_s2 = b_s * b_s
+        b_dv += tl.dot(b_s2.to(b_q.dtype), tl.trans(b_do), allow_tf32=False)
+        b_ds = tl.dot(b_v, b_do, allow_tf32=False) * scale
+        if i_v == 0:
+            b_ds += b_dz[None, :] * scale
+        else:
+            b_ds = b_ds
+        b_dk += tl.dot((2 * b_ds * b_s).to(b_q.dtype), tl.trans(b_q), allow_tf32=False)
+
+    tl.debug_barrier()
+    o_q, o_k = tl.arange(0, BTS), tl.arange(0, BTL)
+    for i in range(i_c*BTL, (i_c+1)*BTL, BTS):
+        p_q = tl.make_block_ptr(q + i_bh * T*K, (K, T), (1, K), (i_k*BK, i), (BK, BTS), (0, 1))
+        p_do = tl.make_block_ptr(do + i_bh * T*V, (V, T), (1, V), (i_v*BV, i), (BV, BTS), (0, 1))
+        p_dz = dz + i_bh * T + i + tl.arange(0, BTS)
+        b_q = tl.load(p_q, boundary_check=(0, 1))  # [BD, BQ]
+        b_do = tl.load(p_do, boundary_check=(0, 1)).to(b_q.dtype)
+        b_dz = tl.load(p_dz, mask=(i + tl.arange(0, BTS)) < T)
+        # [BK, BQ]
+        m_s = o_k[:, None] <= o_q[None, :]
+        b_s = tl.dot(b_k, b_q, allow_tf32=False) * scale
+        b_s2 = b_s * b_s
+        b_s = tl.where(m_s, b_s, 0)
+        b_s2 = tl.where(m_s, b_s2, 0)
+
+        b_ds = tl.dot(b_v, b_do, allow_tf32=False)
+        if i_v == 0:
+            b_ds += b_dz[None, :]
+        else:
+            b_ds = b_ds
+        b_ds = tl.where(m_s, b_ds, 0) * scale
+        # [BK, BD]
+        b_dv += tl.dot(b_s2.to(b_q.dtype), tl.trans(b_do), allow_tf32=False)
+        b_dk += tl.dot((2 * b_ds * b_s).to(b_q.dtype), tl.trans(b_q), allow_tf32=False)
+        o_q += BTS
+
+    p_dk = tl.make_block_ptr(dk + (i_bh + B * H * i_v) * T*K, (T, K), (K, 1), (i_c*BTL, i_k*BK), (BTL, BK), (1, 0))
+    p_dv = tl.make_block_ptr(dv + (i_bh + B * H * i_k) * T*V, (T, V), (V, 1), (i_c*BTL, i_v*BV), (BTL, BV), (1, 0))
+    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))
+    return
+
+
+@triton.jit(do_not_specialize=['T'])
+def parallel_rebased_bwd_kernel(
+    q,
+    k,
+    v,
+    do,
+    dz,
+    dq,
+    dk,
+    dv,
+    scale,
+    T,
+    B: tl.constexpr,
+    H: tl.constexpr,
+    K: tl.constexpr,
+    V: tl.constexpr,
+    BTL: tl.constexpr,
+    BTS: tl.constexpr,
+    BK: tl.constexpr,
+    BV: tl.constexpr
+):
+    i_kv, i_c, i_bh = tl.program_id(0), tl.program_id(1), tl.program_id(2)
+    NV = tl.cdiv(V, BV)
+    i_k = i_kv // (NV)
+    i_v = i_kv % (NV)
+    i_h = i_bh % H
+    _parallel_rebased_bwd_dq(
+        i_bh,
+        i_c,
+        i_k,
+        i_v,
+        i_h,
+        q,
+        k,
+        v,
+        do,
+        dz,
+        dq,
+        scale,
+        B=B,
+        H=H,
+        T=T,
+        K=K,
+        V=V,
+        BTL=BTL,
+        BTS=BTS,
+        BK=BK,
+        BV=BV
+    )
+    tl.debug_barrier()
+    _parallel_rebased_bwd_dkv(
+        i_bh,
+        i_c,
+        i_k,
+        i_v,
+        i_h,
+        q,
+        k,
+        v,
+        do,
+        dz,
+        dk,
+        dv,
+        scale,
+        B=B,
+        H=H,
+        T=T,
+        K=K,
+        V=V,
+        BTL=BTL,
+        BTS=BTS,
+        BK=BK,
+        BV=BV
+    )
+
+
+class ParallelBasedFunction(torch.autograd.Function):
+
+    @staticmethod
+    @input_guard
+    @autocast_custom_fwd
+    def forward(ctx, q, k, v, scale):
+        BTL, BTS = 128, 32
+        assert BTL % BTS == 0
+        # assert q.shape[-1] % 16 == 0
+        BK = min(128, triton.next_power_of_2(k.shape[-1]))
+        BV = min(128, triton.next_power_of_2(v.shape[-1]))
+        BK, BV = max(BK, 16), max(BV, 16)
+        B, H, T, K, V = *k.shape, v.shape[-1]
+        num_stages = 2
+        num_warps = 4
+        NK = triton.cdiv(K, BK)
+        NV = triton.cdiv(V, BV)
+        grid = (NK * NV, triton.cdiv(T, BTL), B * H)
+
+        assert NK == 1, "will encounter some synchronization issue if not."
+
+        o = torch.empty(NK, B, H, T, V, device=q.device)
+        z = torch.empty(NK, B, H, T, device=q.device)
+        parallel_rebased_fwd_kernel[grid](
+            q,
+            k,
+            v,
+            o,
+            z,
+            scale,
+            T=T,
+            B=B,
+            H=H,
+            K=K,
+            V=V,
+            BTL=BTL,
+            BTS=BTS,
+            BK=BK,
+            BV=BV,
+            num_warps=num_warps,
+            num_stages=num_stages
+        )
+        ctx.save_for_backward(q, k, v)
+        ctx.scale = scale
+        return o.sum(0).to(q.dtype), z.sum(0).to(q.dtype)
+
+    @staticmethod
+    @input_guard
+    @autocast_custom_bwd
+    def backward(ctx, do, dz):
+        q, k, v = ctx.saved_tensors
+        scale = ctx.scale
+        BTL, BTS = 64, 32
+        assert BTL % BTS == 0
+        BK = min(128, triton.next_power_of_2(k.shape[-1]))
+        BV = min(128, triton.next_power_of_2(v.shape[-1]))
+        BK, BV = max(BK, 16), max(BV, 16)
+        B, H, T, K, V = *k.shape, v.shape[-1]
+        num_stages = 2
+        num_warps = 4
+        NK = triton.cdiv(K, BK)
+        NV = triton.cdiv(V, BV)
+        grid = (NK * NV, triton.cdiv(T, BTL), B * H)
+
+        assert NK == 1, "will encounter some synchronization issue if not"
+
+        dq = torch.empty(NV, B, H, T, K, dtype=q.dtype, device=q.device)
+        dk = torch.empty(NV, B, H, T, K, dtype=q.dtype, device=q.device)
+        dv = torch.empty(NK, B, H, T, V, dtype=q.dtype, device=q.device)
+
+        parallel_rebased_bwd_kernel[grid](
+            q,
+            k,
+            v,
+            do,
+            dz,
+            dq,
+            dk,
+            dv,
+            scale,
+            T=T,
+            B=B,
+            H=H,
+            K=K,
+            V=V,
+            BTL=BTL,
+            BTS=BTS,
+            BK=BK,
+            BV=BV,
+            num_warps=num_warps,
+            num_stages=num_stages
+        )
+
+        return dq.sum(0).to(q.dtype), dk.sum(0).to(k.dtype), dv.sum(0).to(v.dtype), None
+
+
+def parallel_rebased(
+    q: torch.Tensor,
+    k: torch.Tensor,
+    v: torch.Tensor,
+    eps: float = 1e-5,
+    use_scale: bool = True,
+    use_normalize: bool = True,
+    return_both: bool = False,
+    head_first: bool = True
+):
+    assert q.shape[-1] <= 128, "only support feature dim up to 128"
+    if use_scale:
+        scale = q.shape[-1] ** -0.5
+    else:
+        scale = 1
+    if not head_first:
+        q, k, v = map(lambda x: x.transpose(1, 2), (q, k, v))
+    o, z = ParallelBasedFunction.apply(q, k, v, scale)
+    if return_both:
+        return o, z
+    if use_normalize:
+        o = o / (z[..., None] + eps)
+    if not head_first:
+        o = o.transpose(1, 2)
+    return o.to(q.dtype)