[Dev] Add GQA backward example (#205)

- Introduce `example_gqa_bwd.py` demonstrating the backward pass of FlashAttention with pipelined execution.
- Implement forward and backward functions for FlashAttention, including preprocessing and postprocessing steps.
- Enhance argument parsing for batch size, heads, context size, and dimensions.
- Include a reference implementation for validation and performance benchmarking.

examples/flash_attention/example_gqa_bwd.py

+# Copyright (c) Microsoft Corporation.
+# Licensed under the MIT License.
+
+import torch
+import torch.nn.functional as F
+import tilelang
+from tilelang import cached
+from tilelang.autotuner import *
+import tilelang.language as T
+import argparse
+
+
+def flashattn_fwd(batch, heads, seq_len, dim_qk, dim_v, is_casual, block_M, block_N, groups=1):
+    scale = (1.0 / dim_qk)**0.5 * 1.44269504  # log2(e)
+    head_kv = heads // groups
+    q_shape = [batch, seq_len, heads, dim_qk]
+    k_shape = [batch, seq_len, head_kv, dim_qk]
+    v_shape = [batch, seq_len, head_kv, dim_v]
+    dtype = "float16"
+    accum_dtype = "float"
+
+    @T.prim_func
+    def flash_fwd(
+            Q: T.Buffer(q_shape, dtype),  # type: ignore
+            K: T.Buffer(k_shape, dtype),  # type: ignore
+            V: T.Buffer(v_shape, dtype),  # type: ignore
+            Output: T.Buffer([batch, seq_len, heads, dim_v], dtype),  # type: ignore
+            lse: T.Buffer([batch, heads, seq_len], accum_dtype),  # type: ignore
+    ):
+        with T.Kernel(T.ceildiv(seq_len, block_M), heads, batch, threads=128) as (bx, by, bz):
+            Q_shared = T.alloc_shared([block_M, dim_qk], dtype)
+            K_shared = T.alloc_shared([block_N, dim_qk], dtype)
+            V_shared = T.alloc_shared([block_N, dim_v], dtype)
+            acc_s = T.alloc_fragment([block_M, block_N], accum_dtype)
+            acc_s_cast = T.alloc_fragment([block_M, block_N], dtype)
+            acc_o = T.alloc_fragment([block_M, dim_v], accum_dtype)
+            scores_max = T.alloc_fragment([block_M], accum_dtype)
+            scores_max_prev = T.alloc_fragment([block_M], accum_dtype)
+            scores_scale = T.alloc_fragment([block_M], accum_dtype)
+            scores_sum = T.alloc_fragment([block_M], accum_dtype)
+            logsum = T.alloc_fragment([block_M], accum_dtype)
+
+            T.annotate_layout({Q_shared: tilelang.layout.make_swizzled_layout(Q_shared)})
+            T.copy(Q[bz, bx * block_M:(bx + 1) * block_M, by, :], Q_shared)
+            T.fill(acc_o, 0)
+            T.fill(logsum, 0)
+            T.fill(scores_max, -T.infinity(accum_dtype))
+            loop_range = (
+                T.ceildiv(
+                    (bx + 1) * block_M, block_N) if is_casual else T.ceildiv(seq_len, block_N))
+            for k in T.Pipelined(loop_range, num_stages=1):
+                T.copy(K[bz, k * block_N:(k + 1) * block_N, by // groups, :], K_shared)
+                if is_casual:
+                    for i, j in T.Parallel(block_M, block_N):
+                        acc_s[i, j] = T.if_then_else(bx * block_M + i >= k * block_N + j, 0,
+                                                     -T.infinity(acc_s.dtype))
+                else:
+                    T.clear(acc_s)
+                T.gemm(Q_shared, K_shared, acc_s, transpose_B=True, policy=T.GemmWarpPolicy.FullRow)
+                T.copy(V[bz, k * block_N:(k + 1) * block_N, by // groups, :], V_shared)
+                T.copy(scores_max, scores_max_prev)
+                T.reduce_max(acc_s, scores_max, dim=1, clear=False)
+                for i in T.Parallel(block_M):
+                    scores_scale[i] = T.exp2(scores_max_prev[i] * scale - scores_max[i] * scale)
+                for i, j in T.Parallel(block_M, dim_v):
+                    acc_o[i, j] *= scores_scale[i]
+                for i, j in T.Parallel(block_M, block_N):
+                    acc_s[i, j] = T.exp2(acc_s[i, j] * scale - scores_max[i] * scale)
+                T.copy(acc_s, acc_s_cast)
+                T.gemm(acc_s_cast, V_shared, acc_o, policy=T.GemmWarpPolicy.FullRow)
+                T.reduce_sum(acc_s, scores_sum, dim=1)
+                for i in T.Parallel(block_M):
+                    logsum[i] = logsum[i] * scores_scale[i] + scores_sum[i]
+            for i, j in T.Parallel(block_M, dim_v):
+                acc_o[i, j] /= logsum[i]
+            T.copy(acc_o, Output[bz, bx * block_M:(bx + 1) * block_M, by, :])
+            for i in T.Parallel(block_M):
+                logsum[i] = T.log2(logsum[i]) + scores_max[i] * scale
+            T.copy(logsum, lse[bz, by, bx * block_M:(bx + 1) * block_M])
+
+    return flash_fwd
+
+
+def flashattn_bwd_preprocess(batch, heads, seq_len, dim_v):
+    dtype = "float16"
+    accum_dtype = "float"
+    shape = [batch, seq_len, heads, dim_v]
+    blk = 32
+
+    @T.prim_func
+    def flash_bwd_prep(
+            O: T.Buffer(shape, dtype),  # type: ignore
+            dO: T.Buffer(shape, dtype),  # type: ignore
+            Delta: T.Buffer([batch, heads, seq_len], accum_dtype),  # type: ignore
+    ):
+        with T.Kernel(heads, T.ceildiv(seq_len, blk), batch) as (bx, by, bz):
+            o = T.alloc_fragment([blk, blk], dtype)
+            do = T.alloc_fragment([blk, blk], dtype)
+            acc = T.alloc_fragment([blk, blk], accum_dtype)
+            delta = T.alloc_fragment([blk], accum_dtype)
+            T.clear(acc)
+            for k in range(T.ceildiv(dim_v, blk)):
+                T.copy(O[bz, by * blk:(by + 1) * blk, bx, k * blk:(k + 1) * blk], o)
+                T.copy(dO[bz, by * blk:(by + 1) * blk, bx, k * blk:(k + 1) * blk], do)
+                for i, j in T.Parallel(blk, blk):
+                    acc[i, j] += o[i, j] * do[i, j]
+            T.reduce_sum(acc, delta, 1)
+            T.copy(delta, Delta[bz, bx, by * blk:(by + 1) * blk])
+
+    return flash_bwd_prep
+
+
+def make_dq_layout(dQ):
+    # atomicAdd can not be vectorized, so we need to reorder dq to match the 8x8 gemm fragment
+    return T.Layout(dQ.shape,
+                    lambda b, l, h, d: [b, l // 8, h, d // 8, (d % 2), 4 * (l % 8) + (d % 8) // 2])
+
+
+def flashattn_bwd_postprocess(batch, heads, seq_len, dim_qk):
+    dtype = "float16"
+    accum_dtype = "float"
+    shape = [batch, seq_len, heads, dim_qk]
+    blk = 64
+
+    @T.prim_func
+    def flash_bwd_post(
+            dQ: T.Buffer(shape, accum_dtype),  # type: ignore
+            dQ_out: T.Buffer(shape, dtype),  # type: ignore
+    ):
+        with T.Kernel(T.ceildiv(seq_len, blk), heads, batch, threads=128) as (bx, by, bz):
+            T.annotate_layout({dQ: make_dq_layout(dQ)})
+            T.copy(
+                dQ[bz, bx * blk:(bx + 1) * blk, by, :],
+                dQ_out[bz, bx * blk:(bx + 1) * blk, by, :],
+            )
+
+    return flash_bwd_post
+
+
+def flashattn_bwd(batch, heads, seq_len, dim_qk, dim_v, is_casual, block_M, block_N, groups=1):
+    sm_scale = (1.0 / dim_qk)**0.5
+    scale = (1.0 / dim_qk)**0.5 * 1.44269504  # log2(e)
+    head_kv = heads // groups
+    q_shape = [batch, seq_len, heads, dim_qk]
+    k_shape = [batch, seq_len, head_kv, dim_qk]
+    v_shape = [batch, seq_len, head_kv, dim_v]
+    dtype = "float16"
+    accum_dtype = "float"
+
+    @T.prim_func
+    def flash_bwd(
+            Q: T.Buffer(q_shape, dtype),  # type: ignore
+            K: T.Buffer(k_shape, dtype),  # type: ignore
+            V: T.Buffer(v_shape, dtype),  # type: ignore
+            dO: T.Buffer([batch, seq_len, heads, dim_v], dtype),  # type: ignore
+            lse: T.Buffer([batch, heads, seq_len], accum_dtype),  # type: ignore
+            Delta: T.Buffer([batch, heads, seq_len], accum_dtype),  # type: ignore
+            dQ: T.Buffer(q_shape, accum_dtype),  # type: ignore
+            dK: T.Buffer(k_shape, dtype),  # type: ignore
+            dV: T.Buffer(v_shape, dtype),  # type: ignore
+    ):
+        with T.Kernel(heads, T.ceildiv(seq_len, block_M), batch, threads=256) as (bx, by, bz):
+            K_shared = T.alloc_shared([block_M, dim_qk], dtype)
+            dsT_shared = T.alloc_shared([block_M, block_N], dtype)
+            q = T.alloc_shared([block_N, dim_qk], dtype)
+            V_shared = T.alloc_shared([block_M, dim_v], dtype)
+            qkT = T.alloc_fragment([block_M, block_N], accum_dtype)
+            dsT = T.alloc_fragment([block_M, block_N], accum_dtype)
+            qkT_cast = T.alloc_fragment([block_M, block_N], dtype)
+            dsT_cast = T.alloc_fragment([block_M, block_N], dtype)
+            lse_shared = T.alloc_shared([block_N], accum_dtype)
+            delta = T.alloc_shared([block_N], accum_dtype)
+            do = T.alloc_shared([block_N, dim_v], dtype)
+            dv = T.alloc_fragment([block_M, dim_v], accum_dtype)
+            dk = T.alloc_fragment([block_M, dim_qk], accum_dtype)
+            dq = T.alloc_fragment([block_N, dim_qk], accum_dtype)
+            dv_shared = T.alloc_shared([block_N, dim_v], dtype)
+            dk_shared = T.alloc_shared([block_N, dim_qk], dtype)
+
+            T.annotate_layout({
+                dQ: make_dq_layout(dQ),
+                K_shared: tilelang.layout.make_swizzled_layout(K_shared),
+                dv_shared: tilelang.layout.make_swizzled_layout(dv_shared),
+                dk_shared: tilelang.layout.make_swizzled_layout(dk_shared),
+            })
+
+            T.copy(K[bz, by * block_M:(by + 1) * block_M, bx // groups, :], K_shared)
+            T.copy(V[bz, by * block_M:(by + 1) * block_M, bx // groups, :], V_shared)
+            T.clear(dv)
+            T.clear(dk)
+            loop_st = T.floordiv(by * block_M, block_N) if is_casual else 0
+            loop_ed = T.ceildiv(seq_len, block_N)
+            for k in T.Pipelined(loop_st, loop_ed, num_stages=1):
+                T.copy(Q[bz, k * block_N:(k + 1) * block_N, bx, :], q)
+                T.clear(qkT)
+                T.gemm(K_shared, q, qkT, transpose_B=True, policy=T.GemmWarpPolicy.FullRow)
+                T.copy(lse[bz, bx, k * block_N:(k + 1) * block_N], lse_shared)
+                for i, j in T.Parallel(block_M, block_N):
+                    qkT[i, j] = T.exp2(qkT[i, j] * scale - lse_shared[j])
+                if is_casual:
+                    for i, j in T.Parallel(block_M, block_N):
+                        qkT[i, j] = T.if_then_else(by * block_M + i <= k * block_N + j, qkT[i, j],
+                                                   0)
+                T.copy(dO[bz, k * block_N:(k + 1) * block_N, bx, :], do)
+                T.clear(dsT)
+                T.gemm(V_shared, do, dsT, transpose_B=True, policy=T.GemmWarpPolicy.FullRow)
+                T.copy(qkT, qkT_cast)
+                T.gemm(qkT_cast, do, dv, policy=T.GemmWarpPolicy.FullRow)
+
+                T.copy(Delta[bz, bx, k * block_N:(k + 1) * block_N], delta)
+
+                for i, j in T.Parallel(block_M, block_N):
+                    dsT_cast[i, j] = qkT[i, j] * (dsT[i, j] - delta[j]) * sm_scale
+                T.gemm(dsT_cast, q, dk, policy=T.GemmWarpPolicy.FullRow)
+
+                T.copy(dsT_cast, dsT_shared)
+                T.clear(dq)
+                T.gemm(dsT_shared, K_shared, dq, transpose_A=True)
+                for i, j in T.Parallel(block_N, dim_qk):
+                    if k * block_N + i < seq_len:
+                        T.atomic_add(dQ[bz, k * block_N + i, bx, j], dq[i, j])
+
+            for i, j in T.Parallel(block_M, dim_v):
+                T.atomic_add(dV[bz, by * block_M + i, bx // groups, j], dv[i, j])
+            for i, j in T.Parallel(block_M, dim_qk):
+                T.atomic_add(dK[bz, by * block_M + i, bx // groups, j], dk[i, j])
+
+    return flash_bwd
+
+
+class _attention(torch.autograd.Function):
+
+    @staticmethod
+    def forward(ctx, q, k, v, causal, groups=1):
+        BATCH, N_CTX, H, D_HEAD_QK = q.shape
+        D_HEAD_V = v.shape[-1]
+        block_M = 64
+        block_N = 64
+        mod = cached(flashattn_fwd, [3, 4], BATCH, H, N_CTX, D_HEAD_QK, D_HEAD_V, causal, block_M,
+                     block_N, groups)
+        o, lse = mod(q, k, v)
+        ctx.save_for_backward(q, k, v, o, lse)
+        ctx.causal = causal
+        return o
+
+    @staticmethod
+    def backward(ctx, do):
+        q, k, v, o, lse = ctx.saved_tensors
+
+        def maybe_contiguous(x):
+            if x.stride(-1) != 1:
+                return x.contiguous()
+            return x
+
+        do, q, k, v, o = [maybe_contiguous(x) for x in (do, q, k, v, o)]
+        block_M = 128
+        block_N = 128
+        mod_prep = cached(flashattn_bwd_preprocess, [2], BATCH, H, N_CTX, D_HEAD_V)
+        mod_post = cached(flashattn_bwd_postprocess, [1], BATCH, H, N_CTX, D_HEAD_QK)
+        delta = mod_prep(o, do)
+        mod = cached(flashattn_bwd, [6, 7, 8], BATCH, H, N_CTX, D_HEAD_QK, D_HEAD_V, ctx.causal,
+                     block_M, block_N, groups)
+        dq, dk, dv = mod(q, k, v, do, lse, delta)
+        dq = mod_post(dq)
+        return dq, dk, dv, None, None
+
+
+attention = _attention.apply
+
+
+def ref_program(Q, K, V, is_causal, groups=1):
+    # Q: [B, T, HQ, D_QK]
+    # K: [B, T, HK, D_QK]
+    # V: [B, T, HV, D_V]
+    # HQ = HKV * groups
+    assert Q.size(2) == K.size(
+        2) * groups, f"Q.size(2): {Q.size(2)}, K.size(2): {K.size(2)}, groups: {groups}"
+    assert Q.size(2) == V.size(
+        2) * groups, f"Q.size(2): {Q.size(2)}, V.size(2): {V.size(2)}, groups: {groups}"
+
+    dim_qk = Q.size(-1)
+    K = K.repeat_interleave(groups, dim=2)
+    V = V.repeat_interleave(groups, dim=2)
+    scores = torch.einsum('bqhd,bkhd->bhqk', Q, K)
+    scores = scores / torch.sqrt(torch.tensor(dim_qk, dtype=scores.dtype))
+    if is_causal:
+        seq_len = Q.size(1)
+        mask = torch.tril(torch.ones(seq_len, seq_len, device=scores.device))
+        mask = mask.unsqueeze(0).unsqueeze(0)
+        scores = scores.masked_fill(mask == 0, float('-inf'))
+    attention_weights = F.softmax(scores, dim=-1)
+    output = torch.einsum('bhqk,bkhd->bqhd', attention_weights, V)
+    return output
+
+
+if __name__ == "__main__":
+    parser = argparse.ArgumentParser()
+    parser.add_argument('--batch', type=int, default=8, help='Batch size')
+    parser.add_argument('--h', type=int, default=32, help='Number of heads')
+    parser.add_argument('--n_ctx', type=int, default=1024, help='Context size')
+    parser.add_argument('--d_head_qk', type=int, default=192, help='Head dimension for Q/K')
+    parser.add_argument('--d_head_v', type=int, default=128, help='Head dimension for V')
+    parser.add_argument('--casual', type=bool, default=False, help='Casual flag')
+    parser.add_argument('--groups', type=int, default=16, help='groups')
+    args = parser.parse_args()
+    BATCH, H, N_CTX, D_HEAD_QK, D_HEAD_V, groups = args.batch, args.h, args.n_ctx, args.d_head_qk, args.d_head_v, args.groups
+    casual = args.casual
+    flops_per_qk = 2.0 * BATCH * H * N_CTX * N_CTX * D_HEAD_QK
+    flops_per_v = 2.0 * BATCH * H * N_CTX * N_CTX * D_HEAD_V
+    total_flops = 3 * flops_per_qk + 2 * flops_per_v
+    if casual:
+        total_flops *= 0.5
+    Q = (
+        torch.empty(BATCH, N_CTX, H, D_HEAD_QK, dtype=torch.half,
+                    device="cuda").normal_().requires_grad_())
+
+    head_kv = H // groups
+    K = (
+        torch.empty(BATCH, N_CTX, head_kv, D_HEAD_QK, dtype=torch.half,
+                    device="cuda").normal_().requires_grad_())
+    V = (
+        torch.empty(BATCH, N_CTX, head_kv, D_HEAD_V, dtype=torch.half,
+                    device="cuda").normal_().requires_grad_())
+    dO = (
+        torch.empty(BATCH, N_CTX, H, D_HEAD_V, dtype=torch.half,
+                    device="cuda").normal_().requires_grad_())
+    O = attention(Q, K, V, casual, groups)
+    O.backward(dO, retain_graph=True)
+    dQ, Q.grad = Q.grad.clone(), None
+    dK, K.grad = K.grad.clone(), None
+    dV, V.grad = V.grad.clone(), None
+
+    O_ref = ref_program(Q, K, V, casual, groups)
+    O_ref.backward(dO, retain_graph=True)
+    dQ_ref, Q.grad = Q.grad.clone(), None
+    dK_ref, K.grad = K.grad.clone(), None
+    dV_ref, V.grad = V.grad.clone(), None
+
+    assert torch.allclose(O, O_ref, rtol=1e-2, atol=1e-2)
+    assert torch.allclose(dV, dV_ref, rtol=1e-2, atol=1e-2)
+    assert torch.allclose(dK, dK_ref, rtol=1e-2, atol=1e-2)
+    assert torch.allclose(dQ, dQ_ref, rtol=1e-2, atol=1e-2)
+
+    def run():
+        O_ref.backward(dO, retain_graph=True)
+
+    def run1():
+        O.backward(dO, retain_graph=True)
+
+    from tilelang.profiler import do_bench
+
+    latency = do_bench(run, warmup=500)
+    print("torch: {:.2f} ms".format(latency))
+    print("torch: {:.2f} TFlops".format(total_flops / latency * 1e-9))
+    latency = do_bench(run1, warmup=500)
+    print("tilelang: {:.2f} ms".format(latency))
+    print("tilelang: {:.2f} TFlops".format(total_flops / latency * 1e-9))