Lots of NaNs. Let me debug. The NaN is happening somewhere - probably in the matmul or softmax. Let me run a single shape with more debug info. Shape 2 passed basic correctness (max_abs=0.16 is within 0.02? no, 0.16 > 0.02). So that's also failing tolerance. Let me debug step by step. First, let me check shape 2 with smaller seq_len.

I need to handle the case where HKV is smaller than 16/G. Let me make BLOCK_M adaptive. For our test shapes, BLOCK_M=16 works. For my test case (HKV=2, G=4, H=8 < 16), BLOCK_M=8 would be needed. Let me make BLOCK_M adaptive. The constraint: BLOCK_M >= G (so we can include all G heads per kv head in a block), BLOCK_M is a multiple of G, BLOCK_M is a power of 2, BLOCK_M is in [G, 16]. For our shapes: - G=4, HKV>=4 (so H=HKV*G >= 16). BLOCK_M=16. - G=8, HKV>=2 (so H>=16). BLOCK_M=16. Let me just make BLOCK_M = min(16, H) rounded up to a multiple of G. For our shapes, H>=16, so BLOCK_M=16. For H<16 (e.g., H=8), BLOCK_M=8 (or H). Actually, let me make BLOCK_M = min(16, next_pow2(G)) and assert HKV * G >= BLOCK_M. For our shapes, this is always true. For my test case (H=8, G=4, HKV=2): BLOCK_M = min(16, next_pow2(4)) = min(16, 4) = 4. BLOCK_KV = 1. HKV_BLKS = 2. Hmm, but BLOCK_M=4 might be too small for tl.dot. Hmm. Let me think. For our test shapes, the only thing that matters is correctness. For arbitrary shapes, we might fail. Let me just handle our test shapes correctly and add a fallback for small shapes. Actually, the simplest thing: pick BLOCK_M = min(16, G) if H < 16, else 16. For our shapes, BLOCK_M=16. Let me code: ```python g_pow2 = max(1, 1 << (G - 1).bit_length()) # next power of 2 >= G BLOCK_M = min(16, g_pow2) if H >= 16 else min(16, g_pow2, H) ``` Hmm, but for H=8, G=4, g_pow2=4, BLOCK_M=min(16, 4, 8)=4. That might fail tl.dot. OK let me just enforce: HKV * G >= 16, and use BLOCK_M=16. If not, error out. For our test shapes, HKV * G = 32, 32, 64, 32, 16. All >= 16. Good. For my debug test (H=8, G=4, HKV=2), HKV*G=8 < 16. I should use a different debug test. Let me just use a valid test shape: H=32, HKV=8, G=4 (shape 1). Or H=16, HKV=4, G=4 (shape 5). Or H=64, HKV=8, G=8 (shape 3). Actually let me just use shape 1 directly:

All shapes pass with very small error. Now let me run the actual check.py to confirm:

With CHUNK=64, all shapes have at least 1.36 waves, which is good. With CHUNK=128, shape 0 and 4 are at 0.68 waves. With CHUNK=256, shape 0 and 4 are at 0.34 waves. Let me also think about whether there's a per-shape optimal CHUNK. Or use a heuristic like CHUNK=64 always. Actually, with CHUNK=64, the work per program is small (4 pages). The reduction kernel overhead is larger (more partials to reduce). But the per-program K/V load is smaller, which might allow better cache behavior. Let me also think about a different approach: what if I do BLOCK_KV=1 to get more parallelism? With BLOCK_KV=1, BLOCK_M=4 (G=4), and HKV_BLKS=HKV. For shape 0: HKV=8, HKV_BLKS=8, SPLIT_K=4 (CHUNK=256), total = 8*8*4 = 256 programs. 1.36 waves. Good. For shape 1: HKV=8, HKV_BLKS=8, SPLIT_K=8, total = 32*8*8 = 2048. Good. For shape 2: HKV=8, HKV=8, BLOCK_M=8, HKV_BLKS=8, SPLIT_K=16, total = 4*8*16 = 512. Good. For shape 3: HKV=8, HKV=8, HKV_BLKS=8, SPLIT_K=8, total = 16*8*8 = 1024. Good. For shape 4: HKV=4, HKV_BLKS=4, SPLIT_K=8, total = 8*4*8 = 256. Good. So with BLOCK_KV=1, all shapes have decent parallelism (>= 1.36 waves). Let me try this. The downside: BLOCK_M=4 or 8 might be inefficient for tensor cores. Let me check. Actually, with BLOCK_M=4, the matmul output is (4, 16). Tensor cores typically require M >= 16. Triton might pad to 16 internally, giving the same compute as BLOCK_M=16 with dummies. Or it might use a different code path. Let me try BLOCK_KV=1 with BLOCK_M=G and see. Actually wait, with BLOCK_M=4, the matmul (4, 128) @ (128, 16) has M=4. If Triton pads to 16, the compute is 16*16*128 = 32768 ops. Same as BLOCK_M=16 with dummies. Per-program memory load: - BLOCK_M=16, BLOCK_KV=4: K load (16, 4, 128) = 16KB. Q load (16, 128) = 4KB. Total 20KB. - BLOCK_M=4, BLOCK_KV=1: K load (16, 1, 128) = 4KB. Q load (4, 128) = 1KB. Total 5KB. So BLOCK_KV=1 has 4x less data per program. With 4x more programs, total data is the same. Compute per program per page: - BLOCK_M=16: 4 separate (4, 128) @ (128, 16) batched = 4 * 4 * 16 * 128 = 32K ops (with M=4 padded to 16 = 128K ops). - Wait, with 3D batched and BLOCK_KV=4, we have 4 matmuls of (4, 128) @ (128, 16). Each is 4*16*128 = 8K effective, 16*16*128 = 32K with padding. 4 matmuls = 32K effective, 128K with padding. - BLOCK_M=4 (BLOCK_KV=1): 1 matmul of (4, 128) @ (128, 16) = 4*16*128 = 8K effective, 16*16*128 = 32K with padding. So per-program compute is 4x less for BLOCK_M=4. With 4x more programs, total compute is the same. For memory-bound code, both are equivalent. The choice affects only parallelism (more programs = better for small workloads). Let me try BLOCK_KV=1 (BLOCK_M=G) and see. Actually wait, with BLOCK_M=4, the issue is whether Triton can do a 3D batched matmul with batch=1 and M=4. Let me think. Actually no, with BLOCK_KV=1, there's no 3D batched needed. Just a 2D matmul of (BLOCK_M, BLOCK_D) @ (BLOCK_D, BLOCK_N). M=BLOCK_M=4 or 8. The question is: can tl.dot handle M=4? I think Triton pads to 16 internally for tensor cores. So it works but has 4x compute waste. For G=4, BLOCK_M=4 padded to 16: 4x waste. Same as BLOCK_M=16 with 12 dummies. For G=8, BLOCK_M=8 padded to 16: 2x waste. Less than BLOCK_M=16 with 8 dummies. So for G=8 (shape 3), BLOCK_M=8 is more compute-efficient. Hmm. Let me think. Actually wait, the matmul is (M, K) @ (K, N). For BLOCK_M=8, the matmul is (8, 128) @ (128, 16) = (8, 16). With M=8 padded to 16 internally, compute is 16*16*128 = 32K. Effective compute is 8*16*128 = 16K. 2x waste. For BLOCK_M=16 with 8 dummies (where the dummies use wrong K): compute is 16*16*128 = 32K. 8/16 = 50% waste (from dummies, not from padding). So same as BLOCK_M=8 with 2x padding waste. Hmm, both are similar. For G=4: - BLOCK_M=4 with padding: 4x waste, 32K compute. - BLOCK_M=16 with 12 dummies: 4x waste, 32K compute. Same. So both are equivalent in compute. The difference is in parallelism. For G=4, BLOCK_KV=4 (BLOCK_M=16): HKV_BLKS = HKV/4. For shape 0, HKV=8, HKV_BLKS=2. For G=4, BLOCK_KV=1 (BLOCK_M=4): HKV_BLKS = HKV. For shape 0, HKV_BLKS=8. With SPLIT_K=4 (shape 0), total = 8*8*4 = 256 (BLOCK_KV=1) vs 8*2*4 = 64 (BLOCK_KV=4). 4x more parallelism with BLOCK_KV=1. So BLOCK_KV=1 is better for parallelism. Let me use it. But wait, with BLOCK_M=G=4, the matmul output is (4, 16). I need to use it correctly in the softmax. Let me re-examine. The softmax state is (BLOCK_M,) for m and l, and (BLOCK_M, BLOCK_D) for acc. All of these are (4,) or (4, 128) for BLOCK_M=4. No dummies. All valid. The store is to o_partial[b, h_start:h_start+BLOCK_M, split, :] and lse_partial[b, h_start:h_start+BLOCK_M, split]. For BLOCK_M=4, h_start=hkv*G, and the store covers 4 query heads. All valid. Great, so BLOCK_M=4 is clean. Let me code it. Hmm wait, will Triton handle M=4 for tl.dot? Let me check. Actually, looking at Triton documentation more carefully, I think for tensor core operations, the M dim must be a multiple of 16 (for bf16). For M < 16, Triton might: 1. Pad to 16 internally (waste compute) 2. Use a different code path (slower, no tensor cores) 3. Fail to compile Let me just try and see. Actually, I realize there might be another issue. tl.dot requires the inputs to have specific shapes for tensor cores. For (4, 128) @ (128, 16), the M=4 might be padded to 16 internally, but I'm not sure. Let me just write the code and try. Actually, let me first try a different optimization: increase num_warps. The default is 4. For BLOCK_M=16, num_warps=4 might be suboptimal. Let me try num_warps=8. Also, let me consider: maybe I should pre-compute the offs_kv array differently. With BLOCK_KV=4, the kv heads are [hkv_start, hkv_start+1, hkv_start+2, hkv_start+3]. These are 4 consecutive heads. For shape 0, hkv_blk=0: kv heads 0..3. hkv_blk=1: kv heads 4..7. So all 8 kv heads are covered. OK let me think about the overall strategy. Let me first measure with a larger CHUNK and see. Actually, let me just try BLOCK_KV=1 (BLOCK_M=G) and see if Triton supports it. If yes, the parallelism improves. If not, I'll need a different approach. Let me also think about a third approach: process multiple pages per iteration (BLOCK_N=32 or 64). This reduces the loop overhead and might improve the matmul efficiency. For BLOCK_N=32, we process 2 pages per iteration. The K/V load is (32, BLOCK_KV, BLOCK_D). The matmul is (BLOCK_KV, G, BLOCK_N=32). The block table needs 2 page indices per iteration. Actually for simplicity, let me stick with BLOCK_N=P=16 and tune CHUNK. Let me try smaller CHUNK (e.g., 128) first. Actually, let me just run a few experiments to find the best CHUNK. Hmm, before that, let me also think about num_warps. For BLOCK_M=16, BLOCK_N=16, BLOCK_D=128: - K load: 16*128*4 = 8KB (BLOCK_KV=4) per warp. With 4 warps, each handles 2KB. That's small. - Matmul: 16*16*128 = 32K ops per warp. 4 warps = 8K ops each. That's tiny. Maybe num_warps=2 is better. Or num_warps=8 to use more parallelism. For BLOCK_M=16, BLOCK_N=16, BLOCK_D=128, the matmul is 16*16*128 = 32K ops. Each warp can do 16*16*16 (one 16x16x16 tensor core op) = 4K ops. So 8 ops per warp. With 4 warps, 2 ops each. Tiny. To use tensor cores efficiently, we want larger matmuls. Larger BLOCK_M or BLOCK_N or BLOCK_D. Hmm. Let me try BLOCK_M=32, BLOCK_N=32. Matmul is 32*32*128 = 128K ops. 4x more compute per matmul. With BLOCK_KV=8 (G=4) and HKV=8, HKV_BLKS=1. With BLOCK_M=32, we cover all 32 query heads in one program. For shape 0 with BLOCK_M=32: HKV_BLKS=1, SPLIT_K=4, total = 8*1*4 = 32 programs. Bad parallelism. Hmm. BLOCK_M=32 is worse for parallelism. OK let me just try CHUNK=128 (smaller than 256) and see if it helps shape 0 and 4. Actually, let me also try a completely different idea: avoid the 3D batched matmul. With BLOCK_KV=1, the matmul is 2D (BLOCK_M, BLOCK_D) @ (BLOCK_D, BLOCK_N). No 3D needed. Let me restructure the kernel for BLOCK_KV=1 (and BLOCK_M=G). Wait, but with BLOCK_M=G=4 or 8, the matmul might not be efficient on tensor cores. Let me check by trying. Actually, you know what, let me just try BLOCK_KV=1 and see. Plan: - BLOCK_KV = 1 - BLOCK_M = G (4 or 8) - 2D matmul: (BLOCK_M, BLOCK_D) @ (BLOCK_D, BLOCK_N) = (BLOCK_M, BLOCK_N) If Triton doesn't support M=4 or 8, I'll fall back to 3D batched with BLOCK_M=16. Let me write this. Actually, I realize there's another approach. What if I use a 2D matmul but expand the M dim to cover multiple kv heads? Like BLOCK_M = BLOCK_KV * G with BLOCK_KV=1 and G=4, so BLOCK_M=4. Then I pad BLOCK_M to 16 (with dummies). The matmul is (16, 128) @ (128, 16) = (16, 16). 12 dummies. But the dummies would have wrong K (since BLOCK_KV=1, all dummies use the same K as the valid 4 heads). So this is wrong. To use BLOCK_M=16 with BLOCK_KV=1 correctly, I'd need to load K for all 16 query heads' kv heads, which is BLOCK_KV=4 (since G=4 and 16/4=4 kv heads). That's what I had before. OK let me just try BLOCK_M=G (BLOCK_KV=1) and see. Hmm wait, actually let me think about another angle. What if I use BLOCK_M=16 and BLOCK_KV=4 (as before) but improve the memory access pattern? The K load is (BLOCK_N=16, BLOCK_KV=4, BLOCK_D=128). In memory, this is 16*4*128 = 8192 bf16 = 16KB. The access pattern is: for each (n, kv), we read 128 contiguous bf16 values. The K for (n, kv) is at offset KV + block_id * stride_kvb + n * stride_kvp + (hkv_start + kv) * stride_kvh + d * stride_kvd. For coalesced access, we want consecutive threads to read consecutive memory addresses. The default Triton layout for tl.load is row-major in the last dim. So for our load, threads should access d in [0, 128) for the same (n, kv). That's contiguous, so coalesced. Across (n, kv), the access is less contiguous. For different kv, we have stride_kvh * 2 bytes = 2*D*2 = 512 bytes apart. For different n, stride_kvp * 2 bytes = Hkv * 2 * D * 2 = Hkv * 4 * D bytes. For Hkv=8, D=128, that's 4096 bytes. So consecutive n are 4KB apart. The memory access pattern: 128 contiguous bytes (for one n, one kv, all d), then 4KB to the next n (same kv), then 512 bytes to the next kv (same n). Hmm, the kv dimension is interleaved within the n dimension. Actually, in memory, the layout is (n_outer, kv_inner, d). So for one n, we have all kv's d. Then the next n. So we read 4*128*2 = 1KB per n, then jump 4KB to the next n. For coalesced access, the 32 threads in a warp should read 32 contiguous bytes. With 4 kv heads and 128 d, that's 1024 bytes per n, which is 32 transactions (32 bytes each). 8 warps could handle this in parallel. With 4 warps, 16 transactions per warp. Should be fine. Hmm I think the access pattern is OK. Let me focus on parallelism. OK let me just try CHUNK=128 and see if it helps. Also let me try num_warps=8. Actually, I think the bottleneck for shape 0 might be the kernel launch overhead and the per-program work being too small. With 0.34 waves, most SMs are idle in the second wave, but the first wave might be quick. Let me just run experiments. Let me also consider: what if I make the K and V loads use vectorized loads (e.g., load 4 bf16 at a time = 8 bytes)? Triton handles this automatically based on alignment. Let me check the access pattern. The K data is (BLOCK_N=16, BLOCK_D=128) bf16. The inner dim is 128 * 2 = 256 bytes. With bf16, each thread can load 8 bytes (4 elements). 256/8 = 32 elements per thread. With 32 threads in a warp, we can load 32*8 = 256 bytes per cycle. That's one row of 128 bf16. For BLOCK_N=16 rows, we need 16 row-loads. With multiple warps, this can be parallelized. OK I think the access pattern is fine. Let me just try smaller CHUNK. Actually, I realized there's something else I should consider. The reduction kernel might be slow. For shape 0 with B=8, H=32, the reduction has 8*32=256 programs. For each program, it loads SPLIT_K=4 LSEs and 4 partial outputs (each 128 elements). So it reads 4*4 + 4*128*2 = 1032 bytes per program. Total: 256*1032 = 264KB. That's tiny, fast. The reduction should not be the bottleneck. OK let me try smaller CHUNK. CHUNK=128 means SPLIT_K=8 for shape 0 and 16 for shape 4. Let me also try CHUNK=64 for max parallelism. Actually, let me also consider: with smaller CHUNK, the reduction kernel has more partials to combine. For SPLIT_K=16, the reduction reads 16*4 + 16*128*2 = 4144 bytes per program. For shape 4 with B=8, H=16, that's 8*16*4144 = 530KB. Still tiny. Let me try CHUNK=128 first.

num_stages=3 gives 0.4844, very similar to num_stages=2 (0.4843). No improvement. Let me revert and try a different optimization. Let me think about what could improve performance: 1. Better access pattern - load K and V together 2. Larger BLOCK_N - process multiple pages per iteration 3. Better split-K heuristic 4. Reduce overhead in the kernel Let me also look at the bandwidth utilization: - Shape 0: 758 GB/s (42% peak) - low parallelism - Shape 1: 1344 GB/s (75% peak) - good - Shape 2: 991 GB/s (55% peak) - decent - Shape 3: 1022 GB/s (57% peak) - decent - Shape 4: 490 GB/s (27% peak) - bad For shape 1, we're getting 75% peak which is great. For shapes 0 and 4, we're getting < 50%. These are the small batch / small D cases. The issue with shape 0 and 4 is low parallelism. We need either: 1. More programs (more splits or more kv heads per batch) 2. More work per program (larger BLOCK_N or BLOCK_M) Let me try a per-shape BLOCK_KV. For shapes with low parallelism, use smaller BLOCK_KV to get more programs. For shape 0 (B=8, HKV=8, G=4, BLOCK_KV=4 -> HKV_BLKS=2): if I use BLOCK_KV=2, HKV_BLKS=4, BLOCK_M=8. Total = 8*4*8 = 256 programs. 1.36 waves. Better. But BLOCK_M=8 might be inefficient. Let me try. Actually wait, I realize with BLOCK_KV=2, the 3D batched matmul has batch=2, M=G=4. The matmul size is (2, 4, 128) @ (2, 128, 16) = (2, 4, 16). Per-batch matmul is M=4 (padded to 16), 2x waste. With BLOCK_KV=4, batch=4, M=4. Per-batch is 4x waste (M=4 padded to 16). Total: 4 * 4x = 16x waste... wait that's the same as before. Hmm. Let me think about it differently. The total compute is: - BLOCK_KV=4: 4 matmuls of (4, 128) @ (128, 16), each M=4 padded to 16 = 4 * 16*16*128 = 128K ops per page. - BLOCK_KV=2: 2 matmuls of (4, 128) @ (128, 16), each M=4 padded to 16 = 2 * 16*16*128 = 64K ops per page. - BLOCK_KV=1: 1 matmul of (4, 128) @ (128, 16), M=4 padded to 16 = 16*16*128 = 32K ops per page. So BLOCK_KV=1 has 4x less compute per program. With 4x more programs, total compute is the same. For memory-bound, both are equivalent. The difference is parallelism. For our shapes: - BLOCK_KV=1, shape 0: HKV=8, HKV_BLKS=8, SPLIT_K=8, total = 8*8*8 = 512 programs. 2.72 waves. - BLOCK_KV=1, shape 4: HKV=4, HKV_BLKS=4, SPLIT_K=16, total = 8*4*16 = 512 programs. 2.72 waves. But wait, BLOCK_M=4 with tensor cores - the matmul is 4x waste from padding. Is that what Triton does, or does it use a different code path? Let me try BLOCK_KV=1 and see. Actually, I realize the matmul waste isn't the issue. The matmul is fast on tensor cores. The waste is hidden by the memory load. Let me also think: with BLOCK_M=4, the Q load is 4*128*2 = 1KB. The K load is 4*16*128*2 = 16KB. The matmul is tiny. The softmax state is 4*4 + 4*128*4 = 528 bytes. All very small. For BLOCK_M=16 (BLOCK_KV=4), the K load is 16*16*128*2 = 64KB. The matmul is 4x bigger. The softmax state is 16*4 + 16*128*4 = 8.2KB. So BLOCK_M=4 has 4x less memory load per program, and 4x more programs. Total memory load is the same. Hmm. Let me just try BLOCK_KV=1 with BLOCK_M=4 and see. Actually, I realize the kernel has BLOCK_M as a constexpr, and the matmul is templated on it. Let me just change the BLOCK_KV and see. Let me revert num_stages first.

num_stages=4 is similar. Let me think about other optimizations. Let me try a different approach. What if I process multiple pages per iteration (BLOCK_N=32)? This would reduce loop overhead. Actually let me first check: what's the theoretical peak for this kernel? For shape 0: - Memory: 4MB K + 4MB V = 8MB? Wait let me recompute. - B=8, L=1024, HKV=8, D=128. K size = 8*1024*8*128*2 = 2MB. V = 2MB. Total = 4MB. - At 1.8 TB/s, 4MB takes 2.2us. - Q size = 8*32*128*2 = 64KB. Tiny. - Output size = 64KB. Tiny. - Total memory: 4MB + 128KB = 4.1MB. - Time at peak: 4.1MB / 1.8 TB/s = 2.3us. We achieve 44us. So we're 19x slower than peak. That's a lot. What's wrong? Possibilities: 1. L2 cache misses 2. Insufficient parallelism (only 128 programs = 0.68 waves) 3. The matmul compute is overlapping with memory but not efficiently 4. The reduction kernel adds overhead Let me check the reduction kernel time. For shape 0, B=8, H=32, SPLIT_K=8. The reduction has 256 programs. Each reads 8*4 + 8*128*2 = 2KB. Total: 512KB. At peak bandwidth, 0.3us. But kernel launch overhead is ~5us, so the reduction might take 5-10us. If the attention kernel is 34us and reduction is 10us, total is 44us. The attention kernel is the bottleneck. To speed up the attention kernel, I need to: 1. Increase parallelism (more programs) 2. Improve memory access pattern 3. Reduce per-program overhead Let me try increasing parallelism by using BLOCK_KV=1 (which we already tried, was worse) or smaller CHUNK (which we tried, was marginal). Let me also try a different strategy: skip the reduction kernel when SPLIT_K=1, and write directly to output. For our shapes, SPLIT_K is always > 1, so this doesn't help. Hmm. Let me think about why BLOCK_KV=1 was slower. With BLOCK_KV=1, BLOCK_M=4: - 4x more programs - Each program does 1/4 the work - Total work is the same If kernel launch overhead is significant, 4x more programs = 4x more launch overhead. That could explain the slowdown. For shape 0 with 128 programs, launch overhead per program might be ~1us. Total: 128us. Hmm, that's a lot. Wait, kernel launch overhead is per-launch, not per-program. We launch 1 kernel with 128 programs. Launch overhead is ~5us. So 4x more programs (512 for shape 0 with BLOCK_KV=1) shouldn't increase launch overhead. But it might increase scheduling overhead. Hmm, let me think. Actually, the per-program runtime includes the overhead of starting up. With more programs, the scheduler has more work to do. OK let me try a different approach: increase BLOCK_M to reduce the number of programs. For shape 0 with BLOCK_M=32, BLOCK_KV=8 (G=4). HKV_BLKS=1. Total = 8*1*8 = 64 programs. Less parallelism. But each program does 2x the work (loads 2x the K/V). Total work is the same. Hmm, 64 programs is too few. For BLOCK_M=16 (current) with BLOCK_KV=4, we have 128 programs for shape 0. That's 0.68 waves. The ideal is 188 programs (1 wave). If I use BLOCK_KV=2, BLOCK_M=8, HKV_BLKS=4, total = 8*4*8 = 256 programs. 1.36 waves. Better parallelism. But BLOCK_M=8 might have matmul issues (M=8 is small). With 3D batched, M=4 padded to 16, so 2 batches * 16*16*128 = 64K ops. Same as before, just less work per program (per page). Hmm, with smaller BLOCK_M, the matmul is "smaller" (per batch) but we have more batches. Total compute is the same. OK let me try BLOCK_KV=2 (BLOCK_M=8) and see. Actually wait, I had BLOCK_KV=4 (BLOCK_M=16) before. With BLOCK_KV=2, BLOCK_M=8 (G=4). For shape 0: HKV_BLKS=4, SPLIT_K=8, total = 8*4*8 = 256 programs. Let me try this. Hmm actually, looking at the kernel, I realize the matmul uses BLOCK_M=16, not BLOCK_M=8. The constexpr BLOCK_M is what determines the matmul shape. With BLOCK_M=8, the matmul would be (8, 128) @ (128, 16). M=8 padded to 16, compute = 16*16*128 = 32K ops per batch. With 2 batches, 64K ops per matmul. Hmm. Wait, the 3D batched matmul is (BLOCK_KV, G, BLOCK_N) = (2, 4, 16) for the score. Per batch, M=G=4, so the matmul is (4, 128) @ (128, 16) padded to (16, 128) @ (128, 16) = 32K ops. 2 batches = 64K ops per matmul. Vs BLOCK_KV=4: (4, 4, 16). Per batch, 32K ops. 4 batches = 128K ops per matmul. So BLOCK_KV=2 has half the compute per program per page. With 2x more programs, total compute is the same. For memory: BLOCK_KV=2 reads (BLOCK_N, 2, BLOCK_D) = (16, 2, 128) = 8KB per K load. Same as before, just less per program. With 2x more programs, total is the same. So BLOCK_KV=2 should be similar perf. The difference is parallelism: 256 vs 128 programs. Let me try. Actually, for shape 0 the bigger issue is that we only have 0.68 waves. With BLOCK_KV=2, we have 1.36 waves. The second wave has 68 programs running, which uses ~36% of SMs. Hmm. The wave imbalance is a fundamental issue for low-parallelism shapes. We can't avoid it without more programs. Wait, actually the wave issue is that we have FEWER programs than SMs. With 128 programs and 188 SMs, the first wave has 128 programs running, the second wave has 0 programs. So 60 SMs are idle in the second wave. But there's no second wave; we just have 128 programs that finish in one wave. So the question is: how long does one wave take? If each program takes T time, the total time is T (since they all run in parallel). For shape 0 at 44us, each program takes 44us. The work per program is 64 pages * (16KB load + matmul). The 16KB load at 1.8 TB/s = 9ns. So memory load per program is 64 * 9ns = 576ns. Plus matmul compute (small). So theoretical per-program time is ~600ns. We're at 44us = 73x slower. Why? Let me think. Maybe the matmul is taking longer than I think. The 3D batched matmul with M=4 padded to 16 is 16*16*128 = 32K ops per batch. 4 batches = 128K ops per matmul. For the QK matmul. Plus the PV matmul = 128K ops. Total 256K ops per page. Per program: 64 pages * 256K = 16M ops. At 1650 TFLOPS bf16 (Blackwell), 16M ops = 0.01us. Tiny. For memory: 64 pages * 16KB = 1MB per program. At 1.8 TB/s = 0.55us per program. So the bottleneck should be memory, at 0.55us per program. With 128 programs in 1 wave, total is 0.55us. We're at 44us. 80x slower. There's something fundamentally wrong. Let me profile. Actually, maybe the issue is that the 3D batched matmul with M=4 padded to 16 is actually doing 4x more compute than the minimum. So per page: 4 * 16*16*128 = 128K ops for QK, 128K for PV = 256K ops. Per program: 64 * 256K = 16M ops. At 1650 TFLOPS = 10ns. Still small. Hmm, the compute shouldn't be the bottleneck. Let me check: what's the actual SASS code generated by Triton? Or use ncu to profile. Actually let me first check if the kernel is doing what I think it's doing. Let me add a printf to see. Wait, let me also think about the launch overhead. For 128 programs in 1 wave on 188 SMs, we have 60 idle SMs. If the wave time is 44us, that's a long time per program. For one program: 64 page iters. Per iter: 1 block_id load, 2 K/V loads (16KB), 1 matmul, 1 softmax update. The K/V load is 16KB = 8 cache lines (128 bytes each). At 1.8 TB/s, this should be 8 * 128B / 1.8TB/s = 0.6ns. But with latency, it might be more. Actually, the 1.8 TB/s is the DRAM bandwidth. The cache lines come from L2 (or L1 if cached). The L2 bandwidth is much higher than DRAM. So if the K/V is L2-resident, the load is fast. For one program reading 1MB of K/V, the L2 (128MB) can hold it all. So all reads are L2 hits. L2 bandwidth on Blackwell is much higher than DRAM (maybe 5-10 TB/s?). So 1MB / 5TB/s = 0.2us per program. Still, 44us is way slower than 0.2us. Let me check if the kernel is even running correctly. Maybe there's a bug. Actually, let me check the kernel by running it once and timing carefully. Hmm, the benchmark reports 44us. Let me trust that and think about why. Maybe the issue is that the L2 cache is being thrashed by the multiple programs. With 128 programs each reading 1MB, the total L2 traffic is 128MB, exceeding the 128MB L2 capacity. So some reads miss L2 and go to DRAM. 128 programs * 1MB = 128MB. L2 is 128MB. So the L2 is full, and reads might evict earlier data. Effective bandwidth is somewhere between L2 and DRAM. For shape 0, if half the reads hit L2 (5 TB/s) and half miss (1.8 TB/s), average is ~3 TB/s. 128MB / 3 TB/s = 43us. That matches our 44us! So the bottleneck is L2/DRAM bandwidth, but the L2 is being thrashed. The fix is to either: 1. Reduce L2 traffic (process the data in smaller chunks to fit in L2) 2. Increase parallelism (so each program reads less, and L2 can hold more) 3. Better cache reuse (each K/V value is read by multiple query heads, but we already share within a program) Hmm. With BLOCK_KV=4, we share K/V across 4 query heads. With BLOCK_KV=1, we don't share. So BLOCK_KV=4 should have less L2 traffic (4x less). But I tried BLOCK_KV=1 earlier and it was slower. Maybe because the total number of programs is 4x more, leading to more L2 thrashing. Actually, with BLOCK_KV=1, the total programs is 4x more, but each program reads 4x less data. Total data is the same. L2 traffic is the same. So no difference in L2 behavior. The difference might be in scheduling overhead or kernel runtime overhead. OK let me try a different approach: reduce the K/V load per iteration by using BLOCK_N=8 (half a page). Then we'd have 2 page iters per program (for shape 0 with 64 pages, 128 iters total). Wait, BLOCK_N=8 means we process 8 tokens at a time, but the page boundary is at 16. Hmm, this gets complex. Let me try a different approach: pre-compute the K^T layout. Or use shared memory for the K data. Actually, the simplest optimization: use vectorized loads. Triton should do this automatically for contiguous data, but let me check. For the K load, the address is `block_id * stride_kvb + offs_n * stride_kvp + offs_kv * stride_kvh + offs_d`. With contiguous offs_d (innermost), each thread loads 1 bf16 (or 4 bf16 with vectorization). The default Triton layout should be coalesced. Hmm, let me also think about the page boundary. For BLOCK_N=16 = P, each iteration processes exactly 1 page. The K data is contiguous within the page. So coalesced access is good. For BLOCK_N=32 (2 pages), the K data spans 2 pages, which might be non-contiguous in memory. This could hurt coalescing. OK let me try a different angle. Let me look at the SASS or use ncu. Actually, let me first try a more aggressive optimization: use shared memory explicitly (tl.atomic or similar) to reduce DRAM traffic. Hmm, that's complex. Let me just try a few more configs. Let me try: keep BLOCK_KV=4 but increase num_stages to 3 or 4 for better pipelining. Wait, I already tried num_stages=3 and 4. Both are similar to 2. Let me try: pre-compute the K^T outside the loop, so the permute is done once per page (not inside the loop). Actually, the permute is done inside the loop already (per page). The permute is a layout change, not a data movement. So it should be free. Hmm. Let me try to look at the kernel SASS. Actually, let me first try to simplify the kernel and see if I can find the bottleneck. Let me also think: maybe the issue is that I'm doing too much work per program. For shape 0, each program processes 64 pages. If I split the sequence into more pieces (more splits), each program does less work, and we have more parallelism. With CHUNK=64, SPLIT_K=16 for shape 0, total = 8*2*16 = 256 programs. 1.36 waves. Each program does 64/2 = 32 pages. Half the work per program. Let me try. Actually I already tried CHUNK=64 and it was 0.4829, similar to CHUNK=128 (0.4843). Hmm. Let me think. OK let me try a fundamentally different approach: write the K data layout differently, or use a different memory access pattern. Or, let me just accept the current performance and look at other things. Actually, let me look at the problem.yaml more carefully. The metric is "peak_fraction" = achieved_bandwidth / peak_bandwidth. We have 0.48, which is "OK" (>= 0.1). The goal is to push it higher. Let me also think: the reference SOTA is 70-85% of peak HBM bandwidth. So 50-70% is reasonable for a custom kernel. We're at 48%, which is OK but could be better. Let me try one more optimization: use ldmatrix or async copy for the K/V loads. This requires inline PTX. Hmm, that's complex. Let me first try a simpler optimization: increase the per-program work to amortize the launch overhead. For shape 0 with BLOCK_KV=8 (BLOCK_M=32, G=4). HKV_BLKS=1, SPLIT_K=8, total = 8*1*8 = 64 programs. 0.34 waves. Worse. For BLOCK_M=16 (current), we have 128 programs for shape 0. 0.68 waves. If I use a smaller CHUNK (e.g., 32), SPLIT_K=32, total = 8*2*32 = 512 programs. 2.72 waves. Lots of parallelism, but more reduction overhead. Let me try CHUNK=32. Actually wait, I realize for shape 0, the K/V data per program is 1MB. If I split into CHUNK=32, each program does 32 tokens = 2 pages. Per program: 2 * 16KB = 32KB of K/V. Total programs: 8*2*32 = 512. Total K/V: 512 * 32KB = 16MB. Wait, that's less than the 8MB total! Hmm, let me recompute. Total K = 2MB, V = 2MB. Total = 4MB. Wait, I think I had a mistake earlier. Let me recompute. KV cache shape: (num_blocks, P, Hkv, 2*D) = (num_blocks, 16, 8, 256). For B=8 batches, L=1024, num_blocks = 8 * 64 = 512. Each block is 16*8*256*2 = 64KB. Total KV cache = 512 * 64KB = 32MB. Per batch: 32MB / 8 = 4MB. Per program (1 batch, 4 kv heads, 1 split = 128 tokens): 4 kv * 128 tokens * 128 D * 4 bytes (K+V) = 256KB. Total: 128 programs * 256KB = 32MB. Matches the total KV cache. OK so total KV is 32MB, not 4MB. I was wrong earlier. For 32MB at 1.8 TB/s, time = 17.7us. We're at 44us, so 40% of peak. Matches what I thought. Hmm. The L2 is 128MB, so the entire KV cache fits. So all reads should be L2 hits (after the first read). L2 bandwidth on Blackwell is much higher than DRAM (maybe 5-10 TB/s?). If L2 bandwidth is 5 TB/s, 32MB / 5 TB/s = 6.4us. We're at 44us, so 7x slower. For L2 bandwidth 10 TB/s, 32MB / 10 TB/s = 3.2us. 14x slower. So even with L2 hits, we're much slower than expected. There must be some other bottleneck. Let me think. Maybe the matmul compute is the bottleneck. For 32MB KV, the matmul is 4 * B * H * L * D = 4 * 8 * 32 * 1024 * 128 = 134M ops. At 1650 TFLOPS, 134M / 1650T = 0.08us. Tiny. Hmm, so compute is not the bottleneck. Maybe it's the L1/shared memory bandwidth. For each K/V load, we read 16KB into shared memory (for the matmul). If shared memory bandwidth is limited, this could be the bottleneck. Shared memory bandwidth on Blackwell is high (TB/s), but the matmul reads from shared memory multiple times (for each output element). For (16, 16, 128) matmul, each K element is read 16 times. Total reads from shared memory: 16 * 16 * 128 = 32K elements per matmul. With 4 warps, 8K elements per warp. Hmm, this might be the bottleneck. Let me think. Actually, for tensor cores, the K is read once per warp and broadcast. So the K read is 128 elements per warp per matmul. 8 K-dim chunks = 1024 elements per warp per matmul. With 4 warps, 4096 elements total per matmul. At 1 element per cycle, 4096 cycles per matmul. For 64 pages, 64 * 4096 = 262K cycles per program. At 2 GHz, 131us per program. Hmm, that's higher than 44us. Wait, that's per warp. With 4 warps in parallel, 131/4 = 33us. Closer to 44us. Hmm, but the SM runs warps in parallel. The matmul takes 33us per SM. With 128 programs on 188 SMs, we have 0.68 waves, so total is ~22us per wave. But we're seeing 44us. Maybe the matmul is not as efficient as I think. Let me try a bigger matmul. For BLOCK_M=32, BLOCK_N=32, BLOCK_D=128, the matmul is (32, 128) @ (128, 32). 4x larger. Per matmul: 32*32*128 = 128K ops. With M=4 padded to 16 and 8 batches, 8 * 16*32*128 = 524K ops. 4x more compute per matmul. With 16 iters (CHUNK=128, BLOCK_N=32), 16 * 524K = 8M ops per program. Same total compute. Hmm. OK let me just try a different config and see. Let me try CHUNK=32 with num_warps=2.

The kernel name was wrong. Let me check the actual name.

Interesting! Let me analyze the ncu output: - Time: 33-34us (per kernel) - DRAM bytes/sec: ~1 TB/s (out of 1.8 TB/s peak = 56% peak) - L1TEX bytes/sec: ~1 TB/s - Warps active: 4.16% (low!) - SM cycles active: 60% (decent) - Tensor core cycles: N/A (no tensor cores used!) - Block size: 64 threads (1 warp per block, not 4 warps!) - Grid size: 128 The block size is 64 threads = 2 warps. But I specified num_warps=2, which should give 2 warps = 64 threads. Hmm but my last change was num_warps=2. Let me check. Wait, I changed num_warps=2 earlier. So 2 warps = 64 threads. That matches. Warps active is 4.16%, which is low. With 2 warps per program and 128 programs on 188 SMs, the theoretical max warps is 188*64=12032, but active is 4.16% * 12032 = 500 warps. Hmm, 500/128 = 3.9 warps active per program? That doesn't make sense. Actually, "sm__warps_active" is the average fraction of time that warps are eligible to issue. 4.16% means most of the time, no warps are eligible. This is a stall. Tensor core cycles is N/A, meaning no tensor cores are being used. This is the issue! The matmul is not using tensor cores. Why? With BLOCK_M=16, BLOCK_N=16, BLOCK_D=128, the matmul is (16, 128) @ (128, 16). The M=16, N=16, K=128 are all multiples of 16, so tensor cores should be usable. But the matmul is 3D batched with batch=4, M=4. M=4 is not a multiple of 16, so tensor cores can't be used directly. Triton might fall back to FMA (fused multiply-add) for the M=4 case. This is the issue! The 3D batched matmul with M=G=4 is not using tensor cores. The matmul is done with regular FMA, which is much slower. So I need to avoid the M=4 case. Options: 1. Use BLOCK_M=16 with dummies (but dummies use wrong K) - WRONG 2. Use BLOCK_M=16 with 3D batched (M=4 padded to 16, but maybe Triton handles it differently) 3. Use a flat 2D matmul where M=16 is real 4. Use BLOCK_KV=1 (BLOCK_M=4) with a different matmul approach Wait, the 3D batched matmul with M=4 might still use tensor cores if Triton pads internally. Let me check. Looking at the metric, "sm__pipe_tensor_op_hmma_cycles_active" is N/A. This means the tensor cores are NOT being used at all. So Triton is falling back to FMA for the 3D batched matmul with M=4. The FMA throughput is much lower than tensor cores. This is why the kernel is slow. To fix this, I need to make M >= 16 for the matmul. Options: 1. Use BLOCK_M=16 with BLOCK_KV=4, but correctly load K for all 4 kv heads. - The matmul becomes (16, 128) @ (128, 16) flat. M=16, uses tensor cores. - But the K data for query head m is K[block, m//G, :], which is not contiguous in the loaded K. - We need to "scatter" the K data so that the matmul is correct. 2. Use BLOCK_M=16, BLOCK_KV=4, with 3D batched matmul that uses tensor cores. - This might require padding the inner M dim. 3. Use 4 separate 2D matmuls, each with M=4 padded to 16. Same as 3D batched but more overhead. 4. Use 4 separate 2D matmuls without padding, M=4. Same issue as 3D batched. 5. Use a flat 2D matmul with BLOCK_M=16, but arrange the K data correctly. Let me think about option 5. The matmul is (16, 128) @ (128, 16) = (16, 16). For row m of Q, the K column should be the K for query head m's kv head. If I lay out K as (BLOCK_D, BLOCK_N) where for each kv head, the K data is at a specific column range... no, that doesn't work for a flat matmul. Alternative: use a flat matmul (BLOCK_M, BLOCK_D) @ (BLOCK_D, BLOCK_N), but the K data is duplicated/gathered for each query head. For query head m, the K row is K_orig[block, hkv(m), :]. To do a flat matmul, I'd need K_per_head of shape (BLOCK_M, BLOCK_D) where K_per_head[m, :] = K_orig[block, m // G, :]. This is a "block-diagonal" gather. Not directly supported in tl.dot. Hmm. The cleanest way is to use 3D batched matmul with proper M padding. Let me check if there's a way to force Triton to use tensor cores for the 3D batched matmul. Actually, let me look at the actual generated SASS or PTX. Maybe Triton is using tensor cores but the metric isn't capturing it. Wait, the metric "sm__pipe_tensor_op_hmma_cycles_active" is N/A. N/A usually means the metric isn't applicable or no activity. For our kernel, it should be applicable (we have matmuls). N/A might mean 0 cycles. So no tensor cores. The matmul is using FMA. That's slow. Let me think about how to fix this. Option: use a 2D matmul with BLOCK_M=16 and proper K layout. For BLOCK_M=16, BLOCK_KV=4 (G=4), the K for query head m (in [hkv*4, hkv*4+4)) is K_orig[block, hkv, :]. The K data is the same for all 4 query heads in a group. If I lay out K_per_head as (BLOCK_M, BLOCK_D) where K_per_head[m, :] = K_orig[block, m // G, :], then the matmul is (16, 128) @ (128, 16). M=16, uses tensor cores. To create K_per_head, I can load K_orig as (BLOCK_KV, BLOCK_D) and then "expand" it to (BLOCK_M, BLOCK_D) by repeating each row G times. In Triton: ```python k_kv = tl.load(...) # (BLOCK_KV, BLOCK_D) for one page # k_per_head[m, d] = k_kv[m // G, d] k_per_head = tl.reshape(tl.broadcast_to(k_kv[None, :, :], (G, BLOCK_KV, BLOCK_D)), (BLOCK_M, BLOCK_D)) ``` Wait, this requires the k_kv data to be broadcast/expanded. Triton supports `tl.broadcast_to` for shape expansion. Let me try this. Actually, in Triton, the typical way to do this is: ```python # Load K for BLOCK_KV heads: (BLOCK_KV, BLOCK_D) k = tl.load(...) # shape (BLOCK_KV, BLOCK_D) # Expand to (G, BLOCK_KV, BLOCK_D) then to (BLOCK_M, BLOCK_D) = (G * BLOCK_KV, BLOCK_D) k = k[None, :, :] # (1, BLOCK_KV, BLOCK_D) k = tl.broadcast_to(k, (G, BLOCK_KV, BLOCK_D)) # (G, BLOCK_KV, BLOCK_D) k = tl.reshape(k, (BLOCK_M, BLOCK_D)) # (G * BLOCK_KV, BLOCK_D) = (BLOCK_M, BLOCK_D) ``` Wait, broadcast_to from (1, BLOCK_KV, BLOCK_D) to (G, BLOCK_KV, BLOCK_D) replicates the first dim. So k[g, kv, d] = k_kv[kv, d] for all g. Then reshape to (G * BLOCK_KV, BLOCK_D) = (BLOCK_M, BLOCK_D) where k[m, d] = k_kv[m // G, d]. Yes, this is what we want. But the order of elements in the reshape matters. For (G, BLOCK_KV, BLOCK_D) reshaped to (G * BLOCK_KV, BLOCK_D), the first G elements are k_kv[0, :], k_kv[1, :], ..., k_kv[BLOCK_KV-1, :], k_kv[0, :], k_kv[1, :], .... Wait, that's not right. The reshape of (G, BLOCK_KV, BLOCK_D) to (G*BLOCK_KV, BLOCK_D) is: the first G elements of the new shape correspond to the first element of G (i.e., k[0, :, :]). So the new[0:G, :] = old[0, :, :]. Then new[G:2G, :] = old[1, :, :], etc. So new[m, :] = old[m // G, m % G, :]. But old is broadcast_to'd from (1, BLOCK_KV, BLOCK_D) to (G, BLOCK_KV, BLOCK_D), so old[g, kv, :] = k_kv[kv, :]. Thus new[m, :] = k_kv[m // G, :]. But m % G in this case is 0 to G-1, and m // G is 0 to BLOCK_KV-1. For BLOCK_KV=4, G=4, m=0..15: - m=0: m//G=0, m%G=0 - m=1: m//G=0, m%G=1 - ... - m=15: m//G=3, m%G=3 So new[m, :] = k_kv[m//G, :]. Correct. But wait, the reshape might not be in the order I think. Triton's reshape is row-major by default. For (G, BLOCK_KV, BLOCK_D) reshaped to (G * BLOCK_KV, BLOCK_D), the row-major order is: - new[0, :] = old[0, 0, :] - new[1, :] = old[0, 1, :] - ... - new[BLOCK_KV-1, :] = old[0, BLOCK_KV-1, :] - new[BLOCK_KV, :] = old[1, 0, :] - ... So new[m, :] = old[m // BLOCK_KV, m % BLOCK_KV, :]. NOT m // G. Hmm. Let me think. I want new[m, :] = old[m // G, ...], not old[m // BLOCK_KV, ...]. For old of shape (G, BLOCK_KV, BLOCK_D), the row-major order is: - old[0, 0, :], old[0, 1, :], ..., old[0, BLOCK_KV-1, :], old[1, 0, :], ... So new[m, :] = old[m // BLOCK_KV, m % BLOCK_KV, :]. For BLOCK_KV=4, G=4, BLOCK_M=16: - m=0: new[0, :] = old[0, 0, :] - m=1: new[1, :] = old[0, 1, :] - ... - m=3: new[3, :] = old[0, 3, :] - m=4: new[4, :] = old[1, 0, :] - ... - m=15: new[15, :] = old[3, 3, :] So new[m, :] = old[m // 4, m % 4, :]. This is what we want (since G=4, m//G = m//4). OK so the reshape works out correctly! The layout is: - new[0, :] = old[0, 0, :] = k_kv[0, :] (since broadcast) - new[1, :] = old[0, 1, :] = k_kv[1, :] - new[2, :] = old[0, 2, :] = k_kv[2, :] - new[3, :] = old[0, 3, :] = k_kv[3, :] - new[4, :] = old[1, 0, :] = k_kv[0, :] - new[5, :] = old[1, 1, :] = k_kv[1, :] - ... So for query heads 0..3, K is k_kv[0..3]. For query heads 4..7, K is k_kv[0..3] (same). For query heads 8..11, K is k_kv[0..3]. For query heads 12..15, K is k_kv[0..3]. Wait, that's 4 query heads per group, all using the same K_kv. So we have 4 groups, each using the same K_kv[0..3]? That doesn't make sense. For GQA with G=4, query heads 0..3 use kv head 0, heads 4..7 use kv head 1, etc. So heads 0..3 should use k_kv[0], heads 4..7 should use k_kv[1], etc. Hmm. Let me re-examine. For BLOCK_KV=4, the K data loaded is for kv heads [hkv_start, hkv_start+1, hkv_start+2, hkv_start+3]. So k_kv[0] is for kv head hkv_start, k_kv[1] for hkv_start+1, etc. For query heads 0..15, they use kv heads hkv_start..hkv_start+3. Specifically: - Query head 0, 1, 2, 3 use kv head hkv_start - Query head 4, 5, 6, 7 use kv head hkv_start+1 - Query head 8, 9, 10, 11 use kv head hkv_start+2 - Query head 12, 13, 14, 15 use kv head hkv_start+3 So new[m, :] should be k_kv[m // G, :]. With my reshape, new[0, :] = old[0, 0, :] = k_kv[0, :]. Good. new[1, :] = old[0, 1, :] = k_kv[1, :]. Bad! Should be k_kv[0, :]. new[2, :] = old[0, 2, :] = k_kv[2, :]. Bad. new[3, :] = old[0, 3, :] = k_kv[3, :]. Bad. new[4, :] = old[1, 0, :] = k_kv[0, :]. Good. So the layout is wrong! I need a different arrangement. I want: new[0] = k_kv[0], new[1] = k_kv[0], new[2] = k_kv[0], new[3] = k_kv[0], new[4] = k_kv[1], new[5] = k_kv[1], ... This is the transpose of what broadcast_to gives me. Alternative: use (BLOCK_KV, G, BLOCK_D) instead of (G, BLOCK_KV, BLOCK_D). Then reshape to (BLOCK_M, BLOCK_D): - old[kv, g, d] = k_kv[kv, d] (broadcast in g) - new[m, :] = old[m // G, m % G, :] For BLOCK_KV=4, G=4: - m=0: new[0, :] = old[0, 0, :] = k_kv[0, :] - m=1: new[1, :] = old[0, 1, :] = k_kv[0, :] - m=2: new[2, :] = old[0, 2, :] = k_kv[0, :] - m=3: new[3, :] = old[0, 3, :] = k_kv[0, :] - m=4: new[4, :] = old[1, 0, :] = k_kv[1, :] - ... Yes! This is what we want. So the layout is (BLOCK_KV, G, BLOCK_D) reshaped to (BLOCK_M, BLOCK_D). So: ```python # Load K for BLOCK_KV heads: (BLOCK_KV, BLOCK_D) k_kv = tl.load(...) # (BLOCK_KV, BLOCK_D) # Expand to (BLOCK_KV, G, BLOCK_D) k_expanded = k_kv[:, None, :] # (BLOCK_KV, 1, BLOCK_D) k_expanded = tl.broadcast_to(k_expanded, (BLOCK_KV, G, BLOCK_D)) # (BLOCK_KV, G, BLOCK_D) # Reshape to (BLOCK_M, BLOCK_D) k = tl.reshape(k_expanded, (BLOCK_M, BLOCK_D)) # (BLOCK_M, BLOCK_D) ``` This gives k[m, :] = k_kv[m // G, :]. Correct! Then the matmul is (BLOCK_M=16, BLOCK_D=128) @ (BLOCK_D=128, BLOCK_N=16) = (16, 16). M=16, N=16, K=128. Uses tensor cores. But wait, the K data is now (BLOCK_M, BLOCK_D) with each row duplicated G times. The matmul is 4x more work than the minimum (since we compute the same K * Q for G query heads). Hmm. Actually, the matmul is (16, 128) @ (128, 16) = (16, 16). The Q is (16, 128). For rows 0..3 (group 0), Q is different per row, but K is the same. So S[0..3, n] = Q[0..3, :] @ K[0, :]. This computes the correct score for each of the 4 query heads (since Q is different per head). The matmul compute is 16*16*128 = 32K ops. With M=16 (real, no padding), this is the minimum. No waste. For the original 3D batched approach with M=4 padded to 16, the compute was 4 * 16*16*128 = 128K ops. 4x more. So the new approach (2D matmul with K broadcast) is 4x less compute. Plus uses tensor cores. Win-win! Let me rewrite the kernel. ```python @triton.jit def _attn_kernel(...): ... # Load Q: (BLOCK_M, BLOCK_D) q = tl.load(q_ptrs) for p in range(...): # Load K for BLOCK_KV heads: (BLOCK_KV, BLOCK_D) for one page k_kv = tl.load(k_addr_kv, ...) # (BLOCK_KV, BLOCK_D) # Expand to (BLOCK_M, BLOCK_D) by broadcasting each kv head's K to G query heads k_expanded = k_kv[:, None, :] # (BLOCK_KV, 1, BLOCK_D) k_expanded = tl.broadcast_to(k_expanded, (BLOCK_KV, G, BLOCK_D)) k = tl.reshape(k_expanded, (BLOCK_M, BLOCK_D)) # Same for V v_kv = tl.load(v_addr_kv, ...) v_expanded = v_kv[:, None, :] v_expanded = tl.broadcast_to(v_expanded, (BLOCK_KV, G, BLOCK_D)) v = tl.reshape(v_expanded, (BLOCK_M, BLOCK_D)) # Flat 2D matmul: (16, 128) @ (128, 16) = (16, 16) s = tl.dot(q, tl.trans(k)) ... # PV: (16, 16) @ (16, 128) = (16, 128) acc = acc * alpha[:, None] + tl.dot(p.to(tl.bfloat16), v) ... ``` Wait, but the K load is (BLOCK_KV, BLOCK_D) per page, not (BLOCK_N, BLOCK_KV, BLOCK_D). Let me reconsider. For one page, the K data is (P, Hkv, D) for all kv heads. We want K for BLOCK_KV specific kv heads, which is (P, BLOCK_KV, D). In the previous 3D batched approach, we loaded (P, BLOCK_KV, D) and permuted to (BLOCK_KV, P, D) for the matmul. In the new approach, we want to load (P, BLOCK_KV, D), then for each token, broadcast the K to (BLOCK_KV*G, D) for the matmul. Hmm, this is more complex. Let me think. For one page (P tokens, BLOCK_KV kv heads, D dims): - K_pkv shape: (P, BLOCK_KV, D) - For each token t, K_pkv[t, :, :] is (BLOCK_KV, D) - K for BLOCK_KV heads - We want to broadcast to (BLOCK_M, D) where each kv head is repeated G times - Result K_p[t, m, :] = K_pkv[t, m // G, :] The full K shape: (P, BLOCK_M, D). The memory is 4x larger than the K_pkv (P, BLOCK_KV, D). For one page, K memory: 16 * 4 * 128 * 2 = 16KB. For 4 pages (CHUNK=128, P=16, 4 pages per chunk? wait CHUNK=128 means 128 tokens = 8 pages). Wait, let me recheck. For BLOCK_N=16, each iter processes 1 page (P=16 tokens). With CHUNK=128, we have 8 iters per chunk. For SPLIT_K=8, total iters per program = 64 (for shape 0 with L=1024). For each iter, K load is (P, BLOCK_KV, D) = 16 * 4 * 128 * 2 = 16KB. With the new approach, K is (P, BLOCK_M, D) = 16 * 16 * 128 * 2 = 64KB. 4x more. But the matmul is now 4x less compute (since M=16 is real, no padding). Hmm, total memory bandwidth: - Old: 16KB per K load. 64 iters * 16KB = 1MB per program. - New: 64KB per K load. 8 iters per chunk (CHUNK=128, P=16) * 64KB = 512KB per program. Wait, the iter count is the same, but each iter processes P=16 tokens. So per iter K load is 64KB, not 16KB. Wait, I confused myself. Let me re-clarify. In the 3D batched approach, the loop is over pages, and each iter processes 1 page. K load per iter is (P, BLOCK_KV, D) = 16KB. In the new 2D approach with broadcast, the loop is still over pages, but the K load is (P, BLOCK_M, D) = 64KB per iter (since we materialize the broadcast). The matmul size: in 3D batched, it's (BLOCK_KV=4, G=4, BLOCK_N=16) -> 4 matmuls of (4, 128) @ (128, 16), each M=4 padded to 16 = 32K ops, total 128K ops. In 2D with broadcast, it's (16, 128) @ (128, 16) = 32K ops. So the 2D approach has 4x more memory (64KB vs 16KB per K load) but 4x less compute. For memory-bound, 4x more memory is bad! Hmm. So the 2D broadcast approach increases memory traffic 4x. That defeats the purpose. Let me think of a better approach. Option: don't materialize the broadcast in memory. Use a custom matmul that handles the broadcast logically but doesn't actually duplicate the data. In Triton, the matmul is fixed: (M, K) @ (K, N). The K data must be (K, N). We can't have a "broadcast" in the K dim. But wait, we can have a "broadcast" in the M dim. The Q data is (M, K) where M rows may share the same K. But the matmul still computes (M, N) by multiplying each row of Q with the corresponding row of K (after K^T). If Q has rows that are "duplicates" (because the same K is used), the matmul still computes them. We can't avoid this in the matmul. So the 2D approach with broadcast always increases compute by G times. For G=4, 4x more compute. Hmm. For memory-bound, this is bad. But the 2D approach uses tensor cores (16x faster for the matmul), so the 4x more compute might be hidden. Let me estimate: - 3D batched: 128K ops per matmul per page. Without tensor cores, ~256K cycles per matmul (for 4 warps). Per page: 512K cycles. Per program (64 pages): 32M cycles. At 2 GHz: 16us per program. - 2D with broadcast: 32K ops per matmul per page. With tensor cores (1 op/cycle/lane for tensor core), 32K / 128 threads = 256 cycles per matmul. Per page: 512 cycles. Per program: 32K cycles. At 2 GHz: 16us per program. Wait same? Hmm, let me recompute. For tensor cores, the throughput is 1 matmul per cycle per warp (for 16x16x16). For (16, 128) @ (128, 16), we have K=128, so 8 K-chunks. Each chunk is a 16x16x16 matmul, 1 cycle. 8 cycles per matmul per warp. With 4 warps in parallel, 2 cycles per warp. 2 cycles per matmul per warp. For 64 pages, 2 matmuls per page (QK and PV), 2 * 64 = 128 matmuls. 2 * 128 = 256 cycles per warp. At 2 GHz, 128ns per warp. So the matmul is ~128ns per program with tensor cores. Plus memory load (16KB * 64 = 1MB per program at 5 TB/s L2 = 0.2us). Total per program: ~0.3us. For 128 programs in 1 wave on 188 SMs, total: 0.3us per wave. Wave time = 0.3us. But we're at 34us. 100x slower than estimated. There's a big gap. The tensor cores should make this fast, but the current 3D batched approach doesn't use them. OK let me try the 2D broadcast approach. Even with 4x more memory, the tensor cores should make it much faster overall. For 2D broadcast: 64KB per K load, 64 iters, total 4MB per program. 128 programs = 512MB. Hmm, that's way more than the 32MB total KV. Wait, this would be a 16x duplication (since G=4 and we're loading K for each group, even though the data is the same). That's terrible. Actually no, the K data for one (page, kv_head) is loaded once per program. With BLOCK_KV=4, we load K for 4 kv heads per page. The 2D broadcast approach materializes this as (P, BLOCK_M, D) = 4x larger, but the underlying data is the same (just broadcast in registers). In Triton, the broadcast is a register operation, not a memory operation. The K is loaded as (P, BLOCK_KV, D) and then broadcast to (P, BLOCK_M, D) in registers. The memory load is still (P, BLOCK_KV, D) = 16KB per page. Wait, I need to think about this more carefully. The "broadcast to" in Triton is a layout change, not a data duplication. The data in memory is still (P, BLOCK_KV, D) = 16KB. The "expanded" tensor in registers has more elements, but the underlying memory load is the same. So the memory load is 16KB per K load, same as before. The compute is 4x less (32K vs 128K per matmul). And tensor cores are used. Wait, but if the broadcast is a layout change, then the matmul (BLOCK_M, BLOCK_D) @ (BLOCK_D, BLOCK_N) where the K is "broadcast" in registers... how does this work physically? In the GPU, the matmul is computed by warps. Each warp computes a tile of the output. For (16, 16) output, 1 warp can compute the whole thing. The warp reads Q and K from shared memory or registers. If K is "broadcast" in registers, each thread holds some elements of K. For (BLOCK_D=128, BLOCK_N=16) K, with 32 threads per warp, each thread holds 128*16/32 = 64 elements. If the K is broadcast (4 query heads share the same K), each thread holds the same 64 elements for the 4 query heads. Hmm, but the matmul is (M=16, K=128) @ (K=128, N=16). The output is (16, 16). Each output element is a dot product of a row of Q and a column of K. For 4 query heads sharing the same K, 4 output elements share the same K column. The compute can be done as: for each K column, compute 4 partial dot products (one per query head) and write 4 output elements. The total compute is 16*16*128 = 32K MACs. With tensor cores doing 16*16*16 per cycle, that's 8 cycles per matmul. For 64 pages, 2 matmuls per page = 128 matmuls = 1024 cycles per warp. At 2 GHz, 0.5us per warp. So with tensor cores, the compute is ~0.5us per warp. For 4 warps in parallel, 0.5us per program. Plus memory load (1MB at 5 TB/s = 0.2us). Total: 0.7us per program. For 128 programs in 1 wave: 0.7us per wave. We're at 34us. Still 50x slower. Hmm. There must be some other bottleneck. Let me just try the 2D broadcast approach and see. Actually wait, I realize the broadcast might be tricky. Let me think about how to do it. For the K data, we load (BLOCK_N, BLOCK_KV, BLOCK_D) (per page). For the matmul, we want (BLOCK_M, BLOCK_D) = (BLOCK_KV * G, BLOCK_D). The mapping is K_per_head[m, d] = K_pkv[t, m // G, d]. In Triton, we can do: ```python k_pkv = tl.load(...) # (BLOCK_N, BLOCK_KV, BLOCK_D) # Permute to (BLOCK_KV, BLOCK_N, BLOCK_D) k_kvn = tl.permute(k_pkv, (1, 0, 2)) # (BLOCK_KV, BLOCK_N, BLOCK_D) # Broadcast to (BLOCK_KV, G, BLOCK_N, BLOCK_D) k_kvgn = tl.broadcast_to(k_kvn[:, None, :, :], (BLOCK_KV, G, BLOCK_N, BLOCK_D)) # Reshape to (BLOCK_M, BLOCK_N, BLOCK_D) where BLOCK_M = BLOCK_KV * G k_mnd = tl.reshape(k_kvgn, (BLOCK_M, BLOCK_N, BLOCK_D)) # Permute to (BLOCK_N, BLOCK_M, BLOCK_D) for the matmul k_nmd = tl.permute(k_mnd, (1, 0, 2)) # (BLOCK_N, BLOCK_M, BLOCK_D) ``` This is complex. Let me think if there's a simpler way. Actually, the matmul is (M, K) @ (K, N). For Q @ K^T: - Q: (BLOCK_M, BLOCK_D) - K^T: (BLOCK_D, BLOCK_N) - Output: (BLOCK_M, BLOCK_N) The K^T for this matmul is the transpose of K. K has shape (BLOCK_M, BLOCK_D) (per head, after broadcast). So K^T has shape (BLOCK_D, BLOCK_M). But wait, we have BLOCK_M rows, and for each row, BLOCK_D elements. After broadcast, the K data is logically (BLOCK_M, BLOCK_D) where each row is duplicated G times. The matmul computes S[m, n] = Q[m, :] @ K[m, :]. But K[m, :] is the same for G consecutive m (e.g., m=0,1,2,3 all have the same K row). So S[m, n] is the same for m=0,1,2,3 (if Q[m, :] is the same, but Q is different per m, so S is different). Wait, the matmul is Q @ K^T, not Q @ K. Let me re-derive. S[m, n] = sum_d Q[m, d] * K[n, d] For each (m, n), the K row is K[n, :]. The K data has BLOCK_M rows, where row m' is the K for query head m'. But for the matmul, we need K[n, :] for each n (token), which is the K for token n. In the broadcast approach, we have K_per_head of shape (BLOCK_M, BLOCK_D) where K_per_head[m, d] = K_original[m // G, d]. The matmul Q @ K_per_head^T would give S[m, n] = sum_d Q[m, d] * K_per_head[n, d] = sum_d Q[m, d] * K_original[n // G, d]. But that's wrong! The K for token n and query head m is K_original[n, m // G, d] (the K for the kv head corresponding to query head m, at token n). So I confused two things. Let me re-clarify. The K data has dimensions (token, kv_head, d). For the matmul, the K for (query_head, token) is K_data[token, query_head // G, :]. The matmul S[m, n] = Q[m, :] @ K_data[n, m // G, :]. To do this with a 2D matmul, we'd need K_per_head of shape (BLOCK_M, BLOCK_D) where K_per_head[m, d] = K_data[n, m // G, d] for each n. But K_per_head depends on n! So we can't pre-compute it. For the matmul Q @ K^T, K is (BLOCK_N, BLOCK_D) and K^T is (BLOCK_D, BLOCK_N). For each row of K^T (i.e., each column of K), we need K[:, d] for a specific d. The matmul S[m, n] = sum_d Q[m, d] * K[n, d]. For this to use the correct K, we need K[n, d] = K_data[n, m // G, d]. So K depends on both n and m. In a 2D matmul, K is (BLOCK_N, BLOCK_D), independent of m. So we can't have K depend on m. Hmm. So the 2D matmul approach doesn't work for GQA directly. The 3D batched matmul handles this by having a separate K for each group of query heads. The batch dim is the kv head, and within each batch, the K is the same for all G query heads. For tensor core efficiency, we want M >= 16. With M = G = 4 padded to 16, we have 12 dummy rows that are wasted. With M = G = 4 not padded, no tensor cores. The fundamental issue: tensor cores need M >= 16, but for GQA with small G, we naturally have small M. Workarounds: 1. Process multiple kv heads per program (BLOCK_KV > 1). For BLOCK_KV=4, M=BLOCK_KV*G=16. No padding needed. 2. Use FMA (no tensor cores). Slower but works. 3. Use 2D matmul with broadcast (but as I showed, this is wrong for GQA). Wait, let me re-examine option 3. I think I confused myself. For BLOCK_KV=4, G=4, BLOCK_M=16. The Q is (16, 128) where rows 0..3 use kv head 0, 4..7 use kv head 1, etc. The K data is (BLOCK_N, BLOCK_KV, BLOCK_D) = (16, 4, 128). K[n, kv, d] is the K for kv head kv at token n. For the matmul S[m, n] = Q[m, :] @ K[n, m // 4, :]. In a 2D matmul Q @ K^T, the K is (BLOCK_N, BLOCK_D). For each row m, the K row is K^T[:, n] = K[n, :]. If we use a single K (BLOCK_N, BLOCK_D) for all 16 query heads, then S[m, n] = Q[m, :] @ K_single[n, :]. This is wrong because K should depend on m. To fix this, we can use the 3D batched matmul: S_3d[kv, g, n] = Q_3d[kv, g, :] @ K[n, kv, :] = Q[m, :] @ K[n, m//G, :]. This is correct. Or, we can do 4 separate 2D matmuls (one per kv head), each with M=G=4. This is also correct but has the M=4 padding issue. Or, we can use a flat 2D matmul with M=16 and "fake" the K layout. Specifically, we want a K tensor of shape (BLOCK_M, BLOCK_D) where K[m, d] = K_data[n, m // G, d] for each n. But K depends on n, so we can't have a single K tensor. Wait, but in the matmul, K is (BLOCK_N, BLOCK_D). For each row n, K[n, :] is the K for that token. The K for query head m at token n is K_data[n, m // G, :]. So K_per_query_head[m, n, :] = K_data[n, m // G, :]. For a 2D matmul, K is (BLOCK_N, BLOCK_D) = K_per_query_head[m', :] for some fixed m'. So we can only compute S[m', n] correctly for one m' at a time. To compute S for all 16 query heads, we need 16 separate 2D matmuls (or a 3D batched one with batch=16). Or we can have K depend on m via a 3D structure. So 2D matmul doesn't work directly for GQA. The 3D batched is the way to go. For tensor core efficiency, M should be >= 16. With M=4 (in 3D batched), Triton falls back to FMA. To use tensor cores with M=4, we can either: - Pad M to 16 (waste compute) - Combine multiple kv heads in the batch (BLOCK_KV=4, M=16) The second is what I'm doing. But the 3D batched matmul (4, 4, 16) has batch=4, M=4. Triton might still fall back to FMA because the per-batch M is 4. Let me check if there's a way to force tensor cores. Maybe `tl.dot` with `input_precision='ieee'` or some other option. Or, maybe the issue is that the 3D batched matmul doesn't support tensor cores for small M. Let me look at the Triton source or docs. Actually, let me just try a different approach: do 4 separate 2D matmuls (one per kv head) with M=4. If Triton doesn't use tensor cores, the FMA throughput might still be OK. Wait, that's what I had. And it's slow. OK let me think about this differently. The matmul is 32K MACs per page (for BLOCK_M=16, BLOCK_N=16, BLOCK_D=128). With FMA at 1 MAC/cycle/lane (for fp32 or bf16), 32K / 32 lanes = 1K cycles per warp. Per program (4 warps, 64 pages): 16K cycles per warp. At 2 GHz, 8us per warp. For 4 warps in parallel, 8us per program. For 128 programs in 1 wave, 8us per wave. Plus memory (0.2us per program * 128 = 0.2us per wave). Total: 8us per wave. But we're at 34us. 4x slower than the FMA estimate. Hmm, FMA is also slower than I thought. Let me reconsider. For bf16, the FMA might be 2 ops per cycle per lane (since each bf16 fits in 1 fp16 register, and Blackwell has 2x fp16 throughput). So 64K MACs per cycle per warp (32 lanes * 2 MACs). Wait, 32 lanes * 2 MACs/cycle = 64 MACs/cycle/warp. 32K MACs / 64 = 500 cycles per matmul per warp. 2 matmuls per page, 64 pages = 64K cycles per warp. At 2 GHz, 32us per warp. For 4 warps in parallel, 32us per program. Matches the 34us we see! So the FMA is indeed the bottleneck. To speed up, we need tensor cores. For tensor cores, M >= 16 is needed (for bf16). With M=G=4, we can't use tensor cores. The fix: use a 2D matmul with M=16 and a "smart" K layout that handles GQA. But as I showed, this requires the K to depend on m, which a 2D matmul can't do directly. UNLESS we use a different approach: for the matmul, we have Q of shape (16, 128) and K of shape (16, 128). For row m of Q, the K row should be K_data[n, m // G, :]. If we lay out K_data as (P, BLOCK_KV, BLOCK_D) and reshape to (P * BLOCK_KV, BLOCK_D) = (64, 128), then for query head m, the K row is K_data_reshaped[n + P * (m // G), :]. For the matmul Q @ K^T, the K^T is (128, 16) where for column n, K^T[:, n] = K_data_reshaped[n + P * 0, :] (if all 16 query heads use kv head 0). But we want K^T[:, n] = K_data_reshaped[n + P * (m // G), :], which depends on m. In a 2D matmul, K^T doesn't depend on m. So we can only compute one m at a time. Hmm. Unless we use a different K layout where the K for different m is concatenated. For example, K_combined of shape (P * BLOCK_KV, BLOCK_D) = (64, 128). For query head m at token n, the K row is K_combined[n + P * (m // G), :]. For a 2D matmul, K is (BLOCK_N, BLOCK_D). We want a K of shape (16, 128) where row m has the K for query head m. We can construct this: K_for_matmul[m, :] = K_combined[n + P * (m // G), :] for each n. But this depends on n. For the matmul, K is the same for all m (it's just the K matrix). So we can't have m-dependent K. OK so 2D matmul really doesn't work for GQA. The 3D batched is the way to go, but M=4 prevents tensor cores. Workaround: pad M to 16 in the 3D batched matmul. This means computing 12 dummy rows per batch. Wait, in 3D batched matmul (BLOCK_KV=4, G=4, BLOCK_N=16), the M dim is G=4. If I pad G to 16, the matmul is (BLOCK_KV=4, G_padded=16, BLOCK_N=16). The matmul compute is 4 * 16*16*128 = 128K ops. With dummy rows, the output is (4, 16, 16) with 12 dummies per batch. For 64 pages, 2 matmuls = 16M ops per program. At 128K ops/cycle (tensor core throughput for 4 warps), 125 cycles per page per warp. 64 pages = 8K cycles per warp. At 2 GHz, 4us per program. For 128 programs in 1 wave, 4us. Plus memory 0.2us. Total 4us. Much better than 34us! But the 12 dummy rows add compute. Total compute is 4x more than the minimum. Still, with tensor cores, this is 4us which is fine. Let me try padding G to 16 in the 3D batched matmul. So G_padded = max(G, 16) = 16 for all our shapes (G=4 or 8, so 16 is the next multiple of 16). Wait, G=8 would need G_padded=8 (already >= 8). Hmm, actually tensor cores need M >= 16 (or maybe 8 for some configs). Let me just pad to 16. For G=4, G_padded=16. 12 dummies. For G=8, G_padded=16. 8 dummies. The matmul is (BLOCK_KV, G_padded, BLOCK_N). For BLOCK_KV=4, G=4, G_padded=16, BLOCK_N=16: - 4 batches of (16, 128) @ (128, 16) = 4 * 32K = 128K ops per matmul. For 64 pages, 2 matmuls = 16M ops per program. With tensor cores (1 op/cycle/lane for 16x16x16, with 4 warps, 32 lanes per warp, so 4*32 = 128 lanes doing 1 op/cycle = 128 ops/cycle). 16M / 128 = 125K cycles. At 2 GHz, 62us. Hmm, that's worse than 34us. Wait, tensor cores do 16x16x16 = 4096 ops per cycle per warp (per the warp matrix shape). For 4 warps, 4 * 4096 = 16K ops per cycle. 16M / 16K = 1K cycles. At 2 GHz, 0.5us. So with tensor cores, the matmul is 0.5us per program. Way better. OK so the fix is to pad G to 16 in the 3D batched matmul. Let me try. Actually wait, I realize Triton might not allow padding in 3D batched matmul directly. Let me think. If I use BLOCK_M = 16 (padded from G=4), and reshape Q from (BLOCK_M=16, BLOCK_D) to (BLOCK_KV=4, G=16, BLOCK_D), with 12 dummies... the Q data is loaded as (BLOCK_M=16, BLOCK_D) where rows 0..3 are real Q and rows 4..15 are 0 (or some dummy value). Then the 3D batched matmul is (4, 16, 128) @ (4, 128, 16) = (4, 16, 16). M=16, uses tensor cores. The 12 dummies in each batch don't affect the output (their Q is 0, so the dot product is 0). For the K, we have (BLOCK_N, BLOCK_KV, BLOCK_D) = (16, 4, 128). For the matmul, we permute to (BLOCK_KV, BLOCK_N, BLOCK_D) = (4, 16, 128), then K^T is (4, 128, 16). The matmul: Q_padded (4, 16, 128) @ K^T (4, 128, 16) = (4, 16, 16). The 12 dummies in Q give 0 output. The 4 real Q rows (per batch) give the correct score. The output is (4, 16, 16). We take the first G=4 rows of each batch: (4, 4, 16). Reshape to (16, 16) for the softmax. Let me code this. ```python # Load Q with dummies: (BLOCK_M_padded, BLOCK_D) where BLOCK_M_padded = 16 q_padded = tl.load(q_ptrs_padded) # (16, 128), with dummies = 0 # Reshape to (BLOCK_KV, G_padded, BLOCK_D) = (4, 16, 128) q3_padded = tl.reshape(q_padded, (BLOCK_KV, G_PADDED, BLOCK_D)) # (4, 16, 128) # Load K: (BLOCK_N, BLOCK_KV, BLOCK_D) = (16, 4, 128) k = tl.load(k_addr, ...) # Permute to (BLOCK_KV, BLOCK_N, BLOCK_D) = (4, 16, 128) k3 = tl.permute(k, (1, 0, 2)) # K^T: (4, 128, 16) kt3 = tl.permute(k3, (0, 2, 1)) # Matmul: (4, 16, 128) @ (4, 128, 16) = (4, 16, 16) s3_padded = tl.dot(q3_padded, kt3) # Take only the first G=4 rows: (4, 4, 16) s3 = s3_padded[:, :G, :] # Reshape to (BLOCK_M, BLOCK_N) = (16, 16) s = tl.reshape(s3, (BLOCK_M, BLOCK_N)) ``` The issue: in Triton, can I do `s3_padded[:, :G, :]` to slice? I don't think Triton supports arbitrary slicing. I'd need to use `tl.split` or similar. Alternative: store all 16 rows and just compute the softmax over the 16 rows. The softmax gives uniform for the 12 dummies (since their score is 0 for all valid tokens). Then acc is updated for all 16 rows, but only the 4 real rows are stored. Wait, the matmul output is (4, 16, 16) for s3_padded. If I compute the softmax over all 16 rows (per batch), the 12 dummies have score 0 for all n (since their Q is 0). The softmax gives uniform 1/BLOCK_N for the dummies. The weighted sum is mean(V). For the 4 real rows, the softmax gives the correct attention weights. After the matmul, I have s3 of shape (4, 16, 16) (or (BLOCK_KV, G_PADDED, BLOCK_N)). I can compute the softmax over the last dim (axis=2). The output is (4, 16, 16) where rows 4..15 are uniform (for the dummies). Then for the PV matmul, the input is (4, 16, 16) and V is (4, 16, 128). Output is (4, 16, 128). I take the first G=4 rows: (4, 4, 128). Reshape to (16, 128). But how do I "take the first G rows" in Triton? I can store all 16 rows and just not use the 12 dummies in the final output. The acc for dummies is mean(V), which I don't write out. OK let me try this approach. ```python # Load Q padded: (BLOCK_M, BLOCK_D) where BLOCK_M = BLOCK_KV * G_PADDED = 16 # Rows 0..3 are real (for G=4), rows 4..15 are 0 q_ptrs = ... + offs_m[:, None] * stride_qh + ... # offs_m is h_start + tl.arange(0, BLOCK_M) where BLOCK_M = 16 # But we only have G=4 real heads per program (since hkv_blk covers 4 kv heads * 4 = 16 heads) # Wait, h_start = hkv_start * G, and hkv_start = hkv_blk * BLOCK_KV = 0 (for hkv_blk=0) # So h_start = 0, and the 16 heads are heads 0..15 # All 16 are real! No dummies needed. # Wait, with BLOCK_M=16 and H=32, hkv_blk=0 covers heads 0..15. All real. # hkv_blk=1 covers heads 16..31. All real. # So for our shapes, all 16 heads per program are real. No dummies. # But for the 3D batched matmul, we want (BLOCK_KV, G, BLOCK_N) = (4, 4, 16) # The matmul is M=4, which doesn't use tensor cores. # To force tensor cores, we need to pad G to 16. But then we have 12 dummies in each batch. # The dummies' Q is 0, so their score is 0, and their softmax is uniform. # But all 16 query heads are real! There are no dummy query heads. # So we can't pad with dummies (since there are no extra heads to use as dummies). # Hmm. The only way to pad is to use a BLOCK_M > H, but H=32 for our shapes, and BLOCK_M=16 < 32. # Wait, I'm confusing BLOCK_M with the actual heads. Let me re-think. # For hkv_blk=0, we process kv heads 0..3. The query heads are 0..15 (4 per kv head). # We have 16 query heads, all real. BLOCK_M=16 covers them all. # For the 3D batched matmul, the batch is BLOCK_KV=4 (one per kv head), and within each batch, we have G=4 query heads. # The matmul is (4, 4, 128) @ (4, 128, 16). Per-batch M=4. No tensor cores. # To use tensor cores, we need per-batch M >= 16. We can pad by adding 12 dummy heads per batch. # But we don't have 12 dummy heads (all 16 are real). # Hmm. Can we use a different batch structure? # Option: have BLOCK_KV=1 and pad G to 16. Then for each kv head, we have 4 real heads and 12 dummies. # But BLOCK_KV=1 means 8x more programs (for shape 0), which we tried and was slower. # Wait, the 8x more programs means 8x less work per program. The total work is the same. With more programs, we have more parallelism, but the per-program overhead is higher. # Actually, the issue with BLOCK_KV=1 was the kernel runs longer, not the same. Let me re-think. # With BLOCK_KV=1, the 3D batched matmul is (1, 4, 128) @ (1, 128, 16). M=4. Same FMA issue. # So BLOCK_KV=1 also has the FMA issue. # The only way to use tensor cores is to have M >= 16. With BLOCK_KV=4, M=16 in the 3D batched matmul. But this requires (BLOCK_KV=4, G=4, BLOCK_N=16) -> 4 batches of M=4, no tensor cores. # Hmm. Wait, in 3D batched matmul, the M dim is G=4, not BLOCK_KV*G. So even with BLOCK_KV=4, the per-batch M is G=4. # To get per-batch M=16, we need G >= 16 (not in our shapes) or pad G to 16 with dummies. # Padding G to 16: have 4 real heads and 12 dummies. The dummies' Q is loaded as 0 (since we only have 4 real heads per program). # Wait, but with BLOCK_M=16, we have 16 query heads. If we reshape to (BLOCK_KV=4, G=4, BLOCK_D), the 16 heads are all real. To pad to G=16, we'd need 64 query heads (4 kv heads * 16 = 64), but we only have 16. # So padding G to 16 in this structure doesn't work. # Hmm. Let me think differently. # What if I use a different BLOCK_M? For example, BLOCK_M=64 (covering 4 kv heads * 16 padded G). Then I have 64 query heads, with 16 real and 48 dummies. # But H=32 for our shapes. So 64 > H. We can't have 64 heads in a program. # UNLESS we process multiple batches per program. For shape 0 with B=8, we could have 2 batches per program, giving 64 heads. But then we have less parallelism. # Hmm, this is getting complex. # Let me think about a different approach: use a 2D matmul with M=16, but load K differently. # The K for query head m is K_data[block, m // G, :]. If I lay out K as (BLOCK_N, BLOCK_D) where for each n, K[n, :] is the K for query head 0 (kv head 0), then the matmul Q @ K^T gives S[m, n] = Q[m, :] @ K[n, :] = Q[m, :] @ K_data[block, 0, :]. This is correct for query heads 0..3 (which use kv head 0) but wrong for query heads 4..15. # To fix, I'd need K to be different for different m. But a 2D matmul can't do this. # UNLESS I do 4 separate matmuls, each with M=4. But M=4 doesn't use tensor cores. # Or, I can do 4 separate matmuls, each with M=4 padded to 16 (12 dummies). This uses tensor cores but has 4x more compute. # Wait, that's what I want! 4 matmuls of (16, 128) @ (128, 16) with 12 dummies each = 4 * 32K = 128K ops per matmul. With tensor cores, this is fast. # For 64 pages, 2 matmuls per page = 8 matmuls total. Each is 128K ops. Total: 1M ops per program. # Wait, 2 matmuls per page (QK and PV), each is 128K ops. Per program (64 pages): 64 * 2 * 128K = 16M ops. # With tensor cores (16K ops/cycle for 4 warps), 16M / 16K = 1K cycles. At 2 GHz, 0.5us per program. # For 128 programs in 1 wave, 0.5us per wave. Plus memory 0.2us. Total 0.7us. Way better than 34us. # But the dummies add 12/4 = 3x compute. Still, with tensor cores, it's fast. # Let me try this approach: 4 separate matmuls (or batched 3D) with M padded to 16. # For the 3D batched matmul (4, 4, 16) -> pad to (4, 16, 16). The M dim is padded from 4 to 16. The 12 dummies have Q=0. # For the Q data: we have (BLOCK_M=16, BLOCK_D) = 16 real query heads. Reshape to (BLOCK_KV=4, G=4, BLOCK_D) = (4, 4, 128). Pad to (4, 16, 128) by adding 12 dummies per batch. # But the dummies are 0, and the 4 real Q are the original. The reshape from (16, 128) to (4, 4, 128) gives Q_3d[0, 0, :] = Q[0, :], Q_3d[0, 1, :] = Q[1, :], ..., Q_3d[3, 3, :] = Q[15, :]. All real. # To pad to (4, 16, 128), we need 12 more rows per batch, total 64 rows. But we only have 16. # Hmm. So we can't pad G to 16 within the (4, 4, 128) structure. # Alternative: restructure as (BLOCK_KV * G_PADDED, BLOCK_D) = (64, 128) and pad to 64 with 48 dummies. But we only have 16 real heads. # So padding within the program doesn't work because we don't have extra real heads. # OK so the constraint is: BLOCK_M = 16 (the real number of query heads per program). We can't pad beyond this. # For the 3D batched matmul (4, 4, 16), M=4. No tensor cores. # For the 3D batched matmul to use tensor cores, we need M=4 to be padded to 16. But the 12 dummies would have Q=0, and the matmul would compute (12 dummies) * K for each batch, giving 0 output (since Q=0). The total compute is 4x more, but with tensor cores, it's fast. # Wait, can I just lie to Triton about the M dim? I.e., declare M=16 but only fill in 4 real rows. The other 12 rows are "junk" but Triton doesn't know. # In Triton, tl.dot takes the actual tensor, not a declared size. So I can't lie about the size. # UNLESS I use padding explicitly: create a tensor of shape (16, 128) where rows 0..3 are real and rows 4..15 are 0. Then use this in the matmul. # But the (16, 128) tensor requires 16 query heads, and I only have 4. So I need to construct it with 12 dummies. # Wait, with BLOCK_M=16 and BLOCK_KV=4, the 16 query heads are all real (heads 0..15 for hkv_blk=0). All real, no dummies. # For the 3D batched matmul, I want to reshape (16, 128) to (BLOCK_KV=4, G=4, BLOCK_D) = (4, 4, 128). This is correct: Q_3d[kv, g, :] = Q[kv*4 + g, :]. # The matmul (4, 4, 128) @ (4, 128, 16) has per-batch M=4. No tensor cores. # To force tensor cores, I can pad the inner G dim. But G=4 is the actual group size. I can't change it. # UNLESS I use a different matmul structure. What if I use 4 separate 2D matmuls, each with M=4 padded to 16? # For each kv head in [0, 4): # Q_sub = Q_3d[kv, :, :] # (4, 128) # Pad to (16, 128) with 12 zeros # K_sub = K_3d[kv, :, :] # (BLOCK_N, 128) after permute # S_sub = Q_sub_padded @ K_sub^T # (16, 16) with 12 dummies # Take the first 4 rows: S_sub_real = S_sub[:4, :] # In Triton, I can do this with 4 separate tl.dot calls. But the loop overhead is high. # Or, I can use a single 3D batched matmul (4, 4, 128) @ (4, 128, 16) and accept FMA. # Hmm. Let me think about whether FMA is really 4x slower than tensor cores for our matmul size. # For (4, 128) @ (128, 16) with FMA: # 4*16*128 = 8K MACs per matmul # FMA throughput: 32 lanes * 2 ops/cycle (bf16) = 64 ops/cycle/warp # 8K / 64 = 128 cycles per matmul per warp # 4 matmuls (one per kv head) per page = 512 cycles per page per warp # 64 pages = 32K cycles per warp # At 2 GHz, 16us per warp # With 4 warps in parallel, 16us per program. Matches our 34us (close enough). # For tensor cores (16, 128) @ (128, 16): # 16*16*128 = 32K MACs per matmul # Tensor core throughput: 16x16x16 per cycle = 4K ops/cycle/warp # 32K / 4K = 8 cycles per matmul per warp # 4 matmuls per page = 32 cycles per page per warp # 64 pages = 2K cycles per warp # At 2 GHz, 1us per warp. With 4 warps, 1us per program. # So tensor cores are 16x faster. If we can use them, the kernel would be 16x faster. # The issue: tensor cores need M >= 16. With M=4, they don't work. # Let me see if there's a way to use tensor cores with M=4. # Option 1: use larger BLOCK_N. For BLOCK_N=64, the matmul is (4, 128) @ (128, 64). M=4, N=64. Still M=4, no tensor cores. # Option 2: use larger BLOCK_M. For BLOCK_M=16, BLOCK_KV=4, BLOCK_N=16: 3D batched (4, 4, 128) @ (4, 128, 16). Per-batch M=4. # Hmm. Let me look at the Triton source for tl.dot with 3D inputs. # Actually, let me just try `input_precision='ieee'` or other options. # Or, let me try a flat 2D matmul with M=16 and see if Triton can use tensor cores even with the "wrong" K. # Wait, I had this idea before. The 2D matmul with M=16 and K duplicated gives wrong results for GQA. So that's not an option. # OK let me think about this from a different angle. The 3D batched matmul in Triton might not support tensor cores for small per-batch M. So I need to use a different structure. # Option: 4 separate 2D matmuls with M=4, but pad each to M=16. This uses tensor cores but with 4x more compute. # Let me try this. The kernel would be: # for kv in range(BLOCK_KV): # Q_sub = q3[kv, :, :] # (G, BLOCK_D) # Q_padded = pad_to_16(Q_sub) # (16, BLOCK_D) with 12 zeros # K_sub = k3[kv, :, :] # (BLOCK_N, BLOCK_D) # S_sub = tl.dot(Q_padded, tl.trans(K_sub)) # (16, BLOCK_N) with 12 dummies # ... # But this is 4 separate matmuls per page, with significant overhead. # Alternatively, I can structure the data so that all 4 kv heads' Q are concatenated, giving a (64, 128) tensor with 48 dummies. Then a single (64, 128) @ (128, 16) matmul = (64, 16) with 48 dummies. # But I need 4 different K's for the 4 kv heads. In a 2D matmul, K is the same for all 64 rows. So the matmul is wrong. # Unless I duplicate the K in a specific way. Specifically, I want K for row m to be K_data[block, m // 16, :]. For m in [0, 16), K = K_data[block, 0, :]. For m in [16, 32), K = K_data[block, 1, :]. For m in [32, 48), K = K_data[block, 2, :]. For m in [48, 64), K = K_data[block, 3, :]. # If I lay out K as (4, 16, BLOCK_D) and broadcast to (64, BLOCK_D) where row m uses K_data[m // 16, :], then the 2D matmul is correct. # In Triton: # K_pkv = tl.load(...) # (BLOCK_N, BLOCK_KV=4, BLOCK_D=128) # K_per_head = K_pkv broadcasted to (BLOCK_N, 64, BLOCK_D) where row m uses K_pkv[:, m // 16, :] # Then Q @ K_per_head^T = (64, 128) @ (128, 16) = (64, 16) # But we only have 16 real query heads. The 48 dummies have Q=0, so their score is 0, and softmax gives uniform. # Hmm wait, the dummies' Q is 0, so Q[m, :] = 0 for m in 16..63. The matmul output S[m, n] = Q[m, :] @ K[n, :] = 0. So all 48 dummies' scores are 0, softmax is uniform. # The output for the 16 real heads is correct (since K is correctly broadcasted). # But we waste compute on 48 dummies. The compute is 64*16*128 = 128K ops per matmul (vs 32K for the minimum with M=16). # Hmm wait, the original 3D batched was 4 * 16*16*128 = 128K ops with M=4 padded to 16. Same compute. # So the 2D matmul with broadcast is equivalent in compute to the 3D batched with M=4 padded. Both have 4x more compute than the M=16 minimum. # The difference: 2D matmul with M=16 might use tensor cores. 3D batched with M=4 doesn't. # Wait, but in the 2D approach, the matmul is (64, 128) @ (128, 16) = (64, 16). M=64, N=16, K=128. M=64 >= 16, uses tensor cores. # In the 3D batched approach, the matmul is (4, 4, 128) @ (4, 128, 16) = (4, 4, 16). Per-batch M=4. No tensor cores. # So 2D is better! # Let me try the 2D approach with broadcast. # The K layout: load K_pkv of shape (BLOCK_N, BLOCK_KV, BLOCK_D). Broadcast to (BLOCK_N, 64, BLOCK_D) where row m uses K_pkv[:, m // 16, :]. Then take the first 16 rows (since we have 16 real heads). # Wait, we have 16 real heads (rows 0..15). For m in [0, 16), m // 16 = 0. So all 16 use K_pkv[:, 0, :]. That's wrong! They should use K_pkv[:, m // 4, :] (since G=4). # Hmm, I had the wrong structure. Let me re-think. # For G=4, BLOCK_KV=4, BLOCK_M=16: # - Query heads 0..3 use kv head 0 # - Query heads 4..7 use kv head 1 # - Query heads 8..11 use kv head 2 # - Query heads 12..15 use kv head 3 # So for m in [0, 4), m // 4 = 0. For m in [4, 8), m // 4 = 1. Etc. # The K for query head m is K_pkv[:, m // 4, :]. # For the 2D matmul, K is (BLOCK_N, BLOCK_D). For each row n, K[n, :] should be the K for query head m at token n. But K is the same for all m in a 2D matmul. # So we can only do the matmul for one m at a time. Or use the broadcast approach. # With broadcast: K_per_head of shape (BLOCK_M=16, BLOCK_D) where K_per_head[m, :] = K_pkv[n, m // 4, :] for each n. But this depends on n. # So K_per_head is a function of n. For a 2D matmul, K is (BLOCK_N, BLOCK_D), which is a 2D tensor. We can have K[n, :] be a function of n. # So K_per_head[n, :] = K_pkv[n, m // 4, :] for some m. But we want this for all m. # Hmm. Let me think. # The matmul S[m, n] = Q[m, :] @ K_pkv[n, m // 4, :]. # If I construct K_combined of shape (BLOCK_M, BLOCK_D) where K_combined[m, :] = K_pkv[n, m // 4, :], then the matmul is S[m, n] = Q[m, :] @ K_combined[m, :]. But this is element-wise, not a matmul. # A matmul is Q @ K^T where K^T is (BLOCK_D, BLOCK_N). For S[m, n] = Q[m, :] @ K^T[:, n], we need K^T[:, n] = K_pkv[n, m // 4, :]. # K^T[:, n] is a function of n (and m). In a 2D matmul, K^T is (BLOCK_D, BLOCK_N) where each column corresponds to a token. K^T[:, n] is the K for token n. # For the matmul to give the correct S[m, n] for all m, K^T[:, n] would need to depend on m. But K^T is the same for all m in a 2D matmul. # So we can only compute S for one m at a time. Unless we use a 3D structure. # Hmm. OK so the 2D matmul really doesn't work for GQA with multiple kv heads. # The only way is to have BLOCK_M = G (one kv head's query heads) and accept the M=G tensor core issue. # Or, use a completely different approach: load K for one kv head at a time, and do 4 separate matmuls. # Let me think about option 4: 4 separate matmuls with M=4 padded to 16. # For each kv head in [0, 4): # Q_sub = Q[m_start:m_start+4, :] # (4, BLOCK_D) # Q_padded = pad_to_16(Q_sub) # (16, BLOCK_D) with 12 zeros # K_sub = K_pkv[:, kv, :] # (BLOCK_N, BLOCK_D) # S_sub = Q_padded @ K_sub^T # (16, BLOCK_N) with 12 dummies # softmax over the 16 rows # acc_sub update # This uses tensor cores (M=16). 4x more compute (due to dummies). Per matmul, 32K ops with tensor cores = 8 cycles/warp. # For 4 matmuls per page, 32 cycles/warp. For 64 pages, 2K cycles/warp. At 2 GHz, 1us/warp. With 4 warps, 1us/program. # For 128 programs in 1 wave, 1us/wave. Plus memory. Total ~2us. # But the kernel has 4 separate matmuls per page, which is a lot of code. Let me see if I can express this in Triton efficiently. # In Triton, I can do: # Q_padded = ... # (16, BLOCK_D) with dummies # for kv in range(BLOCK_KV): # K_sub = K_pkv[:, kv, :] # (BLOCK_N, BLOCK_D) # S_sub = tl.dot(Q_padded, tl.trans(K_sub)) # (16, BLOCK_N) # ... # But this is a runtime loop over BLOCK_KV, which Triton supports if BLOCK_KV is a constexpr. # Actually, BLOCK_KV=4 is a constexpr. So I can unroll the loop: # for kv in tl.static_range(BLOCK_KV): # ... # Or use 4 separate code blocks. # Let me try this. # Wait, actually, I realize I can use the 3D batched matmul with the M dim padded. If I declare the matmul as (BLOCK_KV, G_PADDED, BLOCK_D) @ (BLOCK_KV, BLOCK_D, BLOCK_N) = (BLOCK_KV, G_PADDED, BLOCK_N), where G_PADDED=16, then M=16 in each batch, uses tensor cores. # For the Q, I need (BLOCK_KV, G_PADDED, BLOCK_D) = (4, 16, 128). I have Q_3d of shape (BLOCK_KV=4, G=4, BLOCK_D=128) = (4, 4, 128). I need to pad to (4, 16, 128). # The padding is: Q_3d_padded[kv, g, :] = Q_3d[kv, g, :] for g in [0, 4), and 0 for g in [4, 16). # In Triton, I can create a (4, 16, 128) tensor with 12 dummies. The dummies are 0. # To do this, I can: # 1. Load Q as (BLOCK_M=16, BLOCK_D). # 2. Reshape to (BLOCK_KV=4, G=4, BLOCK_D) = (4, 4, 128). # 3. Pad to (4, 16, 128) by adding 12 dummies per batch. # The padding in Triton: I can create a zero tensor of shape (4, 16, 128) and copy Q_3d into the first 4 positions of each batch. # Or, I can use a different approach: load Q as (BLOCK_M=64, BLOCK_D) where the first 16 are real and the rest are 0 (masked load). Then reshape to (4, 16, 128). # But BLOCK_M=64 is larger than the actual number of query heads (16). We don't have 64 heads. # Hmm. Let me think. # Actually, with hkv_blk covering 4 kv heads, the query heads are [hkv_start*G, hkv_start*G + BLOCK_M) = [0, 16) for hkv_start=0. All 16 are real. # If I want to load 64 heads (with 48 dummies), I need offs_m to go up to 48. But H=32, so offs_m up to 31. We can't go to 48. # Unless I use hkv_blk differently. For hkv_blk=0, hkv_start=0, heads 0..15. For hkv_blk=1, hkv_start=4, heads 16..31. With hkv_blk=0, the 48 dummies would be at heads 32..47, but H=32, so we can't access them. # So we can't pad with extra query heads in the same program. # OK so the only way to use tensor cores is to: # 1. Have BLOCK_M = 16 (or some multiple of 16) with all real heads. # 2. Use a matmul structure that has M=16. # For option 2, the matmul must be (16, K) @ (K, N). The K dim corresponds to BLOCK_D. The N dim corresponds to BLOCK_N. # For GQA, the K data depends on the query head. In a 2D matmul, K is the same for all query heads. So the matmul is correct only if all 16 query heads use the same K. # For 4 separate matmuls, each with M=4 padded to 16: # - 4 matmuls, each (16, 128) @ (128, 16) # - The Q is the same for all 4 matmuls (since we have 16 real heads and 12 dummies) # - Wait, that doesn't make sense. We want 4 different Q (one per kv head). # Hmm. Let me re-think. # For BLOCK_KV=4, the 16 query heads use 4 different K's (one per kv head). Specifically: # - Heads 0..3 use K_kv0 # - Heads 4..7 use K_kv1 # - Heads 8..11 use K_kv2 # - Heads 12..15 use K_kv3 # If I do 4 separate matmuls, each with M=4 padded to 16: # - Matmul 1: Q[0:4, :] padded to (16, 128) @ K_kv0^T = (16, 16). First 4 rows are real. # - Matmul 2: Q[4:8, :] padded to (16, 128) @ K_kv1^T = (16, 16). First 4 rows are real. # - ... # Each matmul has different Q and different K. The 4 matmuls together give the correct scores for all 16 heads. # In Triton, I can unroll this loop: # for kv in tl.static_range(BLOCK_KV): # Q_sub = Q[kv*G:(kv+1)*G, :] # (G, BLOCK_D) # Q_padded = pad_to_16(Q_sub) # K_sub = K_pkv[:, kv, :] # S_sub = tl.dot(Q_padded, tl.trans(K_sub)) # ... # But Q_padded has 12 dummies, and the matmul is 4x more compute than necessary. # Alternatively, I can use 4 separate 2D matmuls without padding (M=4). No tensor cores, but less compute. # Let me try the padded version and see. # Actually, let me first try a different optimization: increase the matmul size by using a larger BLOCK_N. # For BLOCK_N=64 (4 pages per iter), the matmul is (4, 128) @ (128, 64) = (4, 64). Per-batch M=4. Still no tensor cores. # Hmm. # OK let me just try the 4 separate padded matmuls. The code will be more complex but should be faster due to tensor cores. # Actually wait, I just thought of something. What if I use BLOCK_M=16 and the 3D batched matmul (4, 4, 16), but pass `input_precision='ieee'` to force IEEE compliance? Or some other option to use tensor cores? # Let me look at Triton's dot options. # From Triton docs: # - input_precision: 'tf32', 'tf32x3', 'ieee'. Controls precision of the matmul. # - For bf16 inputs, the matmul is always done with bf16 precision (no tf32 option). # - allow_tf32: for fp32 inputs, whether to use tf32. # For bf16, the matmul uses bf16 tensor cores. The M, N, K constraints depend on the hardware. # For Blackwell SM 12.0, the tensor core shapes for bf16 are likely 16x16x16 or similar. M=4 wouldn't fit. # So no way to force tensor cores for M=4. # The only option is to pad M to 16. # Let me try the 4 separate padded matmuls approach. # Actually, I realize there might be a simpler way. What if I just have BLOCK_M=16 and the matmul is (16, 128) @ (128, 16), but the K is laid out as a "gathered" tensor where each row is the K for the corresponding query head? # For 16 query heads, the K is (16, 128) where row m is K_data[block, m // 4, :]. This is a "block-diagonal" gather. # The matmul Q @ K^T = (16, 128) @ (128, 16) = (16, 16). M=16, uses tensor cores. # To create the gathered K, I can: # 1. Load K_pkv of shape (BLOCK_N, 4, 128) (i.e., (BLOCK_N, BLOCK_KV, BLOCK_D)) # 2. For each row m in [0, 16), K[m, :] = K_pkv[n, m // 4, :] for the current n # But K is (16, 128), one K per query head, not per token. So K doesn't depend on n. The matmul is over BLOCK_N tokens, with K being the same for all tokens... wait no, K is (BLOCK_N, BLOCK_D) in the matmul. # Hmm. Let me re-derive. # The matmul is Q @ K^T where: # - Q is (BLOCK_M=16, BLOCK_D=128) # - K is (BLOCK_N=16, BLOCK_D=128) # - K^T is (BLOCK_D=128, BLOCK_N=16) # - Output S is (BLOCK_M=16, BLOCK_N=16) # S[m, n] = sum_d Q[m, d] * K[n, d] # For GQA, we want S[m, n] = sum_d Q[m, d] * K_data[n, m // 4, d]. # So K[n, d] should be K_data[n, m // 4, d], which depends on m. But K is (BLOCK_N, BLOCK_D), same for all m. # So we can only compute S for one m at a time. Unless we do 4 separate matmuls. # OK so the 2D matmul doesn't work. Let me just try the 4 separate padded matmuls. # In Triton: # Q is (BLOCK_M=16, BLOCK_D=128) # K_pkv is (BLOCK_N, BLOCK_KV=4, BLOCK_D=128) # Unroll over kv: # for kv in tl.static_range(4): # Q_sub = Q[kv*4:(kv+1)*4, :] # (4, 128) # Q_padded = ... # (16, 128) with 12 zeros # K_sub = K_pkv[:, kv, :] # (16, 128) # S_sub = tl.dot(Q_padded, tl.trans(K_sub)) # (16, 16) with 12 dummies # S_sub_real = S_sub[:4, :] # (4, 16) # # softmax over S_sub_real # ... # P_sub_real = ... # (4, 16) # P_sub_padded = ... # (16, 16) with 12 zeros # V_sub = V_pkv[:, kv, :] # (16, 128) # O_sub = tl.dot(P_sub_padded, V_sub) # (16, 128) with 12 dummies # O_sub_real = O_sub[:4, :] # # update acc # This is 4 iterations of (matmul + softmax + matmul). Each iteration is 32K ops with tensor cores. # Per page: 4 * 2 * 32K = 256K ops (vs 128K for the 3D batched). 2x more compute. # Hmm, why 2x more? Because each batch's matmul is (16, 128) @ (128, 16) = 32K, and we have 4 batches = 128K. Same as 3D batched. # Wait, in the 3D batched (4, 4, 128) @ (4, 128, 16) = (4, 4, 16), the compute is 4 * 4 * 16 * 128 = 32K. But with M=4 padded to 16, it's 4 * 16 * 16 * 128 = 128K. # So both have 128K ops. Same. # With tensor cores, 128K ops / 4K ops/cycle/warp = 32 cycles per matmul per warp. For 2 matmuls per page, 64 cycles. For 64 pages, 4K cycles. At 2 GHz, 2us per warp. With 4 warps, 2us per program. # For 128 programs in 1 wave, 2us per wave. Plus memory 0.2us. Total 2.2us. # Let me try this approach. # Actually, I realize there's an even simpler way. Instead of 4 separate matmuls, I can use a single 3D batched matmul where the inner M is padded to 16. # For the 3D batched matmul (BLOCK_KV=4, G_PADDED=16, BLOCK_N=16): # - Q is (4, 16, 128) where the last 12 rows of each batch are dummies (0). # - K is (4, 16, 128) after permute from K_pkv. # - Matmul: (4, 16, 128) @ (4, 128, 16) = (4, 16, 16). M=16, uses tensor cores. # The 12 dummies per batch give 0 output. The 4 real rows give the correct score. # To create Q_padded (4, 16, 128): # - Load Q as (BLOCK_M=16, BLOCK_D=128). # - Reshape to (4, 4, 128). # - Pad to (4, 16, 128) by adding 12 dummies per batch. # The padding: Q_padded[kv, g, :] = Q_3d[kv, g, :] for g in [0, 4), and 0 for g in [4, 16). # In Triton, I can do this with `tl.cat` or by creating a zero tensor and using `tl.where`. # Or, I can load Q with extra rows (48 dummies) at known memory locations (e.g., a zero buffer). But that's complex. # Simpler: create the padded Q in registers using `tl.where` or by using the fact that Triton supports 0-initialization. # Actually, in Triton, I can just create a (4, 16, 128) tensor initialized to 0 and then copy the (4, 4, 128) into the first 4 positions of each batch. # Hmm, this is getting complex. Let me think of a simpler way. # Alternative: use `tl.dot` with `acc` argument to accumulate, and structure the matmul as a sum of smaller matmuls. # Or: use the 4 separate matmuls approach with tl.static_range. # Let me try the 4 separate matmuls. # Actually, I realize there's an even simpler approach. What if I just use a (BLOCK_M, BLOCK_KV, BLOCK_D) tensor and do 3D batched matmul, but with the understanding that Triton might or might not use tensor cores? # Or, what if I just try `tl.dot` with 3D inputs and see if Triton uses tensor cores for M=4? # Looking at the ncu output, `sm__pipe_tensor_op_hmma_cycles_active.avg.pct_of_peak_sustained_elapsed` is N/A. This means no tensor core activity. So Triton is not using tensor cores for the 3D batched matmul with M=4. # So I need to pad M to 16. Let me just do it. # The cleanest way: use 4 separate 2D matmuls with M=4 padded to 16. # In Triton, I can unroll the loop: # for kv in tl.static_range(BLOCK_KV): # ... # And inside, pad the Q to 16 and do the matmul. # Let me code this. # Actually, an even simpler approach: just use a flat 2D matmul with M=16, but accept that 12 of the 16 rows are "duplicates" of the 4 real rows (in the Q sense). For the K, use the correct K for each row. # Specifically, for the matmul: # - Q: (16, 128) where rows 0..3 are real Q for kv 0, rows 4..7 are real Q for kv 1, etc. # - K: (16, 128) where row m is K_data[n, m // 4, :] (correct K for query head m) # The matmul Q @ K^T = (16, 128) @ (128, 16) = (16, 16). M=16, uses tensor cores. # But the K depends on m, so for a 2D matmul, K is the same for all m. We need to construct K such that K^T[:, n] = K_data[n, m // 4, :] for the correct m. # Wait, in a 2D matmul Q @ K^T, K^T is (128, 16) where column n is K[n, :]. For the matmul to give the correct S[m, n] for all m, K[n, :] should be K_data[n, m // 4, :]. But this depends on m. # So K can't be constructed for a 2D matmul. # Hmm. OK the 4 separate matmuls is the way. # Let me code it. # Wait, I just realized something. In the 4 separate matmuls approach, the Q_sub for each kv head is (4, 128), padded to (16, 128). The padding is 12 zeros. The matmul (16, 128) @ (128, 16) = (16, 16). The 12 dummies give 0 output. # But the 4 real Q rows in Q_sub are different for each kv head. So the 4 matmuls are different. # In Triton, I can do this with tl.static_range. Let me code it. # Actually, an even better idea: have BLOCK_M=16 and the matmul be (16, 128) @ (128, 16), but the K is "constructed" for each m. Since K is (BLOCK_N, BLOCK_D), and for each n, K[n, :] = K_data[n, m // 4, :], I can construct K by loading K_pkv and then "expanding" it. # K_pkv has shape (BLOCK_N, BLOCK_KV, BLOCK_D) = (16, 4, 128). I want K of shape (BLOCK_N, BLOCK_D) = (16, 128) where for each (n, d), K[n, d] = K_pkv[n, m // 4, d]. # But m is the query head index, and K is the same for all m in a 2D matmul. So K can only be correct for one m at a time. # Hmm. OK the 2D matmul really doesn't work for GQA with multiple kv heads. # Let me just go with 4 separate padded matmuls. # Or, wait, here's another idea. What if I have BLOCK_M=16 and the matmul is (16, 128) @ (128, 16), and the K is constructed as follows: # - For each n, K[n, :] is "duplicated" 4 times (once per query head group). But the duplication is in the Q sense, not the K sense. # Hmm, this is getting confusing. Let me just code the 4 separate matmuls. # Actually, I realize there's a cleaner way. I can have BLOCK_M=16 and the matmul be (16, 128) @ (128, 16), and the K is (16, 128) where row n is "duplicated" 4 times to fill 16 rows... no, K is (BLOCK_N, BLOCK_D), not (BLOCK_M, BLOCK_D). # OK let me just code the 4 separate matmuls. # Here's the plan: # - Q is loaded as (BLOCK_M=16, BLOCK_D=128). 16 real query heads. # - K_pkv is loaded as (BLOCK_N=16, BLOCK_KV=4, BLOCK_D=128). # - V_pkv is loaded as (BLOCK_N=16, BLOCK_KV=4, BLOCK_D=128). # - For each kv in [0, 4): # - Q_sub = Q[kv*4:(kv+1)*4, :] # (4, 128) # - Q_padded = pad_to_16(Q_sub) # (16, 128) with 12 zeros # - K_sub = K_pkv[:, kv, :] # (16, 128) # - S_sub = tl.dot(Q_padded, tl.trans(K_sub)) # (16, 16) with 12 dummies # - S_sub_real = S_sub[:4, :] # (4, 16) # - Apply scale and mask # - Online softmax update (using a separate m_i_sub, l_i_sub, acc_sub for this kv) # - P_sub = exp(S_sub_real - m_new_sub) # - P_sub_padded = pad_to_16(P_sub) # - V_sub = V_pkv[:, kv, :] # (16, 128) # - O_sub = tl.dot(P_sub_padded, V_sub) # (16, 128) with 12 dummies # - O_sub_real = O_sub[:4, :] # - acc_sub = acc_sub * alpha_sub + O_sub_real # - Combine the 4 kv heads' acc into the full acc # This is 4 separate softmax updates. Each has its own m_i, l_i, acc. # After the loop, we have 4 (m_i, l_i, acc) tuples. We combine them: # - m_total = max of 4 m_i's # - alpha_kv = exp(m_i_kv - m_total) # - l_total = sum(alpha_kv * l_i_kv) # - acc_total = sum(alpha_kv * acc_kv) / l_total # This is correct because the softmax is a per-(batch, head) operation, and each kv head's group of 4 query heads has its own softmax. # Wait, but the softmax is per-query-head, not per-kv-head-group. Let me re-think. # For GQA, each query head has its own softmax over the tokens. The 4 query heads in a GQA group all use the same K (kv head), but their Q is different, so their softmax is different. # So the 4 query heads in a group have 4 different (m_i, l_i, acc) tuples. They're computed independently. # For the 4 separate matmuls approach, each matmul handles 1 query head's softmax. So 4 matmuls per kv head group, each with M=1 padded to 16. 16x more compute than minimum. # Hmm, that's even more compute. Let me re-think. # For the 4 separate matmuls with M=4 (one per kv head), each handles 4 query heads. The 4 query heads in a group have 4 different softmaxes. So we need 4 separate softmax updates per matmul. # This is getting complex. Let me think about it more carefully. # Actually, the 4 query heads in a GQA group can share the same softmax if we want. But that would be incorrect (they have different Q, so different scores, so different softmaxes). # So the correct approach is: each query head has its own softmax. For a kv head group of 4 query heads, we have 4 softmaxes. # In the original 3D batched matmul (4, 4, 16), the softmax is per-query-head (4 query heads per batch, 4 batches = 16 softmaxes total). This is correct. # In the 4 separate matmuls approach, each matmul (4, 128) @ (128, 16) handles 4 query heads, with 4 separate softmaxes. This is also correct. # The difference: in 3D batched, M=4 per batch (no tensor cores). In 4 separate, M=4 per matmul (no tensor cores). # To use tensor cores, we need M >= 16. We can pad M to 16, but then we have 12 dummies per matmul. The 12 dummies' softmaxes are uniform (since their Q is 0). Not used. # So the padded 4 separate matmuls: # - Each matmul: (16, 128) @ (128, 16) = (16, 16) with 12 dummies # - 4 matmuls per page # - Per-page compute: 4 * 16*16*128 = 128K ops (same as 3D batched padded) # - Per-page compute with tensor cores: 128K / 4K = 32 cycles/warp # For 64 pages, 2 matmuls per page = 128 matmuls. Per warp: 32 cycles * 128 = 4K cycles. At 2 GHz, 2us/warp. With 4 warps, 2us/program. # For 128 programs in 1 wave, 2us. Plus memory 0.2us. Total 2.2us. Much better than 34us. # OK let me code the 4 separate padded matmuls. # Hmm wait, actually I realize the 3D batched with padded M might be simpler. Let me try that. # For the 3D batched matmul (4, 16, 128) @ (4, 128, 16) = (4, 16, 16): # - Q_3d_padded[kv, g, :] = Q_3d[kv, g, :] for g in [0, 4), and 0 for g in [4, 16) # - K_3d[kv, n, :] = K_pkv[n, kv, :] # - Output: S_3d[kv, g, n] = sum_d Q_3d_padded[kv, g, d] * K_3d[kv, n, d] # The 12 dummies per batch give S_3d[kv, g, n] = 0 for g in [4, 16). The 4 real rows give the correct score. # To create Q_3d_padded, I can: # 1. Load Q as (16, 128). # 2. Reshape to (4, 4, 128) = Q_3d. # 3. Pad to (4, 16, 128) by adding 12 dummies per batch. # In Triton, I can create a (4, 16, 128) zero tensor and copy Q_3d into the first 4 positions. # Or, I can load Q as (64, 128) with 48 dummies (at offs_m = 16..63), but offs_m > H-1, so we'd be out of bounds. # Wait, offs_m is h_start + tl.arange(0, BLOCK_M_padded) where BLOCK_M_padded=64. h_start = hkv_start * G = 0 for hkv_blk=0. So offs_m = 0..63. With H=32, offs_m > 31 is out of bounds. # I can mask the load: `tl.load(q_ptrs, mask=offs_m[:, None] < H, other=0.0)`. This loads 0 for out-of-bounds rows. # So I can load Q as (64, 128) with 48 dummies, and reshape to (4, 16, 128) for the matmul. # Let me try this. # But wait, for hkv_blk=1, h_start=16, offs_m=16..79. With H=32, offs_m > 31 is out of bounds. So rows 32..79 are 0 (dummies). Rows 16..31 are real. Rows 0..15 are... wait, offs_m starts at h_start=16, so rows are 16..79. Rows 16..31 are real (for hkv_blk=1, heads 16..31). Rows 32..79 are dummies (0). # Hmm, this doesn't match the 3D batched structure. Let me re-think. # For hkv_blk=1 (hkv_start=4), the 4 kv heads are 4, 5, 6, 7. The 16 query heads are 16..31. So h_start = 4 * 4 = 16. The 16 query heads for this program are 16..31, all real. # For the 3D batched matmul, Q_3d[kv, g, :] = Q[kv*4 + g, :] for kv in [0, 4), g in [0, 4). So Q_3d[0, 0, :] = Q[0, :], Q_3d[0, 1, :] = Q[1, :], ..., Q_3d[3, 3, :] = Q[15, :]. All real (for hkv_blk=0, heads 0..15). # For hkv_blk=1, Q[0, :] = head 16, ..., Q[15, :] = head 31. All real. # So for both hkv_blks, the 16 query heads are all real. No dummies in the Q. # To pad to (4, 16, 128), I need 12 dummies per batch. Total 48 dummies. But I have 0 dummies in the Q (all 16 are real). # So I'd need to "create" 48 dummies. I can do this by loading Q at offs_m = h_start + tl.arange(0, 64), but offs_m > 31 is out of bounds. Use mask to load 0. # In Triton: # offs_m_padded = h_start + tl.arange(0, 64) # q_padded = tl.load(q_ptrs_padded, mask=offs_m_padded[:, None] < H, other=0.0) # Reshape to (4, 16, 128) # The first 16 rows are real Q, the next 48 are 0. Reshape to (4, 16, 128) gives: # Q_3d_padded[0, 0, :] = Q[0, :] (real) # Q_3d_padded[0, 1, :] = Q[1, :] (real) # ... # Q_3d_padded[0, 3, :] = Q[3, :] (real) # Q_3d_padded[0, 4, :] = 0 (dummy) # ... # Q_3d_padded[0, 15, :] = 0 (dummy) # Q_3d_padded[1, 0, :] = Q[4, :] (real) # ... # Q_3d_padded[3, 15, :] = 0 (dummy) # So the first 4 rows of each batch are real, the next 12 are 0. This is what we want for padded matmul. # Let me code this. # Actually wait, I realize this is essentially the same as the 4 separate padded matmuls, just expressed as a 3D batched matmul. The compute is the same. # Let me code it and test. # Hmm, but I realize there might be an issue with the reshape. (64, 128) reshape to (4, 16, 128): the first 16 rows become (4, 4, 128) = (BLOCK_KV, G, BLOCK_D). Wait, that's not (4, 16, 128). # Let me re-think. I want (BLOCK_KV=4, G_PADDED=16, BLOCK_D=128). Total elements: 4 * 16 * 128 = 8192. The Q is (64, 128) = 64 * 128 = 8192. So reshape is possible. # Reshape (64, 128) to (4, 16, 128): row 0 of new = old row 0, row 1 = old row 1, ..., row 63 = old row 63. So new[kv*16 + g, d] = old[kv*16 + g, d] for kv in [0, 4), g in [0, 16). # Wait, that's not right. Reshape (64, 128) to (4, 16, 128) in row-major: new[kv, g, d] = old[kv*16 + g, d]? Let me think. # The total number of elements is 64 * 128 = 8192. Reshape to (4, 16, 128) = 4 * 16 * 128 = 8192. OK. # In row-major order, new[kv, g, d] = old[kv * 16 * 128 + g * 128 + d] = old[(kv * 16 + g) * 128 + d]. So new[kv, g, d] = old[(kv*16 + g), d] for the (64, 128) -> (4, 16, 128) reshape. # Hmm wait, reshape (64, 128) to (4, 16, 128): the first dim of the new shape is 4, second is 16, third is 128. The total is 4*16*128 = 8192. # In row-major, the new array is laid out as new[0, 0, 0], new[0, 0, 1], ..., new[0, 0, 127], new[0, 1, 0], ..., new[3, 15, 127]. This corresponds to old[0, 0], old[0, 1], ..., old[0, 127], old[1, 0], ..., old[63, 127]. # So new[kv, g, d] = old[kv * 16 + g, d]. Yes. # So if I load Q as (64, 128) where rows 0..15 are real (hkv_blk=0, heads 0..15) and rows 16..63 are dummies (0), then reshape to (4, 16, 128): # new[0, 0, :] = old[0, :] = Q[0, :] (real) # new[0, 1, :] = old[1, :] = Q[1, :] (real) # ... # new[0, 3, :] = old[3, :] = Q[3, :] (real) # new[0, 4, :] = old[4, :] = Q[4, :] (real) # ... # new[0, 15, :] = old[15, :] = Q[15, :] (real) # new[1, 0, :] = old[16, :] = 0 (dummy) # ... # new[3, 15, :] = old[63, :] = 0 (dummy) # So new[kv, g, :] = Q[kv*16 + g, :] if kv*16 + g < 16, else 0. # For kv=0, g in [0, 16): kv*16 + g in [0, 16). All real. # For kv=1, g in [0, 16): kv*16 + g in [16, 32). All > 16, so all dummies. # For kv=2, 3: all dummies. # Hmm, that's not what I want. I want the first 4 rows of each batch to be real, and the next 12 to be dummies. # So I need: new[kv, g, :] = Q[kv*4 + g, :] if g < 4, else 0. This is a different mapping. # For this, I'd need to construct the (4, 16, 128) tensor explicitly, not via reshape. # In Triton, I can do: # Q_3d = tl.reshape(Q, (BLOCK_KV, G, BLOCK_D)) # (4, 4, 128) # Q_3d_padded = ... pad to (4, 16, 128) ... # The padding: I can use tl.cat or tl.where. # Or, I can use a different approach: have a separate "mask" for the M dim and use it in the matmul. # Actually, in Triton, there's no direct way to do "masked matmul" where some rows of Q are 0. The matmul just multiplies. # Hmm. Let me think. # OK here's another approach: use 4 separate 2D matmuls, each with M=4 padded to 16. This is essentially the same as 3D batched with padded M, but expressed differently. # In Triton: # Q is (16, 128) = (BLOCK_M, BLOCK_D) # K_pkv is (16, 4, 128) = (BLOCK_N, BLOCK_KV, BLOCK_D) # V_pkv is (16, 4, 128) = (BLOCK_N, BLOCK_KV, BLOCK_D) # For each kv in [0, 4): # Q_sub = Q[kv*4:(kv+1)*4, :] # (4, 128) # # Pad to (16, 128) with 12 zeros # Q_padded = tl.cat(Q_sub, tl.zeros((12, 128))) # (16, 128) # K_sub = K_pkv[:, kv, :] # (16, 128) # S_sub = tl.dot(Q_padded, tl.trans(K_sub)) # (16, 16) # S_sub_real = S_sub[:4, :] # take the first 4 rows # ... # The slicing in Triton: I think `Q_sub = Q[kv*4:(kv+1)*4, :]` doesn't work directly. I need to use `tl.split` or similar. # Actually, in Triton, slicing a tensor at runtime is not supported. I can only do constant slicing. # So I need to use static_range and index with constants. # Let me code this with tl.static_range and constant indices. # Q is (16, 128). For kv=0, Q_sub = Q[0:4, :]. For kv=1, Q_sub = Q[4:8, :]. Etc. # In Triton, I can do this with: # for kv in tl.static_range(BLOCK_KV): # if kv == 0: # Q_sub = Q[0:4, :] # elif kv == 1: # Q_sub = Q[4:8, :] # ... # But this is ugly. Let me use a different approach: use 3D batched with padding via broadcasting. # Actually, here's a clean way: use the 3D batched matmul with the G dim padded via broadcasting. # Q_3d = tl.reshape(Q, (BLOCK_KV, G, BLOCK_D)) # (4, 4, 128) # # Pad to (4, 16, 128) by broadcasting G=4 to G=16 with 12 dummies # # Hmm, broadcasting doesn't add dummies. # OK let me just use a different structure. Let me use a 2D matmul with M=16, but have the K be (BLOCK_N, BLOCK_D) where for each n, K[n, :] is the K for "query head 0" (i.e., K_pkv[n, 0, :]). # This gives the correct score for query heads 0..3 (which use kv head 0). For query heads 4..15, the score is wrong. # To fix this, I'd need to do 4 separate 2D matmuls. Each handles 4 query heads with the correct K. # OK let me just code the 4 separate matmuls with M=4 padded to 16. # In Triton, the cleanest way might be: # 1. Load Q as (16, 128). # 2. Load K_pkv as (16, 4, 128). # 3. Permute K to (4, 16, 128) = K_kvn. # 4. For each kv, do a 2D matmul. # For the padding, I can construct a (4, 16, 128) tensor where the first 4 rows of each batch are the real Q and the next 12 are 0. # In Triton: # Q_kvg = tl.reshape(Q, (BLOCK_KV, G, BLOCK_D)) # (4, 4, 128) # # Pad to (4, 16, 128) by adding 12 dummies # Q_kvg_padded = ... # need to construct # Hmm. Let me just use a different approach: have the matmul be (BLOCK_M, BLOCK_D) @ (BLOCK_D, BLOCK_N) with BLOCK_M=16, and the K be (BLOCK_D, BLOCK_N). For the K, I need to "gather" the correct K for each query head. # Wait, I keep going in circles. Let me just code the 4 separate matmuls. # Actually, the simplest fix: increase the G dim by using a larger group size in the matmul. # For BLOCK_KV=4 and G_padded=16, the Q has 4 * 16 = 64 query heads. But we only have 16. So we need 48 dummies. # I can create the dummies by loading Q at offs_m > 15 with mask. The mask returns 0 for out-of-bounds. # Let me code this. # Q_padded is (64, 128). Load with mask. # Reshape to (4, 16, 128). # The first 4 rows of each batch are real, the next 12 are 0. # Wait, let me re-derive the reshape. (64, 128) reshape to (4, 16, 128): # new[kv, g, d] = old[kv * 16 + g, d] # For kv=0: new[0, g, d] = old[g, d] for g in [0, 16). old[0..15] are real (Q[0..15]). So new[0, 0..15, :] = Q[0..15, :]. All real. # For kv=1: new[1, g, d] = old[16 + g, d] for g in [0, 16). old[16..31] are 0 (dummies). So new[1, 0..15, :] = 0. All dummies. # For kv=2, 3: all dummies. # Hmm, this gives only 1 batch with real Q and 3 batches with dummies. Not what I want. # I want each batch to have the first 4 rows real and the next 12 dummies. So: # new[0, 0..3, :] = Q[0..3, :], new[0, 4..15, :] = 0 # new[1, 0..3, :] = Q[4..7, :], new[1, 4..15, :] = 0 # new[2, 0..3, :] = Q[8..11, :], new[2, 4..15, :] = 0 # new[3, 0..3, :] = Q[12..15, :], new[3, 4..15, :] = 0 # So new[kv, g, :] = Q[kv*4 + g, :] if g < 4, else 0. # This is a strided gather. I can construct it explicitly. # In Triton, I can do: # Q_3d = tl.reshape(Q, (BLOCK_KV, G, BLOCK_D)) # (4, 4, 128) # # Q_3d[kv, g, :] = Q[kv*4 + g, :] # # Now I need to pad G from 4 to 16. The padded version has 12 dummies per batch. # To create the padded version, I can use `tl.cat` along the G dim. But `tl.cat` requires the tensors to be the same shape along non-cat dims. # Alternatively, I can use a mask and where: # Q_3d_padded = tl.zeros((4, 16, 128)) # Q_3d_padded[:, :4, :] = Q_3d # but Triton doesn't support this assignment # Hmm. Let me use a different approach: do the 4 separate matmuls without 3D batched. # In Triton: # Q is (16, 128) = (BLOCK_M, BLOCK_D) # K_pkv is (16, 4, 128) = (BLOCK_N, BLOCK_KV, BLOCK_D) # V_pkv is (16, 4, 128) = (BLOCK_N, BLOCK_KV, BLOCK_D) # For each kv in [0, 4): # Q_sub = Q[kv*4:(kv+1)*4, :] # Need to slice, which Triton doesn't support at runtime # ... # I can use tl.static_range and constant indices. Let me do that. # But actually, the simplest is to use the 3D batched matmul and just accept that M=4 doesn't use tensor cores. The performance will be limited by FMA. # Hmm. Let me think about whether there's a way to make the 3D batched matmul use tensor cores. # Actually, I recall that in Triton, you can sometimes get tensor cores for small M by using `input_precision` or other options. Let me try. # Or, I can use a 2D matmul with the K "broadcasted" in a specific way. Let me think. # For 2D matmul (M, K) @ (K, N): # - M = BLOCK_M = 16 # - K = BLOCK_D = 128 # - N = BLOCK_N = 16 # Q is (16, 128). K is (128, 16) = K^T of (16, 128). # For GQA, the K for query head m is K_data[block, m // 4, :]. # If I construct K_combined of shape (128, 16) where K_combined[:, n] = K_data[block, m // 4, :] for the "correct" m... but this depends on m. # Hmm. What if I construct K_combined as (128, 16) where for each (d, n), K_combined[d, n] = K_data[block, 0, d, n]? Then the matmul is correct for query heads 0..3 (which use kv head 0). For other query heads, it's wrong. # To make it correct for all 16 query heads, I'd need 16 different K's, one per query head. But a 2D matmul has only one K. # So 2D matmul doesn't work for GQA with multiple kv heads. # OK let me just go with the 3D batched matmul (current) and accept the FMA performance. The result is 34us per kernel for shape 0, which gives 42% peak. Not great but not terrible. # Actually wait, let me look at the ncu output more carefully. The "sm__pipe_tensor_op_hmma_cycles_active" is N/A. Let me check if it's because no tensor core activity or because the metric isn't applicable. # For Blackwell, tensor cores are accessed via different instructions. The metric name might be different. # Let me try a different metric: sm__inst_executed_pipe_tensor. If it's 0, no tensor cores. # Hmm, I don't have the metric in my output. Let me just trust the result. # Actually, let me also try `input_precision='ieee'` to see if it changes anything. # Or, let me try the 4 separate padded matmuls. It's more code but should be faster. # OK let me just try the 4 separate matmuls with M=4 padded to 16. Here's the plan: # In the kernel: # Q is (BLOCK_M=16, BLOCK_D=128). 16 real query heads. # K_pkv is (BLOCK_N, BLOCK_KV=4, BLOCK_D=128). K for 4 kv heads. # V_pkv is (BLOCK_N, BLOCK_KV=4, BLOCK_D=128). V for 4 kv heads. # For the matmul, I want to do 4 separate matmuls, each (4, 128) @ (128, 16) = (4, 16). With M=4 padded to 16, it's (16, 128) @ (128, 16) = (16, 16). # In Triton, I can use tl.static_range and do 4 iterations. # For each kv in [0, 4): # Q_sub = ... # (4, 128) - the 4 query heads for this kv head # Q_padded = ... # (16, 128) with 12 zeros # K_sub = K_pkv[:, kv, :] # (16, 128) # S_sub = tl.dot(Q_padded, tl.trans(K_sub)) # (16, 16) with 12 dummies # S_sub_real = S_sub[:4, :] # ... # The slicing in Triton: I think I can use `tl.split` or just construct with offsets. # Actually, in Triton, I can use `tl.reshape` and `tl.split`. But for arbitrary slicing, I might need to use `tl.where` and masking. # Hmm, let me look at the Triton API for slicing. # Actually, Triton supports constant slicing via `tl.split` and `tl.reshape`. For example, `tl.reshape(Q, (BLOCK_KV, G, BLOCK_D))` splits the first dim into (BLOCK_KV, G). # For BLOCK_KV=4, G=4, Q of shape (16, 128) reshape to (4, 4, 128). Then Q_3d[kv, g, :] = Q[kv*4 + g, :]. Correct. # For padding G from 4 to 16, I need to add 12 dummies per batch. I can do this with `tl.cat` or by creating a zero tensor and using `tl.where`. # Or, I can avoid the padding by using 4 separate 2D matmuls, each with M=4. M=4 doesn't use tensor cores, but it's simpler. # Hmm. Let me just try the simple approach: 4 separate 2D matmuls with M=4. If it's still slow, I'll add padding. # Actually, I already have the 3D batched matmul, which is equivalent to 4 separate 2D matmuls. The performance is the same. # So the issue is fundamental: M=4 doesn't use tensor cores. To use tensor cores, I need M >= 16. # Let me try the padded approach. Here's the plan: # 1. Load Q as (16, 128). # 2. Reshape to (4, 4, 128) = Q_3d. # 3. Create Q_3d_padded of shape (4, 16, 128) by padding Q_3d along the G dim. # 4. Load K_pkv as (16, 4, 128). # 5. Permute to K_kvn of shape (4, 16, 128). # 6. Matmul: Q_3d_padded @ K_kvn^T = (4, 16, 16). M=16, uses tensor cores. # 7. Take the first 4 rows of each batch: S_3d_real = S_3d_padded[:, :4, :]. Shape (4, 4, 16). # 8. Reshape to (16, 16) for the softmax. # Step 3 (padding) is the tricky part. In Triton, I can do: # - Use `tl.cat` to concatenate Q_3d with a zero tensor along the G dim. # - Use `tl.where` with a mask. # - Use `tl.broadcast_to` and `tl.where`. # Let me try `tl.broadcast_to`: # Q_3d is (4, 4, 128). Broadcast to (4, 16, 128) by adding a dim of size 16/4 = 4? No, that doesn't work. # OK let me use a different approach. I'll create the padded Q directly by loading with extra rows. # Q_padded of shape (64, 128). Rows 0..15 are real Q. Rows 16..63 are 0. # Reshape to (4, 16, 128) = Q_3d_padded. # new[kv, g, d] = old[kv*16 + g, d] # For kv=0: new[0, g, :] = old[g, :] for g in [0, 16). old[0..15] = Q[0..15]. So new[0, 0..15, :] = Q[0..15]. All real. # For kv=1: new[1, g, :] = old[16+g, :] for g in [0, 16). old[16..31] = 0. So new[1, 0..15, :] = 0. All dummies. # For kv=2, 3: all dummies. # Hmm, this gives only 1 batch with real Q. Not what I want. # I want: new[kv, g, :] = Q[kv*4 + g, :] if g < 4, else 0. # So for kv=0: new[0, 0..3, :] = Q[0..3], new[0, 4..15, :] = 0 # For kv=1: new[1, 0..3, :] = Q[4..7], new[1, 4..15, :] = 0 # ... # This is a strided gather. Not a simple reshape. # I can construct it explicitly: # Q_3d_padded = tl.zeros((BLOCK_KV, G_PADDED, BLOCK_D)) # for kv in range(BLOCK_KV): # for g in range(G): # Q_3d_padded[kv, g, :] = Q[kv*G + g, :] # In Triton, this is a series of assignments. Might work. # Or, I can use a 2D approach: have the matmul be (16, 128) @ (128, 16), and the K be (128, 16) where K[:, n] is the K for the "correct" query head for token n. # But as I discussed, K can't depend on m in a 2D matmul. # OK let me just try the 4 separate matmuls with M=4 padded to 16. I'll use tl.static_range and constant indices. # In Triton: # Q is (16, 128) = (BLOCK_M, BLOCK_D) # K_pkv is (16, 4, 128) = (BLOCK_N, BLOCK_KV, BLOCK_D) # V_pkv is (16, 4, 128) = (BLOCK_N, BLOCK_KV, BLOCK_D) # For the matmul, I'll do 4 iterations. Each iteration handles 1 kv head. # In each iteration: # - Q_sub: (4, 128) - the 4 query heads for this kv head # - Q_padded: (16, 128) with 12 zeros # - K_sub: (16, 128) - the K for this kv head # - S_sub: (16, 16) with 12 dummies # - S_sub_real: (4, 16) - the first 4 rows # - Apply mask, softmax # - P_sub: (4, 16) # - P_padded: (16, 16) with 12 zeros # - V_sub: (16, 128) # - O_sub: (16, 128) with 12 dummies # - O_sub_real: (4, 128) - the first 4 rows # - Update acc for this kv head # After the loop, combine the 4 kv heads' acc. # Hmm, the softmax state (m_i, l_i) is per-query-head, not per-kv-head. So each of the 4 query heads in a kv head group has its own softmax state. # In the loop, for each kv, I process 4 query heads. Each has its own (m_i, l_i, acc). So I have 16 separate softmax states, one per query head. # This is correct. # Let me code this. # Actually, I realize I can use the 3D batched matmul with the G dim padded. The padding can be done by loading Q with extra rows (dummies) and reshaping. # Let me try: load Q as (16, 128). Create Q_3d_padded of shape (4, 16, 128) by: # Q_3d_padded[kv, g, :] = Q[kv*4 + g, :] if g < 4, else 0 # In Triton, I can do this with a where and broadcast: # Q_3d = tl.reshape(Q, (BLOCK_KV, G, BLOCK_D)) # (4, 4, 128) # # Pad G to 16 by adding 12 dummies # # Create a (4, 16, 128) tensor # # Use tl.cat or manual construction # Hmm, let me just try a different approach. Let me have the matmul be 2D (BLOCK_M, BLOCK_D) @ (BLOCK_D, BLOCK_N) with BLOCK_M=16, and the K be (BLOCK_D, BLOCK_N). For the K, I'll construct it as follows: # K_data has shape (BLOCK_N, BLOCK_KV, BLOCK_D). For the matmul, K is (BLOCK_D, BLOCK_N). I want K[:, n] = K_data[n, m // 4, :] for each m. But K can't depend on m. # So K[:, n] is the same for all m. I can only pick one m. Let me pick m=0 (or any specific m). Then the matmul is correct for query heads 0..3 and wrong for 4..15. # This is the original 2D matmul approach, which is wrong for GQA. # OK so the only correct approaches are: # 1. 3D batched matmul (current, FMA) # 2. 4 separate 2D matmuls (equivalent to 3D batched, FMA) # 3. 4 separate 2D matmuls with M=4 padded to 16 (tensor cores, 4x compute) # 4. 3D batched matmul with G padded to 16 (equivalent to 3, tensor cores) # Let me try approach 4: 3D batched matmul with G padded to 16. # To pad G from 4 to 16, I need to add 12 dummies per batch. The dummies' Q is 0. # In Triton, I can construct Q_3d_padded as follows: # Q_3d = tl.reshape(Q, (BLOCK_KV, G, BLOCK_D)) # (4, 4, 128) # # Create a mask for the G dim: mask[g] = (g < G) # g_offs = tl.arange(0, G_PADDED) # 0..15 # g_mask = g_offs < G # True for 0..3, False for 4..15 # # Broadcast Q_3d to (4, 16, 128) and mask # Q_3d_broadcast = tl.broadcast_to(Q_3d[:, :, None, :], (BLOCK_KV, G, G_PADDED, BLOCK_D)) # wrong shape # Hmm, this is getting complex. Let me just try a different approach. # Let me use the 4 separate matmuls with M=4 padded to 16. In Triton, I'll use tl.static_range and construct the padded Q for each kv head. # For each kv in [0, 4): # Q_sub = Q[kv*4:(kv+1)*4, :] # (4, 128) # # Pad to (16, 128) with 12 zeros # # In Triton, I can use tl.cat or construct manually # Q_padded = tl.cat(Q_sub, tl.zeros((12, 128), dtype=tl.bfloat16), axis=0) # might not work # K_sub = K_pkv[:, kv, :] # (16, 128) # S_sub = tl.dot(Q_padded, tl.trans(K_sub)) # (16, 16) with 12 dummies # S_sub_real = S_sub[:4, :] # need slicing # Hmm, Triton might not support arbitrary slicing. Let me check. # Actually, Triton supports `tl.split` for splitting a tensor along an axis. For example, `tl.split(Q_reshape)` splits the first dim into two. # But for arbitrary slicing like `Q[0:4, :]`, I'm not sure. # Let me just try and see. # Actually, you know what, let me just try a completely different approach. Let me use a 2D matmul with M=64 (BLOCK_M=64) where 48 rows are dummies. The matmul is (64, 128) @ (128, 16) = (64, 16). M=64, uses tensor cores. # For the K, I need K to be the correct K for each query head. Specifically, for query head m, the K is K_data[block, m // 4, :]. For the 48 dummies (m=16..63), the K can be anything (since Q is 0). # So I construct K_per_head of shape (BLOCK_N, BLOCK_M=64, BLOCK_D) where K_per_head[n, m, :] = K_data[block, m // 4, :] for m in [0, 16), and 0 for m in [16, 64). But K_per_head depends on m, so it's a 3D tensor. # For the 2D matmul, K is (BLOCK_N, BLOCK_D). I need to "flatten" the m dim into the n dim somehow. # Hmm, this doesn't work. # OK let me just go with the 3D batched matmul (current) and accept the FMA performance. The kernel is correct and achieves 42% peak for shape 0. Not great but OK. # Actually wait, let me try one more thing. What if I use BLOCK_M=16 and the 3D batched matmul (4, 4, 16), but with a different layout that Triton can use tensor cores for? # For example, what if I transpose the matmul to (16, 4) @ (4, 128) = (16, 128)? This is a 2D matmul with M=16, K=4. K=4 is too small for tensor cores. # Or (4, 16) @ (16, 128) = (4, 128)? M=4, no tensor cores. # Hmm. K=4 is the bottleneck if I transpose. # What if I do (4, 128) @ (128, 16) = (4, 16) for each of 4 batches? This is what the 3D batched does, and M=4 is the issue. # OK I'll just go with the current approach. Let me see if there are other optimizations. # Actually, let me try one more thing: use `tl.dot` with `out_dtype=tl.float32` and see if it changes the tensor core behavior. # Or, let me try increasing the BLOCK_N to make the matmul bigger. For BLOCK_N=64, the matmul is (4, 128) @ (128, 64) = (4, 64). M=4, still no tensor cores. But the matmul is 4x larger, so more compute per call. # Hmm, doesn't help with the M=4 issue. # Let me try BLOCK_N=64 and see if it helps overall. The matmul compute is 4x more, but the loop has 4x fewer iters. Total compute is the same. # Actually, let me just focus on other optimizations. The current 0.48 peak is OK. Let me see if I can improve the low-parallelism shapes. # For shape 0 and 4, the bottleneck is low parallelism. Let me try: # 1. Use BLOCK_KV=2 (BLOCK_M=8) to get 2x more programs. But M=8 is still small for tensor cores. # 2. Use CHUNK=64 to get 2x more splits. I tried this, marginal improvement. # 3. Combine: use BLOCK_KV=2 and CHUNK=64. # For shape 0 with BLOCK_KV=2, BLOCK_M=8, HKV_BLKS=4, SPLIT_K=16 (CHUNK=64): total = 8*4*16 = 512 programs. 2.72 waves. # Let me try this. # Actually, BLOCK_M=8 means the 3D batched matmul is (2, 4, 16) with M=4. Same M=4 issue. FMA. # So it won't help with the FMA bottleneck, but it might help with parallelism. # Hmm. Let me think about this differently. The FMA throughput is ~64 ops/cycle/warp. The tensor core throughput is ~4K ops/cycle/warp. 64x speedup with tensor cores. # For our matmul (4, 128) @ (128, 16) = (4, 16), the compute is 4*16*128 = 8K MACs. With FMA: 8K/64 = 128 cycles/warp. With tensor cores: 8K/4K = 2 cycles/warp. # So tensor cores would be 64x faster for this matmul. # To use tensor cores, M must be >= 16. With BLOCK_M=16 (padded from G=4), the matmul is (16, 128) @ (128, 16) = 32K MACs. With FMA: 512 cycles/warp. With tensor cores: 8 cycles/warp. 64x speedup. # So padding M to 16 gives 4x more compute but 64x faster per op. Net: 16x faster. # This is worth doing. Let me code the 4 separate matmuls with M=4 padded to 16. # In Triton, the cleanest way: # - Load Q as (16, 128). # - Reshape to (4, 4, 128) = Q_3d. # - Create Q_3d_padded of shape (4, 16, 128) by padding along G. # - Load K_pkv as (16, 4, 128). # - Permute to K_kvn of shape (4, 16, 128). # - Matmul: Q_3d_padded @ K_kvn^T = (4, 16, 16). M=16, tensor cores. # - Take first 4 rows of each batch for the softmax. # For the padding, I'll use a workaround: load Q with extra rows (dummies) and reshape. # Q_dummy is (4*16, 128) = (64, 128). The first 16 rows are real Q, the next 48 are 0. # Reshape to (4, 16, 128): # new[kv, g, d] = old[kv*16 + g, d] # For kv=0: new[0, g, :] = old[g, :] for g in [0, 16). old[0..15] = Q[0..15]. So new[0, 0..15, :] = Q[0..15]. All real. # For kv=1: new[1, g, :] = old[16+g, :] for g in [0, 16). old[16..31] = 0. All dummies. # Hmm, this gives only 1 batch with real Q. Not what I want. # I want each batch to have the first 4 rows real. So the layout should be: # new[kv, g, :] = Q[kv*4 + g, :] if g < 4, else 0 # This is a different mapping. I can construct it explicitly. # In Triton, I can use `tl.split` to split Q into (4, 4, 128) and then pad. # Or, I can use a 2D matmul with the K "constructed" for each m. But as discussed, this doesn't work for GQA. # OK let me just try a different approach. Let me use the 4 separate matmuls with tl.static_range. # In Triton: # Q is (16, 128). K_pkv is (16, 4, 128). V_pkv is (16, 4, 128). # I'll do 4 separate matmuls. For each kv in [0, 4): # Q_sub = ... # (4, 128) - the 4 query heads for this kv head # # I need to extract Q[kv*4:(kv+1)*4, :] # # In Triton, I can use tl.split # Actually, in Triton, I can reshape (16, 128) to (4, 4, 128) and then use tl.split or indexing to get the kv-th slice. # But tl.split splits along the first dim. For (4, 4, 128), tl.split would give 2 tensors of shape (4, 4, 128) each (splitting the first dim of size 4 into 2 of size 2). Not what I want. # Hmm. Let me look at Triton's API for indexing. # Actually, in Triton, you can index a tensor with constants at compile time. For example, `Q[0:4, :]` if the indices are constexpr. # With tl.static_range, I can iterate and use constant indices: # for kv in tl.static_range(BLOCK_KV): # # Q_sub = Q[kv*G:(kv+1)*G, :] # # I need to construct this # Hmm, I'm not sure if Triton supports this directly. Let me just try. # Actually, the simplest workaround: use the 3D batched matmul with M=G=4 (no padding). The FMA is used. The performance is limited but correct. # For our shapes, the FMA performance gives ~40-75% peak. The shapes with high parallelism (1, 3) get 57-75%. The shapes with low parallelism (0, 4) get 27-42%. # The 0.48 geomean is OK. Let me see if I can improve the low-parallelism shapes. # For shape 0, the issue is 0.68 waves. If I could get 1 wave (188 programs), the time would be similar (since each program does the same work), but the SM utilization would be 100% in the first wave. # Hmm, actually, with 0.68 waves, the time is determined by the wave time. Each program takes 44us. With 1 wave (188 programs), the time is still 44us (since each program does the same work). # Wait, that's not right. With more programs, we have more parallelism, but each program does the same work. The wave time is the same. # So the time is determined by the per-program work, not the number of programs (as long as we have at least 1 wave). # For shape 0 at 44us, the per-program work takes 44us. To reduce this, we need to reduce the per-program work. # Per-program work: 64 pages * (16KB K load + 16KB V load + matmul + softmax). # K/V load: 32KB per page. 64 pages = 2MB per program. # At 1.8 TB/s, 2MB = 1.1us. At L2 bandwidth (~5 TB/s), 2MB = 0.4us. # So the memory load is 0.4-1.1us per program. But we're at 44us. 40-100x slower. # The matmul compute (with FMA) is the bottleneck. As I calculated, ~16us per program with FMA. # 16us is closer to 44us but still 3x off. There might be other overheads. # OK let me just try to use tensor cores by padding M. Let me code it. # I'll use the approach: 3D batched matmul (4, 16, 128) @ (4, 128, 16) = (4, 16, 16) with 12 dummies per batch. # To create Q_3d_padded (4, 16, 128), I'll use a workaround: # - Create a (4, 16, 128) zero tensor. # - Copy Q_3d (4, 4, 128) into the first 4 positions of each batch. # In Triton, I can do this with `tl.where`: # g_offs = tl.arange(0, G_PADDED) # 0..15 # g_mask = g_offs < G # True for 0..3 # # Broadcast Q_3d to (4, 16, 128) and mask # Q_3d_padded = tl.where(g_mask[None, :, None], Q_3d_broadcast, 0.0) # where Q_3d_broadcast is Q_3d broadcasted to (4, 16, 128) # But Q_3d is (4, 4, 128). To broadcast to (4, 16, 128), I need to add a dim. Specifically, Q_3d[:, None, :, :] is (4, 1, 4, 128). Broadcast to (4, 16, 4, 128) by adding a dim of 16. Then take the first 4 along the new dim? No, that doesn't work. # Hmm. Let me think. # I want Q_3d_padded[kv, g, :] = Q_3d[kv, g, :] if g < 4, else 0. # This is equivalent to: Q_3d_padded[kv, g, :] = Q_3d[kv, min(g, 3), :]. # I can construct this with: # g_idx = tl.minimum(tl.arange(0, G_PADDED), G - 1) # 0, 1, 2, 3, 3, 3, ..., 3 # Q_3d_padded = Q_3d[:, g_idx, :] # But Triton doesn't support arbitrary indexing like this. # Alternative: use `tl.gather` if available. Let me check. # Triton has `tl.gather` in recent versions. Let me check if it's available. # Actually, a simpler approach: just use a 2D matmul with M=16 and the K be (BLOCK_D, BLOCK_N) where for each (d, n), K[d, n] is the K for "some" query head. Since all 16 query heads use different K's, the matmul is correct for only 1 query head at a time. # But if I do 4 separate 2D matmuls, each with M=4 (no padding), I get the correct result for all 16 query heads. The FMA is used, but the code is simple. # I've already tried this (it's the 3D batched). The performance is limited by FMA. # To use tensor cores, I need M >= 16. With M=4, no tensor cores. With M=16 (padded), tensor cores but 4x more compute. # Net: 16x speedup with padding. # Let me just code the padded version. I'll use a workaround: have the matmul be (BLOCK_M, BLOCK_D) @ (BLOCK_D, BLOCK_N) with BLOCK_M=16, and the K be (BLOCK_D, BLOCK_N) where K is "constructed" for each query head group. # Wait, K is the same for all m in a 2D matmul. So I can only construct K for one m at a time. # Hmm. OK let me just try the 4 separate matmuls with M=4 padded to 16. In Triton, I'll use tl.static_range and manual indexing. # Actually, I just realized: I can have BLOCK_M=16 and the matmul be (16, 128) @ (128, 16), and the K be (128, 16) where K[:, n] is "duplicated" for the 4 query head groups. Specifically, K[:, n] = K_data[n, 0, :] (the K for kv head 0). This gives the correct score for query heads 0..3. # For query heads 4..7, the score is wrong (should use K_data[n, 1, :]). To fix, I'd need to do 4 separate matmuls. # OK let me just try the 4 separate matmuls approach. Here's the plan: # In the kernel, for each kv head in [0, BLOCK_KV): # Extract Q_sub = Q[kv*G:(kv+1)*G, :] # (G, BLOCK_D) # Pad Q_sub to (16, BLOCK_D) with zeros # Extract K_sub = K_pkv[:, kv, :] # (BLOCK_N, BLOCK_D) # Matmul: Q_padded @ K_sub^T = (16, BLOCK_N) with 12 dummies # Take first G rows: S_sub = S[:G, :] # Softmax over S_sub # Extract V_sub = V_pkv[:, kv, :] # Matmul: P_padded @ V_sub = (16, BLOCK_D) with 12 dummies # Take first G rows: O_sub = O[:G, :] # Update acc for this kv head # In Triton, the slicing Q[kv*G:(kv+1)*G, :] requires constant indices. With tl.static_range, the kv is a constant at compile time, so the indices are constants. # Let me code this. # Wait, I realize that in Triton, the slicing might not be directly supported. Let me check. # Looking at Triton docs, `tl.split` is for splitting a tensor along an axis. For example, `tl.split(Q, 2)` splits Q into 2 halves along the first dim. # For Q of shape (16, 128), I want to split into 4 tensors of shape (4, 128) each. I can use `tl.split(Q, 4)` if it supports splitting into 4. # Actually, `tl.split` might only support splitting into 2. Let me check. # From Triton docs: `tl.split(x)` splits a tensor into two equal-sized tensors along the last dim. Hmm, that doesn't help. # OK let me just use a different approach. I'll have the kernel do the 3D batched matmul, but with the G dim padded. I'll construct the padded Q explicitly. # Here's the plan: # 1. Load Q as (16, 128). # 2. Reshape to (4, 4, 128) = Q_3d. # 3. Create Q_3d_padded of shape (4, 16, 128) by padding along the G dim. # - In Triton, I can use `tl.broadcast_to` and `tl.where`: # g_offs = tl.arange(0, G_PADDED) # 0..15 # g_mask = (g_offs < G) # True for 0..3 # # Q_3d is (4, 4, 128). I want to broadcast to (4, 16, 128) by mapping g in [0, 4) to Q_3d[:, g, :] and g in [4, 16) to 0. # # Hmm, this requires indexing Q_3d with g_offs, which is a tensor index. Triton might not support this. # OK let me try yet another approach. Let me have the matmul be a flat 2D matmul (BLOCK_M=16, BLOCK_D=128) @ (BLOCK_D=128, BLOCK_N=16), and the K be (BLOCK_D=128, BLOCK_N=16). For the K, I'll construct it as follows: # K_data has shape (BLOCK_N, BLOCK_KV, BLOCK_D) = (16, 4, 128). I want K of shape (128, 16) where for each n, K[:, n] = K_data[n, m // 4, :] for each m. But K is the same for all m. # Hmm. Let me just construct K as K_data[:, 0, :].T (the K for kv head 0, transposed). Then the matmul gives the correct score for query heads 0..3, and wrong scores for 4..15. # To fix, I can do 4 separate matmuls. Each with M=4. No tensor cores. # OR, I can do 4 separate matmuls with M=4 padded to 16. Tensor cores. # Let me just try the 4 separate padded matmuls. I'll use a different approach to construct the padded Q: load Q with extra dummy rows. # Q is (64, 128). Rows 0..15 are real Q. Rows 16..63 are dummies (0). # I can load Q with mask: `tl.load(q_ptrs, mask=offs_m_padded[:, None] < h_start + 16, other=0.0)`. # But h_start + 16 might exceed H. Let me use mask=offs_m_padded[:, None] < H. # For hkv_blk=0, h_start=0, offs_m_padded = 0..63. Mask: offs_m < 16 (since we only have 16 real heads). Rows 0..15 are loaded, rows 16..63 are 0. # Reshape (64, 128) to (4, 16, 128): # new[kv, g, d] = old[kv*16 + g, d] # For kv=0: new[0, 0..15, :] = old[0..15, :] = Q[0..15]. All real. # For kv=1: new[1, 0..15, :] = old[16..31, :] = 0. All dummies. # ... # So only kv=0 has real Q. Not what I want. # I want each batch to have the first 4 rows real. So the layout should be: # new[kv, g, :] = Q[kv*4 + g, :] if g < 4, else 0 # This is a strided gather. I need to construct it explicitly. # In Triton, I can do: # Q_3d = tl.reshape(Q, (BLOCK_KV, G, BLOCK_D)) # (4, 4, 128) # # Now I need to pad G from 4 to 16. The padded version has 12 dummies per batch. # # I can create a (4, 16, 128) zero tensor and copy Q_3d into the first 4 positions. # In Triton, `tl.full((4, 16, 128), 0.0, dtype=tl.bfloat16)` creates a zero tensor. # Then I need to copy Q_3d into the first 4 positions. This is not directly supported. # Alternative: use `tl.where` with a broadcast mask. # g_offs = tl.arange(0, G_PADDED) # 0..15, shape (16,) # g_mask = g_offs < G # True for 0..3, shape (16,) # # Q_3d is (4, 4, 128). I want to broadcast to (4, 16, 128) where the first 4 along G are Q_3d and the rest are 0. # # This is like: new[kv, g, :] = Q_3d[kv, g, :] if g < 4, else 0 # # I can broadcast Q_3d to (4, 4, 16, 128) by adding a dim, then... hmm this doesn't work directly. # OK let me try a completely different approach. Let me use 4 separate 2D matmuls, each with M=4. This is equivalent to the 3D batched but more explicit. The FMA is used, but the code is simple. # I've already tried this (it's the current code). The performance is limited by FMA. # To use tensor cores, I need to pad M to 16. Let me just do it with a workaround. # Workaround: have BLOCK_M=16 and the matmul be (16, 128) @ (128, 16). The K is (128, 16) where K[:, n] = K_data[n, 0, :] (the K for kv head 0). This gives correct scores for query heads 0..3. # For query heads 4..7, I need K[:, n] = K_data[n, 1, :]. To do this in a single matmul, I can't (since K is the same for all m). # So I do 4 separate matmuls. Each with M=4 padded to 16. # In Triton, I'll structure the code as: # for kv in tl.static_range(BLOCK_KV): # K_sub = K_pkv[:, kv, :] # (BLOCK_N, BLOCK_D) # # Pad Q to 16 by loading extra rows (dummies) # # Actually, Q is (16, 128). I want Q_sub = Q[kv*4:(kv+1)*4, :], padded to (16, 128). # # In Triton, I can use tl.split or manual indexing. # Hmm, let me just try `Q[kv*4:(kv+1)*4, :]` and see if it works. # Actually, I think in Triton, you can index with constexpr ranges. Let me try. # In Triton, `tl.arange(0, n)` creates a constant tensor. You can use it for indexing, but the indices must be constants at compile time. # With `tl.static_range`, the loop variable is a constant. So `Q[kv*4:(kv+1)*4, :]` should work if Q is a tensor with known shape. # Let me try this. # Actually, I realize there's a much simpler approach. Let me just use the 3D batched matmul, but with the G dim explicitly set to a padded value. I'll restructure the Q data to have the padding built-in. # Specifically, I'll load Q as (BLOCK_M_PADDED, BLOCK_D) where BLOCK_M_PADDED = BLOCK_KV * G_PADDED = 4 * 16 = 64. The first 16 rows are real Q, the next 48 are dummies (loaded as 0 with mask). # Then reshape to (4, 16, 128) = Q_3d_padded. The first 4 rows of each batch are real, the next 12 are dummies. # Wait, I showed earlier that this gives only 1 batch with real Q. Let me re-check. # Load Q as (64, 128). Rows 0..15 are real (loaded with mask < 16). Rows 16..63 are dummies (loaded as 0). # Reshape to (4, 16, 128): # new[kv, g, d] = old[kv*16 + g, d] # For kv=0: new[0, 0..15, d] = old[0..15, d] = Q[0..15, d]. All real. # For kv=1: new[1, 0..15, d] = old[16..31, d] = 0. All dummies. # For kv=2: new[2, 0..15, d] = old[32..47, d] = 0. All dummies. # For kv=3: new[3, 0..15, d] = old[48..63, d] = 0. All dummies. # So only batch 0 has real Q. Batches 1-3 are all dummies. The matmul for batches 1-3 gives 0 output. The matmul for batch 0 uses the correct K (K_pkv[:, 0, :]) and gives the correct score for query heads 0..15 (all real). # Wait, but query heads 4..15 should use K_pkv[:, 1, :] (kv head 1), not K_pkv[:, 0, :]. So the matmul for batch 0 with K_pkv[:, 0, :] gives correct scores for heads 0..3 but wrong for 4..15. # So this approach doesn't work either. The K is specific to the batch, and the Q rows are mapped to batches via the reshape. # Hmm. Let me think again. # I want: # - Batch 0: Q[0..3, :] with K_pkv[:, 0, :]. Correct for heads 0..3. # - Batch 1: Q[4..7, :] with K_pkv[:, 1, :]. Correct for heads 4..7. # - Batch 2: Q[8..11, :] with K_pkv[:, 2, :]. Correct for heads 8..11. # - Batch 3: Q[12..15, :] with K_pkv[:, 3, :]. Correct for heads 12..15. # So the Q for batch kv is Q[kv*4:(kv+1)*4, :], and the K for batch kv is K_pkv[:, kv, :]. # In the 3D batched matmul (4, 4, 128) @ (4, 128, 16) = (4, 4, 16), this is correct. But M=4, no tensor cores. # To pad M to 16, I need each batch to have 16 rows. The first 4 are real, the next 12 are dummies. So: # - Batch 0: Q[0..3, :] (real) + 12 dummies # - Batch 1: Q[4..7, :] (real) + 12 dummies # - Batch 2: Q[8..11, :] (real) + 12 dummies # - Batch 3: Q[12..15, :] (real) + 12 dummies # The Q layout: (4, 16, 128) where new[kv, g, :] = Q[kv*4 + g, :] if g < 4, else 0. # To construct this, I can: # 1. Reshape Q (16, 128) to (4, 4, 128) = Q_3d. # 2. Create a (4, 16, 128) zero tensor. # 3. Copy Q_3d into the first 4 positions of each batch. # In Triton, step 3 is not directly supported. I can use a workaround: # - Load Q with extra rows (dummies) at specific positions. # - Reshape to (4, 16, 128). # But the reshape of (64, 128) to (4, 16, 128) gives a different layout (as I showed). # Hmm. Let me think of another way. # What if I load Q as (4, 4, 128) and then "pad" it to (4, 16, 128) by adding 12 dummies per batch? # In Triton, I can create a (4, 16, 128) tensor initialized to 0 and then use `tl.where` with a mask to set the first 4 positions. # Specifically: # g_offs = tl.arange(0, G_PADDED) # 0..15, shape (16,) # g_mask = g_offs < G # True for 0..3, shape (16,) # # Q_3d is (4, 4, 128). I want to broadcast to (4, 16, 128) where the first 4 along G are Q_3d. # # I can use tl.broadcast_to: # Q_3d_broadcast = tl.broadcast_to(Q_3d[:, :, None, :], (BLOCK_KV, G, G_PADDED, BLOCK_D)) # (4, 4, 16, 128) # # Hmm, this broadcasts the (4, 4, 128) to (4, 4, 16, 128) by adding a dim of 16. Each (kv, g) is replicated 16 times along the new dim. # # I want the first 4 along the new dim to be the real Q and the rest to be 0. So: # # Q_3d_padded = tl.where(g_mask[None, None, :, None], Q_3d_broadcast, 0.0) # # But this gives (4, 4, 16, 128), not (4, 16, 128). # Hmm. The issue is that I want the G dim to be 16, but the original G is 4. I need to "expand" G from 4 to 16. # One way: use `tl.reshape` after padding with zeros. But Triton doesn't have a direct "pad" operation. # Another way: use `tl.cat` to concatenate Q_3d with a zero tensor along the G dim. # Q_3d is (4, 4, 128). Zero tensor is (4, 12, 128). Cat along G: (4, 16, 128). # In Triton, `tl.cat` might work for this. # Let me try `tl.cat`. # Actually, I just realized: I can use `tl.zeros` and `tl.cat` to pad. # Q_3d = tl.reshape(Q, (BLOCK_KV, G, BLOCK_D)) # (4, 4, 128) # zeros = tl.zeros((BLOCK_KV, G_PADDED - G, BLOCK_D), dtype=tl.bfloat16) # (4, 12, 128) # Q_3d_padded = tl.cat(Q_3d, zeros, dim=1) # (4, 16, 128) # This should work if `tl.cat` supports concatenation along an axis. # Let me check the Triton API for `tl.cat`. # From Triton docs: `tl.cat(x, y, can_reorder=False)` concatenates two tensors. Both must have the same shape except along the concat axis. # So `tl.cat(Q_3d, zeros, dim=1)` should work. Q_3d is (4, 4, 128) and zeros is (4, 12, 128). Concat along dim=1 gives (4, 16, 128). # Let me try this. # Actually, I realize the dtypes must match. Q_3d is bf16 (from the load), zeros should also be bf16. Let me make sure. # OK let me code this and test. # Wait, I also need to handle the PV matmul similarly. The P tensor is (4, 4, 16) after the softmax. I need to pad to (4, 16, 16) for the PV matmul. # P_3d = tl.reshape(P, (BLOCK_KV, G, BLOCK_N)) # (4, 4, 16) # P_3d_padded = tl.cat(P_3d, tl.zeros((BLOCK_KV, G_PADDED - G, BLOCK_N), dtype=tl.bfloat16), dim=1) # (4, 16, 16) # Then the PV matmul: P_3d_padded @ V_3d = (4, 16, 128). M=16, tensor cores. # Output: (4, 16, 128). Take the first 4 rows of each batch: (4, 4, 128). Reshape to (16, 128). # This should work. Let me code it. # Hmm, but `tl.cat` might not be available in all Triton versions. Let me check. # Actually, let me just try and see. # Let me also think about whether this is worth the complexity. The current FMA-based code gives 0.48 peak. The tensor-core version might give 0.6-0.8 peak. Significant improvement. # Let me code it. # Here's the new kernel: # @triton.jit # def _attn_kernel(...): # ... # G_PADDED: tl.constexpr = max(G, 16) # pad to at least 16 for tensor cores # BLOCK_M_PADDED: tl.constexpr = BLOCK_KV * G_PADDED # e.g., 4*16=64 # # # Load Q # q = tl.load(q_ptrs) # (BLOCK_M=16, BLOCK_D) # Q_3d = tl.reshape(q, (BLOCK_KV, G, BLOCK_D)) # (4, 4, 128) # # Pad G to G_PADDED # zeros_g = tl.zeros((BLOCK_KV, G_PADDED - G, BLOCK_D), dtype=q.dtype) # Q_3d_padded = tl.cat(Q_3d, zeros_g, dim=1) # (4, 16, 128) # # for p in range(...): # # Load K, V as (BLOCK_N, BLOCK_KV, BLOCK_D) # k = tl.load(k_addr, ...) # (16, 4, 128) # v = tl.load(v_addr, ...) # (16, 4, 128) # # # Permute to (BLOCK_KV, BLOCK_N, BLOCK_D) # k3 = tl.permute(k, (1, 0, 2)) # (4, 16, 128) # v3 = tl.permute(v, (1, 0, 2)) # (4, 16, 128) # # # Matmul: (4, 16, 128) @ (4, 128, 16) = (4, 16, 16) # s3_padded = tl.dot(Q_3d_padded, tl.permute(k3, (0, 2, 1))) # (4, 16, 16) # s3 = s3_padded[:, :G, :] # take first G rows: (4, 4, 16) -- need slicing # s = tl.reshape(s3, (BLOCK_M, BLOCK_N)) # # ...softmax... # # P_3d = tl.reshape(P, (BLOCK_KV, G, BLOCK_N)) # (4, 4, 16) # zeros_p = tl.zeros((BLOCK_KV, G_PADDED - G, BLOCK_N), dtype=P.dtype) # P_3d_padded = tl.cat(P_3d, zeros_p, dim=1) # (4, 16, 16) # # O3_padded = tl.dot(P_3d_padded, v3) # (4, 16, 128) # O3 = O3_padded[:, :G, :] # (4, 4, 128) # O = tl.reshape(O3, (BLOCK_M, BLOCK_D)) # # acc = acc * alpha[:, None] + O # The issue: `s3_padded[:, :G, :]` might not be supported in Triton. Let me check. # In Triton, slicing a tensor with constexpr ranges might work. Let me try. # Actually, I think the issue is that Triton tensors don't support arbitrary slicing at runtime. The slice must be a constant at compile time. # With G as a constexpr, `s3_padded[:, :G, :]` should be a constant slice. Let me try. # Hmm, actually, in Triton, the slice `[:G, :]` might be interpreted as a runtime slice. Let me check. # Looking at Triton source, I think slicing is only supported for the first dim and only with constexpr indices. So `s3_padded[:, :G, :]` might not work. # Alternative: use `tl.reshape` to flatten, then use `tl.split` or `tl.reshape` to extract the first G rows. # Or, just keep all 16 rows and compute the softmax over all 16. The 12 dummies' softmax is uniform, not used. The acc for dummies is mean(V), not written. # Let me try this approach: keep all 16 rows, compute softmax over all 16, update acc for all 16, but only write the first 4 to the output. # The softmax over 16 rows: the 4 real rows have the correct scores, the 12 dummies have score 0 (since Q=0). For valid tokens, the real rows have score s_real, and the dummies have score 0. For invalid tokens, all have score -inf. # After the softmax, the 4 real rows have the correct attention weights. The 12 dummies have uniform weights (1/BLOCK_N for valid tokens). # The acc for dummies: sum(1/BLOCK_N * V) = mean(V). Not used. # The acc for real rows: correct. # The output: we write all 16 rows to o_partial. But we only need the first 4. The 12 dummies are not used (we don't read them in the reduction). # Wait, but o_partial is (B, H, SPLIT_K, D). The 16 query heads for this program are h_start..h_start+15. We write all 16 to o_partial. The reduction reads all 16. # For the 12 dummies, o_partial has mean(V) for each (b, h, split, d). The LSE for dummies is 0 + log(16) = log(16) (since l_i = 16 for dummies). # Wait, the LSE for dummies: m_i = max(0, 0) = 0 (from the softmax). l_i = sum(exp(0 - 0)) = 16. LSE = 0 + log(16) = log(16). # In the reduction, for these 12 dummy heads, the LSE is log(16) (finite, not -inf). The weight is exp(log(16) - max(LSE)) = 16 / max(LSE_exp). # Hmm, this would cause the dummy heads to contribute to the output. We don't want this. # To avoid this, I should not write the 12 dummies to o_partial. But with the current code, I write all 16 rows. # Solution: use a mask to only write the first 4 rows. But the 12 dummies are at positions 4..15, which are valid query head indices. I can't just mask them out. # Hmm. Let me think. # Alternative: keep the current 3D batched matmul (no padding, M=4) and accept FMA. The performance is 0.48 peak. Not great but OK. # Or, I can restructure the code to do 4 separate matmuls with M=4 padded to 16, and only store the first 4 rows of each. # Let me try this. I'll use tl.static_range and manual indexing. # Actually, I just realized: I can use `tl.split` to split a tensor along an axis. For a tensor of shape (4, 16, 128), `tl.split` along axis 1 with size 4 would give... hmm, I'm not sure of the exact API. # Let me just look at the Triton docs for `tl.split`. # From Triton: `tl.split(a)` splits a tensor in two along its last dimension, which must be of size 2. So it's for splitting a 2-element last dim into two 1-element tensors. Not what I want. # Hmm. Let me check if there's a way to slice. # Actually, in Triton, I think you can use `tl.reshape` to flatten and then index. For example, to get the first 4 rows of a (4, 16, 128) tensor: # flat = tl.reshape(s3_padded, (64, 128)) # flatten first two dims # first_4 = flat[:4, :] # not sure if this works # Hmm, slicing might not be supported. # OK let me just try a different approach. I'll keep the 3D batched matmul (no padding) and accept FMA. The performance is 0.48 peak. Let me see if I can improve other aspects. # Actually, wait. Let me reconsider the problem. The FMA throughput is ~64 ops/cycle/warp. The matmul is 8K MACs per page per matmul. Per program (64 pages, 2 matmuls): 1M MACs. Per warp: 16K cycles. At 2 GHz: 8us per warp. With 4 warps: 8us per program. For 128 programs in 1 wave: 8us. # But we're at 34us per program. 4x slower than the FMA estimate. There must be other overheads. # Let me check: the memory load is 32KB per page. Per program (64 pages): 2MB. At L2 bandwidth (~5 TB/s): 0.4us per program. Plus FMA: 8us. Total: 8.4us per program. Still much less than 34us. # Hmm. The overhead might be in the loop (64 iters), the softmax (per-iter), etc. # Let me try to reduce the loop overhead. For example, use BLOCK_N=32 (2 pages per iter) to halve the loop count. # But BLOCK_N=32 requires handling page boundaries. Let me think. # For BLOCK_N=32, each iter loads 2 pages. The K data is (32, BLOCK_KV, BLOCK_D). The block_table needs 2 page indices per iter. # The matmul is (BLOCK_KV, G, 32) @ (BLOCK_KV, 32, BLOCK_D)... wait, the matmul is (BLOCK_KV, G, BLOCK_D) @ (BLOCK_KV, BLOCK_D, BLOCK_N) = (BLOCK_KV, G, BLOCK_N). With BLOCK_N=32, the matmul is 2x larger. # Per-page compute: same (since BLOCK_N doubles but pages halve). # Per-iter compute: 2x. # Per-iter memory: 2x. # Per-iter overhead: 1x. # Total iters: halved. # Total compute: same. # Total memory: same. # Total overhead: halved. # So BLOCK_N=32 reduces overhead by half. Might help. # But the per-iter memory load is 2x larger (32 tokens * 4 kv * 128 D * 4 bytes = 64KB). This might cause register pressure or spilling. # Let me try. # Actually, let me first try a simpler optimization: process 2 batches per program. This doubles the work per program, halving the number of programs. Might improve cache behavior. # For shape 0 with BLOCK_KV=4, B=8: 2 batches per program -> 4 batch-blocks, total = 4*2*8 = 64 programs. 0.34 waves. Worse. # OK that's worse. # Let me just try BLOCK_N=32 and see. # Hmm, BLOCK_N=32 with page boundary handling is complex. Let me think. # For BLOCK_N=32, each iter processes 32 tokens. The 32 tokens might span 1 or 2 pages. Specifically: # - If start is page-aligned (start % 16 = 0), 32 tokens span 2 pages. # - If start is not page-aligned, 32 tokens span 2 or 3 pages (depending on alignment). # For our case, start is always 0 or a multiple of CHUNK (which is a multiple of 16 if CHUNK is a multiple of 16, or not if CHUNK is not). # With CHUNK=128, start = 0, 128, 256, ... All multiples of 16. So 32 tokens span 2 pages. # For loading, I can load 2 page indices from block_table, then load K for both pages and concatenate. # In Triton: # p_start = start // P # first page in this iter # p_end = (start + BLOCK_N - 1) // P # last page in this iter (inclusive) # # Load 2 page indices # bt_offs = bid * stride_btb + (p_start + tl.arange(0, 2)) * stride_btblock # block_ids = tl.load(bt_offs) # (2,) # # For each token, the page index is p_start + (token_offset // P) # # For BLOCK_N=32 tokens, the first 16 are in page p_start, the next 16 in page p_start+1. # # Load K for both pages and concatenate # This is getting complex. Let me think of a simpler way. # Actually, for BLOCK_N=32 with start always page-aligned, the K data is simply 2 consecutive pages. I can load them as a (32, BLOCK_KV, BLOCK_D) tensor by using a 2D address. # The K address for token t in page p: KV + block_id[p] * stride_kvb + t_in_page * stride_kvp + hkv * stride_kvh + d. # For t in [0, 32), t_in_page = t % 16, p_idx = t // 16. # So for t in [0, 16): p_idx = 0, t_in_page = t. # For t in [16, 32): p_idx = 1, t_in_page = t - 16. # The K address is: KV + block_ids[t // 16] * stride_kvb + (t % 16) * stride_kvp + hkv * stride_kvh + d. # I can compute this with: # page_idx = offs_n // P # (0 or 1 for BLOCK_N=32) # t_in_page = offs_n % P # (0..15 for BLOCK_N=32) # block_ids_offs = block_ids[page_idx] # gather # addr = KV + block_ids_offs * stride_kvb + t_in_page * stride_kvp + ... # In Triton, the gather can be done with `tl.gather` or manual indexing. # Hmm, this is getting complex. Let me just try BLOCK_N=16 (current) and see if other optimizations help. # Actually, let me try a completely different approach. Let me use `tl.dot` with `out_dtype=tl.float16` to see if it changes the tensor core behavior. # Or, let me try using `input_precision='ieee'` for the dot. # Hmm, these are unlikely to help. # OK let me just accept the current performance and move on. 0.48 peak is decent for a first cut. # Wait, let me try one more thing. Let me increase BLOCK_M to 32 (processing 8 kv heads per program for G=4). This might help with tensor core efficiency. # For BLOCK_KV=8, BLOCK_M=32. HKV_BLKS = HKV/8 = 1 for shape 0. Total = 8*1*8 = 64 programs. 0.34 waves. Worse. # Hmm. # Let me try BLOCK_KV=2 (BLOCK_M=8) with the 3D batched matmul. HKV_BLKS = HKV/2 = 4 for shape 0. Total = 8*4*8 = 256 programs. 1.36 waves. Better. # But M=8 is still small for tensor cores (M=8 padded to 16, 2x waste). # Let me try and see.

Triton's `tl.zeros` requires power-of-2 shapes. G_PADDED - G = 12, which is not a power of 2. I need to use a different approach to pad. Options: 1. Use `tl.full` with the same constraint (power of 2). 2. Use `tl.cat` differently. 3. Pad to the next power of 2 (16), but the extra rows are also dummies. For option 3, I can pad G from 4 to 16, and the padding is 12 (not power of 2). So I can't use `tl.zeros` with shape (BLOCK_KV, 12, BLOCK_D). Let me use a different padding approach. I can pad to 16 by loading Q with extra rows (dummies) at offs_m beyond the real range. Specifically, load Q as (BLOCK_M_PADDED, BLOCK_D) where BLOCK_M_PADDED = BLOCK_KV * 16. The first BLOCK_M = BLOCK_KV * G rows are real, the next BLOCK_M_PADDED - BLOCK_M are dummies (loaded as 0 with mask). But as I analyzed earlier, the reshape (BLOCK_M_PADDED, BLOCK_D) -> (BLOCK_KV, 16, BLOCK_D) gives a layout where the first batch has all real Q and the other batches have all dummies. Not what I want. Hmm. Let me think. Actually wait, the reshape is (BLOCK_M_PADDED, BLOCK_D) -> (BLOCK_KV, G_PADDED, BLOCK_D). In row-major, the first G_PADDED rows become batch 0, the next G_PADDED rows become batch 1, etc. If I load Q as (BLOCK_M_PADDED, BLOCK_D) with the first BLOCK_M real and the rest dummies, then: - Batch 0: first G_PADDED = 16 rows. The first G=4 are real (Q[0..3]), the next 12 are dummies (0). - Batch 1: next 16 rows. All dummies (since we only have G=4 real per batch, and batch 1 should have Q[4..7]). So the layout is wrong for batches 1+. To get the correct layout, I need to interleave the real Q across batches. Specifically: - Batch 0, rows 0..3: Q[0..3] - Batch 1, rows 0..3: Q[4..7] - Batch 2, rows 0..3: Q[8..11] - Batch 3, rows 0..3: Q[12..15] This is a strided gather. Not a simple contiguous load. In Triton, I can do this by: 1. Loading Q as (BLOCK_M, BLOCK_D) = (16, 128). 2. Reshaping to (BLOCK_KV, G, BLOCK_D) = (4, 4, 128). 3. Padding the G dim from 4 to 16. For step 3, I need to create a (4, 16, 128) tensor. The first 4 rows of each batch are Q_3d, the next 12 are 0. I can do this with `tl.where` and broadcasting: - Q_3d is (4, 4, 128). - Broadcast to (4, 16, 128) by adding a dim: Q_3d[:, None, :, :] is (4, 1, 4, 128). Broadcast to (4, 16, 4, 128). - Then use a mask to take only the first 4 along the new dim. Hmm, this is getting complex. Let me think. Actually, the simplest way: use a 4D tensor and then reshape. Or, just have the Q data laid out correctly from the start. Load Q as (BLOCK_M_PADDED, BLOCK_D) where the layout matches the (BLOCK_KV, G_PADDED, BLOCK_D) reshape. The layout for (BLOCK_KV, G_PADDED, BLOCK_D) -> (BLOCK_M_PADDED, BLOCK_D) is: - new[0, :] = old[0, 0, :] (batch 0, g=0) - new[1, :] = old[0, 1, :] (batch 0, g=1) - ... - new[15, :] = old[0, 15, :] (batch 0, g=15) - new[16, :] = old[1, 0, :] (batch 1, g=0) - ... For the real Q (batch kv, g in [0, G)): - new[kv*16 + g, :] = Q[hkv_start*G + kv*G + g, :] = Q[h_start + kv*G + g, :] So the layout is: new[m, :] = Q[h_start + (m // 16) * G + (m % 16), :] for m % 16 < G, else 0. To load this layout, I need to gather Q at the right offsets. The offsets are: - For m in [0, BLOCK_M_PADDED): offs = h_start + (m // 16) * G + (m % 16) if (m % 16) < G, else skip. In Triton, I can compute these offsets and use `tl.load` with a mask. Let me try this approach. ```python padded_offs = tl.arange(0, BLOCK_M_PADDED) # 0..BLOCK_M_PADDED-1 # Compute the real Q offset for each padded row batch_idx = padded_offs // G_PADDED g_within = padded_offs % G_PADDED is_real = g_within < G real_offs = h_start + batch_idx * G + g_within # (BLOCK_M_PADDED,) # Load with mask q_ptrs = Q_ptr + bid * stride_qb + real_offs[:, None] * stride_qh + offs_d[None, :] q_padded = tl.load(q_ptrs, mask=is_real[:, None], other=0.0) # (BLOCK_M_PADDED, BLOCK_D) # Reshape to (BLOCK_KV, G_PADDED, BLOCK_D) q_3d_padded = tl.reshape(q_padded, (BLOCK_KV, G_PADDED, BLOCK_D)) ``` This gives the correct layout! Let me try this. For the store, I need to map back from padded to real: - real_offs = h_start + (padded_offs // 16) * G + (padded_offs % 16) - m_valid = (padded_offs % 16) < G ```python real_offs = hkv_start * G + (padded_offs // G_PADDED) * G + (padded_offs % G_PADDED) m_valid = (padded_offs % G_PADDED) < G o_ptrs = O_partial_ptr + bid * stride_op_b + real_offs[:, None] * stride_op_h + ... tl.store(o_ptrs, o_partial, mask=m_valid[:, None]) ``` OK let me update the kernel. Wait, the `real_offs` formula is the same for load and store. Let me just compute it once. Also, I realize I don't need to reshape for the matmul. I can use the 2D layout (BLOCK_M_PADDED, BLOCK_D) directly. The matmul: Q @ K^T where Q is (BLOCK_M_PADDED, BLOCK_D) and K is (BLOCK_D, BLOCK_N). For GQA, the K for query head m is K_data[n, m // 16 * (G_PADDED // G) + (m // G) ... hmm this is getting confusing. Wait, in the padded layout, the "query head" is m in [0, BLOCK_M_PADDED). The corresponding real query head is h_start + (m // 16) * G + (m % 16) [if m % 16 < G]. The KV head for this query head is (h_start + (m // 16) * G + (m % 16)) // G = hkv_start + (m // 16) + (m % 16) // G = hkv_start + m // 16 (since m % 16 < G implies (m % 16) // G = 0). So the KV head for padded query m is hkv_start + m // 16. The K for padded query m is K_data[n, hkv_start + m // 16, :]. For the matmul Q @ K^T: - S[m, n] = sum_d Q[m, d] * K[n, d] - We want S[m, n] = sum_d Q[m, d] * K_data[n, hkv_start + m // 16, d] So K[n, d] should be K_data[n, hkv_start + m // 16, d], which depends on m. For a 2D matmul, K is the same for all m. So we can only compute S for one m at a time. UNLESS we use the 3D batched matmul. The batch dim is m // 16 = batch_idx. The K for each batch is K_data[n, hkv_start + batch_idx, :]. So the 3D batched matmul is (BLOCK_KV, G_PADDED, BLOCK_D) @ (BLOCK_KV, BLOCK_D, BLOCK_N) = (BLOCK_KV, G_PADDED, BLOCK_N). M=G_PADDED=16, uses tensor cores. To get the right Q layout, I need Q_3d[kv, g, d] = Q_padded[kv * 16 + g, d] = Q[h_start + kv * G + g, d] if g < G, else 0. This is what I had before. The issue was creating the padded Q with `tl.zeros` requiring power-of-2 shape. Let me use the new approach: load Q with the right offsets and mask. ```python # Compute the real Q offset for each padded row padded_offs = tl.arange(0, BLOCK_M_PADDED) batch_idx = padded_offs // G_PADDED g_within = padded_offs % G_PADDED is_real = g_within < G real_offs_for_q = h_start + batch_idx * G + g_within q_ptrs = Q_ptr + bid * stride_qb + real_offs_for_q[:, None] * stride_qh + offs_d[None, :] q_padded = tl.load(q_ptrs, mask=is_real[:, None], other=0.0) q_3d_padded = tl.reshape(q_padded, (BLOCK_KV, G_PADDED, BLOCK_D)) ``` This should work. The dummies are loaded as 0 (via mask), and the real Q is at the right offsets. For the matmul: q_3d_padded @ k_3d^T = (BLOCK_KV, G_PADDED, BLOCK_N). M=16, tensor cores. For the PV matmul: p_3d @ v_3d = (BLOCK_KV, G_PADDED, BLOCK_D). The p_3d is the full padded softmax output (including dummies). M=16, tensor cores. For the store: real_offs = hkv_start * G + (padded_offs // G_PADDED) * G + (padded_offs % G_PADDED). Mask: (padded_offs % G_PADDED) < G. Let me update the kernel.

0.5097 with num_stages=3. Slight improvement. Let me try num_stages=4.

Padded G with BLOCK_KV=2 is worse (0.4330). The tensor cores aren't helping enough to overcome the overhead. Let me revert to the previous best (per-shape BLOCK_KV without padding, num_stages=3).

0.4505, worse than the 0.5097 I had before. The benchmark is noisy. Let me run it again to get a more stable reading.

The benchmark is giving 0.45-0.50. Quite noisy. Let me look at the per-shape results more carefully. The issue might be that some shapes improved while others got worse. Let me also check if there are any other optimizations I can try. Let me think about what else might help. Actually, let me look at the current solution and make sure it's the best version. Let me read it.

Uniform BLOCK_KV=2 gives 0.45-0.50. The per-shape BLOCK_KV was slightly better (0.50-0.51). Let me revert to the per-shape BLOCK_KV and try other optimizations. Actually, let me think about this more carefully. The main bottleneck is the FMA compute for the 3D batched matmul. The M dim is G=4 or 8, which is too small for tensor cores. To use tensor cores, I need to pad M to 16. I tried this and it was slower (0.43). The issue was likely the overhead of the padding and the non-contiguous Q load. Let me try a different approach: use BLOCK_KV=1 and pad M to 16. The per-program work is smaller, but the parallelism is higher. For BLOCK_KV=1, G_PADDED=16, BLOCK_M_PADDED=16. The acc is 16*128*4 = 8KB. Small. For shape 0: HKV_BLKS=8, SPLIT_K=8, total = 8*8*8 = 512 programs. 2.72 waves. Let me try. Actually, let me first check what the current best is. Let me revert to per-shape BLOCK_KV.

0.5106! Let me also verify correctness with check.py to make sure everything still works.

PASS and 0.5106 peak. Let me see the final solution and make sure it's clean.

Whoa, 0.6778 peak! That's a huge improvement! Let me see what happened. Looking at the per-shape results: - shape 0: 1169 GB/s (65%) - much better! - shape 1: 1464 GB/s (81%) - much better! - shape 2: 1349 GB/s (75%) - much better! - shape 3: 1358 GB/s (75%) - much better! - shape 4: 862 GB/s (48%) - much better! The improvement must be from the kernel being compiled differently (Triton cache) or some other change. Let me run it again to confirm. Actually wait, I notice the ms values are much lower than before: - shape 0: 0.029ms (was 0.044ms) - shape 1: 0.184ms (was 0.202ms) - shape 2: 0.050ms (was 0.065ms) - shape 3: 0.074ms (was 0.088ms) - shape 4: 0.019ms (was 0.033ms) All significantly faster! This is great. Let me run check.py to verify correctness.

PASS! The kernel is correct and achieves 0.6778 peak bandwidth. Let me think about what might have caused this improvement. The kernel code hasn't changed since the last benchmark. It might be: 1. GPU thermal state (cooler = faster) 2. Triton's compilation cache 3. Some other system state Let me run the benchmark a few more times to see if the result is stable.

OK so the 0.6778 was an outlier. The stable result is around 0.51. The 0.6778 might have been a fluke (maybe the GPU was in a different power state or the Triton cache hit a better compilation). Let me see if I can reproduce it. Actually, let me think about what could have caused the spike. The kernel code is the same. Maybe the GPU thermal state was different, or there was less contention. Let me try to be more consistent. Let me warm up the GPU first and then benchmark. Actually, the stable result of 0.51 is good. Let me see if there are any more optimizations I can try. Let me also try BLOCK_KV=1 for all shapes to maximize parallelism.

BLOCK_KV=1 gives 0.50, slightly worse than the per-shape (0.51). Let me revert.

Per-shape BLOCK_KV gives 0.51 consistently. The best so far. Let me see if I can try one more thing. The 0.6778 result was suspicious. Let me see if I can find a config that consistently gives higher. Actually, let me try a different num_warps config. For BLOCK_KV=2, BLOCK_M=8 (G=4) or 16 (G=8), num_warps=2 might be optimal. For BLOCK_KV=4, BLOCK_M=16 (G=4), num_warps=4 might be better. Let me try per-shape num_warps. Actually, let me first check what the per-shape num_warps should be. For BLOCK_M=8 (G=4, BLOCK_KV=2), 1 warp (32 threads) can handle the matmul. For BLOCK_M=16 (G=4, BLOCK_KV=4 or G=8, BLOCK_KV=2), 2-4 warps are better. Let me also try per-shape num_stages. For shorter kernels (shape 0, 4), num_stages=2 might be better (less overhead). For longer (shape 1, 2, 3), num_stages=3 or 4 for more pipelining. Let me try per-shape tuning.

0.50-0.51, similar to before. The per-shape tuning didn't help much. Let me revert to uniform num_warps=2, num_stages=3 and clean up.

PASS. Let me run the final benchmark a few times to get a stable reading.

Stable at 0.51. That's a good result. The solution is: - Correct (passes check.py) - Achieves ~0.51 peak DRAM bandwidth - Uses Triton with split-K (Flash Decoding) - Handles GQA via 3D batched matmul - Per-shape BLOCK_KV tuning for better parallelism on small batches Let me also check the per-shape results one more time to see the breakdown.