The profiling shows significant issues: 1. Global load coalescing is very poor (4.0 sectors/request instead of 1.0 ideal) - 4x wasted bandwidth 2. Low occupancy (12.4%) 3. Latency bound - need more parallelism The issue is that my current access pattern isn't coalesced well. When threads access memory like: - Thread 0: reads input[r * BD + 0 * D + 0], input[r * BD + 0 * D + 0], ... for each r - Thread 1: reads input[r * BD + 0 * D + 1], ... Actually, looking at this more carefully: - For a fixed (b, d), threads access r * B * D + b * D + d - The stride between consecutive r values is B*D = 16*48 = 768 elements - This means each thread reads 4 elements with stride 768, which is strided access A better approach would be to have threads read contiguous data. Let me think about this... Actually, for coalescing: - Threads in a warp should access consecutive memory addresses - Currently, thread bd accesses addresses: r*B*D + b*D + d for r=0,1,2,3 - If bd = b*D + d, then thread bd accesses: r*B*D + bd for each r - This means thread 0 accesses: 0, B*D, 2*B*D, 3*B*D - Thread 1 accesses: 1, B*D+1, 2*B*D+1, 3*B*D+1 Wait, that's actually contiguous! Thread 0 reads address 0, thread 1 reads address 1, etc. The issue is that the 4 values each thread needs are strided. Let me reconsider the memory layout: - Input is (R, B, D) = (4, 16, 48) - Memory: [r=0, b=0, d=0-47], [r=0, b=1, d=0-47], ..., [r=1, b=0, d=0-47], ... For each output element (b, d), we need to read: - input[0, b, d] = offset b*D + d - input[1, b, d] = offset B*D + b*D + d - input[2, b, d] = offset 2*B*D + b*D + d - input[3, b, d] = offset 3*B*D + b*D + d So the input for one output is at addresses: bd, BD+bd, 2*BD+bd, 3*BD+bd If threads 0-31 form a warp and access addresses bd=0..31, then: - For r=0: addresses 0..31 - contiguous! ✓ - For r=1: addresses BD..BD+31 - contiguous! ✓ - For r=2: addresses 2*BD..2*BD+31 - contiguous! ✓ - etc. So actually, the accesses should be coalesced! The sectors/request=4.0 suggests something else is happening. Oh wait, I think the issue might be that D=48 and we have 256 threads per block. With B*D=768 total outputs, we have 3 blocks of 256 threads. Actually, let me look at the memory more carefully. If D=48, then for threads 0..31: - b = bd / D, d = bd % D - For bd=0..31: b=0, d=0..31 - These threads all access input[r*B*D + b*D + d] = input[r*768 + d] - So for r=0: addresses 0..31 - contiguous ✓ - For r=1: addresses 768..799 - contiguous ✓ The issue might be that with D=48 not being a multiple of 128 (typical cache line is 128 bytes = 32 floats), the coalescing isn't perfect. Actually, sectors/req=4.0 suggests that each request is fetching 4 cache sectors. A cache line is typically 128 bytes = 32 float32 values. If D=48, and we're accessing d=0..31, we'd get one full cache line per warp. Let me try a different approach - using vectorized loads to improve efficiency. Actually, given the very small problem size (R=4, B=16, D=48), the kernel is likely dominated by launch overhead. The small grid size (3 blocks) means we can't saturate the GPU. Let me try a few optimizations: 1. Use vectorized loads (float4) to read 4 consecutive values 2. Better thread organization Wait - for this problem, each thread needs R=4 strided values. We can't use float4 for that since the values are strided. Let me try a different approach: have each thread process multiple outputs (increase work per thread) to better hide latency. Actually, let me try using shared memory to stage the input data. Hmm, but with such small input sizes, the overhead of the kernel launch itself might dominate. Let me try a different approach: 1. Use a single block 2. Have each thread process multiple outputs 3. Use shared memory to load input Actually, let me first try the simple optimization of using vectorized loads differently. Another approach: process 4 elements at a time by having threads read in a pattern that allows coalescing. Let me try: - Each thread handles 4 outputs (consecutive in d) - Use float4 loads for the input But wait, the input isn't contiguous for the 4 values we need per output. Let me try another approach: use shared memory to transpose the data. Actually, given the problem size is so small, let me try optimizing for reduced kernel launch overhead: 1. Use fewer blocks but more work per block 2. Use warp-level primitives Let me try using warp shuffles for the reduction if R=4 is fixed. Actually, for this very specific problem (R=4 fixed), we could use a specialized approach where: - Each warp handles one batch element - Each thread handles D/32 values (D=48, so each thread handles 1.5 values - we'd need to adjust) Let me try a different approach: - Each thread handles 1 output element - Use vectorized load from each r slice (we could read float4 if D is aligned) Actually wait - I can try: - Use __ldg (read-only cache) for the input loads - Vectorized loads won't help due to strided pattern Let me try using __ldg and see if that helps with the cache behavior. Actually, let me try a completely different approach for this small problem: - Use shared memory to load all input - Each thread then reads from shared memory With input size R*B*D = 4*16*48 = 3072 floats = 12KB, this easily fits in shared memory! Let me try that approach.

Back to ~3.1 μs. The coalescing is still poor (4.0 sectors/request). The issue is that each thread in a warp reads values spaced B*D apart - when thread 0 reads input[0], thread 1 reads input[1], etc., they're reading consecutive memory which IS good. But each thread then reads input[B*D], input[2*B*D], input[3*B*D] which are far apart. Wait, let me re-think the memory layout: - Input is (R, B, D) = (4, 16, 48) with 3072 total elements - The memory is laid out as: input[0,0,0], input[0,0,1], ..., input[0,0,47], input[0,1,0], ..., input[0,15,47], input[1,0,0], ... When thread tid processes element at position tid (where tid = b*D + d), it reads: - input[tid] (element at position tid in linear memory) - input[B*D + tid] (element at position B*D + tid) - input[2*B*D + tid] (element at position 2*B*D + tid) - input[3*B*D + tid] (element at position 3*B*D + tid) For a warp of 32 threads: - Thread 0 reads input[0], thread 1 reads input[1], etc. -> coalesced (first read) - Thread 0 reads input[768], thread 1 reads input[769], etc. -> also coalesced (second read) So actually each of the 4 reads should be coalesced! Why is the profiler showing 4.0 sectors/request? Maybe it's because the L1 cache line is 128 bytes = 32 floats, and each warp reads 32 consecutive floats which should be 1 sector (128 bytes). But wait, I'm using __ldg which goes through the read-only data cache (texture cache). On Hopper, the L2 handles the sector requests. I should try vectorized loads to let the compiler issue wider transactions. Using float4 would let each thread read 4 consecutive elements, and then I could transpose in registers, but since the sum is over R with each thread holding a different b,D pair, I need to think about how the data is actually laid out. If I reinterpret the input as float4, then each thread could load 4 consecutive floats at once. The real issue is that tid = b*D + d, where D = 48. When I use __ldg(&input[r*B*D + tid]), the first warp accesses elements [0..31] across r values, which should be coalesced, but then the second warp accesses [32..63], and the pattern breaks down because D isn't a multiple of 32. Actually, looking more carefully: within a single warp, threads access consecutive tid values and consecutive memory addresses for a given r value, which should be perfectly coalesced. The problem might be that I'm making 4 separate load requests per thread—one for each r value—and the profiler is accumulating these sectors across all requests. Let me try an alternative: use shared memory with a coalesced load pattern. Each thread can load multiple elements with a stride, compute partial sums, then perform a block-wide reduction. Or I could simply use a vectorized load approach since the input is already contiguous in memory. I could reinterpret the input tensor as float4 directly—since R*B*D = 3072 floats, this becomes 768 float4 values, matching exactly the 768 threads I need. Each thread loads one float4 containing four consecutive floats across the R dimension, sums them, then divides by 4. This gives me perfect coalescing with a single vectorized load per thread.