Good, we're on an H100. This is a very powerful GPU. Let me analyze the problem: - Input: h0 of shape (B, H0) = (16, 48) - Stage 1: h0 @ W1 -> (16, 64), then ReLU - Stage 2: h1 @ W2 -> (16, 40), then ReLU - Stage 3: h2 @ W3 -> (16, 24) The dimensions are quite small. With B=16 and hidden dimensions up to 64, this is a very small workload. Let me compute the FLOPs: - Stage 1: 16 * 48 * 64 * 2 = 98,304 FLOPs - Stage 2: 16 * 64 * 40 * 2 = 81,920 FLOPs - Stage 3: 16 * 40 * 24 * 2 = 30,720 FLOPs Total: ~211k FLOPs This is extremely small. The challenge here is that the problem sizes are tiny, so the overhead of kernel launches, memory allocation, etc. can dominate. We need to fuse operations to minimize overhead. Let me write a fused kernel that does all three stages in one kernel. This will: 1. Load the input and all three weight matrices 2. Perform the three matmuls with ReLU fused 3. Write the output For such small sizes, we could have each thread handle one output element, or we could use a more sophisticated tiling approach. Given the small sizes, let's try a simple approach first where each thread computes one output element of h3, but needs to compute the intermediate values. Actually, with 16 batches and 24 output features, we have 384 output elements. With 128 threads per block, we'd need 3 blocks. Each thread would need to: 1. Compute its h3 element 2. This requires computing the relevant h2 values (40 of them) 3. Each h2 requires computing h1 values (64 of them) 4. Each h1 requires computing from h0 (48 values) This would result in a lot of redundant computation. A better approach is to use shared memory to share intermediate results. Let me think about a tiled approach where we process in stages within the kernel: 1. All threads cooperatively compute h1 for all batches, store in shared memory 2. All threads cooperatively compute h2, store in shared memory 3. All threads cooperatively compute h3, store in global memory But with only 384 total output elements and small intermediate sizes, the overhead of synchronization might not be worth it. Actually, let me try a different approach: each thread block handles one batch sample, and we compute all stages within the block. With B=16 batch size, we'd have 16 blocks (one per batch). Each block would have enough threads to parallelize across the output dimensions. Let me try: - 16 blocks (one per batch) - Each block has enough threads to cover the maximum intermediate dimension (64 for h1) - Threads cooperatively compute h1, h2, h3 using shared memory This should work well for this small problem size. Let me write the kernel: ```cpp #include <cuda.h> #include <cuda_runtime.h> // Fused 3-stage MLP kernel // Each block handles one batch sample // Threads cooperatively compute h1, h2, h3 template <int H0, int H1, int H2, int H3> __global__ void fused_mlp_kernel( const float* __restrict__ h0, // [B, H0] const float* __restrict__ W1, // [H0, H1] const float* __restrict__ W2, // [H1, H2] const float* __restrict__ W3, // [H2, H3] float* __restrict__ out, // [B, H3] int batch_size ) { int batch_idx = blockIdx.x; if (batch_idx >= batch_size) return; int tid = threadIdx.x; int block_size = blockDim.x; // Shared memory for intermediate activations extern __shared__ float smem[]; float* h1_shared = smem; // [H1] float* h2_shared = smem + H1; // [H2] // Pointers for this batch const float* h0_batch = h0 + batch_idx * H0; float* out_batch = out + batch_idx * H3; // Stage 1: Compute h1 = relu(h0 @ W1) // Each thread computes some elements of h1 for (int i = tid; i < H1; i += block_size) { float sum = 0.0f; for (int j = 0; j < H0; j++) { sum += h0_batch[j] * W1[j * H1 + i]; } h1_shared[i] = fmaxf(0.0f, sum); } __syncthreads(); // Stage 2: Compute h2 = relu(h1 @ W2) for (int i = tid; i < H2; i += block_size) { float sum = 0.0f; for (int j = 0; j < H1; j++) { sum += h1_shared[j] * W2[j * H2 + i]; } h2_shared[i] = fmaxf(0.0f, sum); } __syncthreads(); // Stage 3: Compute out = h2 @ W3 for (int i = tid; i < H3; i += block_size) { float sum = 0.0f; for (int j = 0; j < H2; j++) { sum += h2_shared[j] * W3[j * H3 + i]; } out_batch[i] = sum; } } ``` This is a simple approach. Let me think about optimizations: 1. The inner loops could benefit from vectorized loads 2. The weight matrices are small enough to potentially fit in shared memory or even registers 3. We could pre-transpose the weight matrices for better memory access patterns Actually, looking at the weight layouts: PyTorch Linear stores weights as [out_features, in_features], but the reference code uses it as a matmul which would be h0 @ W1.T. Let me check the PyTorch convention... Actually, nn.Linear with bias=False computes `input @ weight.T`, so the weight shape stored is (out_features, in_features) = (H1, H0) for W1. But in the computation h0 @ W1 (conceptually), we need the transpose. So W1's stored shape is (H1, H0) and we compute h0 @ W1.T. Hmm, let me re-read the code... The forward function does `self.W1(h0)` which is nn.Linear. So the actual computation is: - h1 = relu(h0 @ W1.weight.T) where W1.weight is (H1, H0) - h2 = relu(h1 @ W2.weight.T) where W2.weight is (H2, H1) - out = h2 @ W3.weight.T where W3.weight is (H3, H2) So I need to transpose the weight access pattern. Let me rewrite: For W1 with shape (H1, H0), computing h0 @ W1.T: - h1[i] = sum over j of h0[j] * W1.weight[i, j] - In row-major, W1.weight[i*H0 + j] Let me rewrite the kernel with correct indexing: Actually, wait. Let me think more carefully. PyTorch's nn.Linear stores weight as (out_features, in_features). So: - W1.weight has shape (H1, H0) - W2.weight has shape (H2, H1) - W3.weight has shape (H3, H2) When computing the forward pass, PyTorch does: - output = input @ weight.T + bias - So h1 = h0 @ W1.T where W1.T has shape (H0, H1) In terms of element access: - h1[b, i] = sum over j of h0[b, j] * W1.weight[i, j] - Since W1.weight is stored as (H1, H0), W1.weight[i, j] is at index i*H0 + j Let me verify: h0 has shape (B, H0), W1.weight has shape (H1, H0). h0 @ W1.T would give shape (B, H1). For element (b, i) of the result: h1[b, i] = sum_{j=0}^{H0-1} h0[b, j] * (W1.T)[j, i] = sum_{j=0}^{H0-1} h0[b, j] * W1.weight[i, j] = sum_{j=0}^{H0-1} h0[b, j] * W1[i*H0 + j] So my original indexing was wrong. Let me fix it: ```cpp // Stage 1: h1[i] = sum_j h0[j] * W1[i*H0 + j] for (int i = tid; i < H1; i += block_size) { float sum = 0.0f; for (int j = 0; j < H0; j++) { sum += h0_batch[j] * W1[i * H0 + j]; } h1_shared[i] = fmaxf(0.0f, sum); } ``` Similarly for W2 and W3: - W2 has shape (H2, H1), so W2[i*H1 + j] for element (i, j) - h2[i] = sum_j h1[j] * W2[i*H1 + j] - W3 has shape (H3, H2), so W3[i*H2 + j] for element (i, j) - out[i] = sum_j h2[j] * W3[i*H2 + j] Let me rewrite with correct indexing and then compile to test. Actually, I realize there might be a more efficient way to do this. The weights are small and could be cached. But let me start with a simple correct implementation and optimize from there. Let me write the full kernel code:

The errors keep occurring without much detail. Let me try a simpler kernel without verbose mode to see if we can get a better error message, or try a different approach altogether. Actually, let me try just using torch operations directly and see if there are issues with the approach. I should focus on getting something to compile first. Let me try without the verbose flag and see if we get more error detail. Or let me try a simpler inline extension.

The kernel is correct. Now let me profile it to understand the performance characteristics and see if there are opportunities for optimization.

The profiling shows significant issues: 1. Very low occupancy (2.8%) due to small batch size 2. Uncoalesced global loads (18.6x wasted BW) - the threads are accessing scattered memory locations 3. High register pressure (66 regs/thread) 4. Latency-bound kernel The main issue is that for each output element in h1, the thread is loading W1[i, j] where j varies, but j is accessed in a loop. This means different threads are loading W1 elements scattered across memory. Let me rethink the approach: - For better coalescing, threads should access consecutive memory locations - The current pattern has thread i loading W1[i, 0], W1[i, 1], etc. which means thread 0 loads W1[0*H0+0], W1[0*H0+1], etc. and thread 1 loads W1[1*H0+0], W1[1*H0+1], etc. - This is actually a strided access pattern when we consider what threads 0-63 load in parallel Actually, let me think about this more carefully. For stage 1: - We have 64 threads (blockDim.x = 64) - Each thread i computes h1[i] for one row of W1 - W1 is stored row-major as [H1, H0] - Thread i accesses W1[i, 0], W1[i, 1], ..., W1[i, H0-1] - But threads 0-63 in a warp access W1[0, j], W1[1, j], ..., W1[63, j] in parallel - These addresses are W1[0*H0+j], W1[1*H0+j], ..., W1[63*H0+j] - The stride between consecutive threads is H0 = 48 elements = 192 bytes - This is NOT coalesced at all! For coalesced access, we need consecutive threads to access consecutive memory addresses. A better approach would be to: 1. Have threads cooperate on computing multiple outputs 2. Use shared memory to cache the weights 3. Or change the thread mapping Let me try a different approach where each thread block handles multiple output elements and we use shared memory more efficiently to cache weights and inputs. Actually, for small matrices like this, we could have each thread handle multiple output elements with better memory access patterns. Or we could cache the weights in shared memory. Let me try a tiled approach where we: 1. Cache the input (h0) in shared memory 2. Process in tiles 3. Have each thread compute multiple outputs Actually, let me try a simpler optimization first - increase threads per block and have each thread compute multiple output elements. Also, use __launch_bounds__ to control register usage. Let me try having all 16 batches processed by a single block with more threads. Actually wait, that won't work well because we need __syncthreads() between stages. Let me think about this differently. The weights are the problem - they're accessed in a strided manner. Let me try: 1. Transpose weights before using them (but this adds overhead) 2. Or change the computation order Actually, the simplest fix is to cache the weights in shared memory. But the weights are large (H1*H0 = 64*48 = 3072 floats = 12KB), which is too large for shared memory if we want to cache all weights. Wait, actually 12KB is fine - we have 228KB per SM. But each block has limited shared memory (227KB max). Let me try caching weights in shared memory. Actually, let me try a different approach - have each warp work together on a single output element, so that the threads in a warp access contiguous memory for the weights. Actually, let me reconsider. The best approach for this small problem might be: 1. Use a single warp per output element 2. Have the warp cooperatively compute the dot product Let me try a version where: - Each warp computes one output element - Threads in a warp access contiguous memory for weights - Use warp reduction for the dot product

Good, let me check correctness for this new approach.

The kernel is correct. Let me profile it to see if it's better optimized, and if it looks good I should submit.

The kernel is correct but slightly regressed in timing. The occupancy improved (2.8% → 12.3%) and coalescing improved (18.6x → 3.2x), but timing got slightly worse (8.67 → 9.70 μs). The main issue is still that this is a very small workload (B=16) and we're latency-bound. The memory access patterns for the weight matrices are still problematic - when different warps in the same block access different rows of W1, they're accessing memory that's far apart. Let me think about whether there's a better approach. The weights are shared across batches, so loading them efficiently is key. Actually, let me reconsider the memory layout: - h0: [B, H0] - each batch accesses a contiguous H0 elements - W1: [H1, H0] - stored row-major - W2: [H2, H1] - W3: [H3, H2] For the first matmul h1 = h0 @ W1.T: - h1[i] = sum_j h0[j] * W1[i,j] - W1[i,j] = W1[i*H0 + j] When warps compute different i values, they access different rows of W1. The rows are H0=48 floats apart. This means consecutive threads in a warp accessing the same column j but different rows i would access addresses that are H0 apart - not coalesced. For better coalescing, I should have threads in a warp access consecutive memory addresses. This means threads should work on the same output row, processing different input columns. Let me try a different approach - have each warp compute ONE output element, but have lanes process different input elements and use shared memory for weights. Actually, wait - the current approach has lanes processing consecutive input elements (j = lane_id and j = lane_id + 32), and all lanes in a warp work on the same output i. This should give coalesced reads from h0 (contiguous) but scattered reads from W1 (since we're reading W1[i*H0 + lane_id] - which is contiguous within a row!). The issue is the weight access pattern. For W1[i*H0 + lane_id], within a warp all lanes access the same row i of W1, but at different columns. This should be coalesced! But when warp_id changes, we access different rows of W1. Multiple warps accessing different rows means we're accessing non-contiguous memory regions. Actually the real problem here is that this workload is TINY. We have: - B=16 batches - 64+40+24 = 128 total output elements - ~6K FLOPs total This is almost nothing for an H100. The kernel overhead dominates. The PyTorch reference makes 3 separate kernel calls. Each kernel call has overhead. My fused kernel has ONE kernel call. So even if the kernel itself isn't super efficient, we avoid 2 kernel launch overheads. Let me submit this version and see how it compares. The correctness passed and this is a fused approach that should beat the PyTorch reference due to reduced kernel launch overhead.