We need to write a custom CUDA kernel that replaces the forward computation of the model above. The model: - nodes: (B, N, D) - pair_feats: (B, N, N, P) - edge_mask: (B, N, N) float {0,1} - hi = nodes.unsqueeze(2).expand(-1,-1,N,-1) => (B,N,N,D) - hj = nodes.unsqueeze(1).expand(-1,N,-1,-1) => (B,N,N,D) - cat [hi, hj, pair_feats] => (B,N,N, 2D + P) - fuse = nn.Sequential(Linear(2D+P, hidden), SiLU(), Linear(hidden, D)) - fused = fuse(cat) => (B,N,N,D) - fused = fused.masked_fill(edge_mask.unsqueeze(-1) < 0.5, neg) where neg = minimum float value - pooled, _ = fused.max(dim=2) => (B,N,D) max over j (the middle dim) - out = nodes + LayerNorm(pooled) => (B,N,D) We need to produce a custom CUDA kernel. The parameters dim=96, pair_dim=16, hidden=192, but we should write generic kernel that handles these sizes, maybe using compile-time constants where possible. We need to keep the weights of the MLP and the LayerNorm parameters inside the ModelNew as nn.Parameter or similar, and load them into the kernel. Since the fused MLP and SiLU and LayerNorm need weights and biases. Plan: In ModelNew.__init__, we will allocate the same parameters as reference: two Linear layers and one LayerNorm. But we need to implement the fused kernel that does everything in one go per output element. However, the max reduction over j requires computing fused(i,j) for all j. Given sizes: B=4, N=24, D=96, P=16, H=192. Not huge. But to be faster than PyTorch reference, we need to fuse the MLP + mask + max reduction + residual + layer norm into a single kernel, avoiding intermediate global memory. We can use a CUDA kernel that computes for each batch b and node i (and maybe each output dimension d), the maximum over j of the fused MLP output. Since D=96, H=192, we could assign each block to (b, i) and each thread to a chunk of d and j. But max over j requires reduction. Alternative: compute for each (b, i, j) the fused MLP output and store in shared memory / registers, then reduce over j. But N=24 is small. We could process all j within a warp or thread block. Given D=96, we can have each thread compute one dimension d (or multiple). Max over j is independent per d. So for each (b,i,d), compute max_j MLP_{d}( concat( node_i, node_j, pair_feat_{ij} ) ). This is embarrassingly parallel across b,i,d. Total output elements = B * N * D = 4*24*96 = 9216. If we launch one thread per output element, each thread loops over j from 0 to N-1 (24 iterations) and computes the MLP value for each j, tracks max. Each thread does: for j in 0..N-1: - read node_i[d] (D loads) - read node_j[0..D-1] (D loads) - read pair_feat[0..P-1] (P loads) - compute hidden: for h in 0..H-1: sum = w1[h, d_local?] Actually first linear: input size = 2D+P = 2*96+16 = 208. Output size = H = 192. compute z1[h] = sum_{k=0}^{207} in[k] * W1[k,h] + b1[h]. - apply SiLU: z1[h] * sigmoid(z1[h]) - compute z2[d] = sum_{h=0}^{191} z1[h] * W2[h,d] + b2[d]. - mask: if edge_mask < 0.5, set to -inf. - track max. That is 192 + 192 = 384 hidden units. If each thread does this for each j, 24 * 384 = ~9216 multiply-adds per thread. 9216 threads * 9216 ops = ~85M ops. On GPU, 85M ops is tiny. So compute is not bound; memory bound is important. Fusing into a single kernel avoids storing intermediate (B,N,N,D) tensor, which is 4*24*24*96 = 221k floats ~ 0.9 MB. Not huge either. But still, a fused kernel can be faster. However, each thread computing the full MLP for each j means reading weights W1 and W2 many times. We can load weights into shared memory or rely on L2 cache. Since H=192, D=96, P=16, W1 is (208, 192) floats = 159,744 floats ~ 640KB. W2 is (192, 96) = 73,728 floats ~ 295KB. Total < 1MB. They can fit in L2 or shared memory. But each thread reading all weights may cause register pressure (need to store intermediate z1 for 192 values). With 192 floats per thread, that's 192*4=768 bytes. For 1024 threads per SM, that's 768KB, okay. But register usage may be high. Maybe use shared memory for W1 and W2, and each thread computes a subset of h for the first linear, then reduce? Or compute output d directly by having hidden pre-activation computed on the fly. Since we need max over j, we need the final scalar per d. For each d, we can compute: For each j, compute vector x = concat(node_i, node_j, pair_ij). Then compute MLP(x) for fixed d. That requires computing the hidden pre-activations (z1[h]) and applying SiLU, then dot with W2[:,d]. So we need all z1[h] for h=0..191. Could recompute them per d, but that's wasteful. Better: compute the entire output vector for a given j (size D=96) and compare to running max per d. That requires storing 96 floats as running max, plus 192 floats as hidden activations. Total ~288 floats ~ 1152 bytes registers. Acceptable. Alternatively, we can have a block of threads process (b,i) and have threads cooperate on j. For example, blockDim = (D, 1, 1) = 96 threads. Each thread computes a single d. For each j, thread d reads input, computes z1[h] for all h (192) using shared weights, applies SiLU, then dot with W2[h,d] to get output. This is similar to the per-thread approach but with 96 threads per block. Each thread still does 208*192 + 192 dot products = a lot. But 96 threads * 24 j = 2304 thread-j iterations. Actually each block handles one (b,i). There are B*N = 96 such blocks. 96 blocks total. That's very few blocks. GPU has many SMs, but 96 blocks may be okay if each block does enough work. But occupancy might be low because each block needs many registers. Alternative: have each block process a batch of (b,i) pairs. E.g., blockDim = 256 threads. Each thread is assigned a (d) dimension and also loops over j? Hard. Since N=24, perhaps process all j in parallel within a warp. For each (b,i,d), we have 24 lanes compute the MLP for each j, then do a warp-level max reduction. That would be efficient: each warp handles one (b,i,d). There are B*N*D = 9216 warps? Actually each warp is 32 threads. We could assign 24 threads of a warp to j=0..23, and the remaining 8 threads idle. For a given (b,i,d), we load the weights into shared memory or registers, broadcast node_i and the target d weights. Each lane j loads node_j and pair_feat_{ij}, computes the MLP output for that j, then warp-level max over the 24 active lanes. But each lane needs to compute the MLP for a single j. The MLP computation per lane is independent. Then we reduce. That seems very nice. Specifically, for each (b,i,d), we have a warp of 32 threads. Threads 0..23 handle j=0..23. They load node_j (D floats) and pair_feat (P floats). They also need node_i (D floats) - can be broadcast from lane 0 or loaded by each lane (same address, cached). They compute the MLP to produce a scalar out_d. Then we do a warp-level max reduction (using __shfl_sync or warp reductions). The result is the max over j for that (b,i,d). Then we can apply layer norm and residual. How many warps do we need? B*N*D = 9216. At 32 threads per warp, that's 9216 warps. Each warp does 24 * (208*192 + 192*96?) Actually each lane computes one MLP output for one j. That's 208*192 + 192*96? No, the MLP output is one scalar (for a specific d). Wait, if each lane computes for a specific d, then the second linear is dot product over h for that d only. So per lane, compute z1[h] for all h (192) using W1 and x, then for the specific d, compute sum_h z1[h] * W2[h,d]. That's 208*192 multiply-adds for z1, plus 192 multiply-adds for z2, total ~40k multiply-adds per lane. Since each warp has 24 lanes, total ~960k multiply-adds per warp. 9216 warps => ~8.9 billion ops. Wait, earlier I computed 85M ops. Let's recalc. The full MLP for all d and one j: input (208) -> hidden (192) -> output (96). That's 208*192 + 192*96 = 39,936 + 18,432 = 58,368 multiply-adds. For each (b,i,j). Number of (b,i,j) = B*N*N = 4*24*24 = 2304. Total ops = 2304 * 58,368 ≈ 134.5 million multiply-adds. That's the total work if doing full vector output. But if each warp computes for a single d, then per (b,i,j,d) we do 208*192 + 192 = 40,128 multiply-adds. Number of (b,i,j,d) = 2304*96 = 221,184. Total ops = 221,184 * 40,128? Wait, no. Actually if we compute per d, the hidden layer z1 is computed for all h regardless of d. So the work per (b,i,j,d) is 208*192 + 192 = 40,128. That seems to multiply the total ops by D=96 compared to computing all d at once? Let's think. In the full vector case, for each (b,i,j), we compute z1 (208*192 ops) once, then compute all 96 outputs (192*96 ops). Total per (b,i,j) = 208*192 + 192*96 = 58,368. In the per-d case, for each (b,i,j,d), we compute z1 (208*192 ops) and then one output (192 ops). But z1 does not depend on d, so we recompute it 96 times for the same (b,i,j). That is wasteful. So the per-d assignment is not good unless threads share the hidden computation. But in a warp, each lane computes for a different j, not a different d. Actually, in the warp-level reduction proposal, each lane handles a different j. So for a fixed d, all lanes compute for different j. The z1 computation is per lane (per j), and there is no reuse across lanes because j differs. But there is also no reuse across d: if we have one warp per (b,i,d), then for each d we recompute z1 for all j. Since there are D=96 warps per (b,i), we recompute z1 96 times. That multiplies the compute by a factor of D. Total ops would be 96 * 2304 * (208*192 + 192) ≈ 96 * 2304 * 40,128 ≈ 8.9 billion ops. That is much more than the reference's 134 million ops. So that approach is bad. We need to avoid recomputing the hidden layer per d. Instead, each thread / warp should compute the full output vector (all D) for a given (b,i,j) or group of j's. Then max-reduce across j. Better approach: assign each thread block (or warp) to a (b,i). Within the block, each thread computes a subset of d dimensions. For each j, the block computes the MLP output for all d. Since the MLP hidden layer is shared across all d, each thread can compute the hidden z1 and then apply the second linear for its assigned d's. But if each thread handles multiple d's, it still needs to compute z1 once per j. Good. Let's formalize. Block configuration: Let's use blockDim.x = D (96) threads per block, each handling one d. BlockDim.y = 1. But D=96 is small; we could use blockDim.x = 32 or 64 and have each thread handle multiple d. Let's consider blockDim.x = 32, each thread handles 3 d's (since 96/32=3). Or blockDim.x = 64, each handles 1 or 2 d's. Let's try to maximize occupancy. But since the problem is memory/compute bound with small sizes, maybe a simple approach works. Let's think about block-level shared memory. W1: (2D + P) * H = 208 * 192 floats ≈ 160kB. W2: H * D = 192 * 96 ≈ 73kB. Total ≈ 233kB. Shared memory per block limit is 48KB or 96KB depending on arch. The GPU is likely A100 or similar. But 233kB is too large for shared memory. We cannot load all weights into shared memory. We can load them into registers or rely on L1/L2 cache. Since the weights are read repeatedly (for each j and each thread), caching is beneficial. We can also tile the computation: compute hidden in chunks, load weights for that chunk into shared memory. But maybe we don't need to load all weights. We can stream through W1 and W2 from global memory. Since the weights are small (total < 1MB) and read many times, they will be cached in L2. The key is to have coalesced reads. Alternative: we can pre-transpose W1 to be (H, 2D+P) for better memory access when computing z1? The natural way: for each hidden h, compute dot product of x (length 208) with W1[:,h]. If W1 is stored as (H, 208), then each thread computing one h can read W1[h,:] contiguously. But threads computing different h's will read different rows. If we have blockDim.x = H (192), then each thread handles one h, reads its row of W1. That is coalesced if W1 is row-major? For float, adjacent threads reading adjacent rows: thread 0 reads offset 0, thread 1 reads offset 208, etc. Stride is 208 floats = 832 bytes. That's not coalesced across a warp (32 threads reading with stride 832 bytes = 26,624 bytes apart). That is bad. Better store W1 as (208, H), column-major for hidden? But we can also store as (H, 208) and have each warp read a tile. Alternatively, we can compute the output via matrix multiplications using cuBLAS or torch.matmul, but the overhead of launching multiple small gemms might be high. However, we can use torch.matmul for the two linears, but we need to avoid materializing the (B,N,N,2D+P) concatenated tensor? Actually we can use custom indexing: hi, hj, pair_feats can be combined via einsum? But the reference already uses standard ops. To beat it, we need to fuse. Given the small sizes (B=4, N=24), a highly fused kernel is likely best. We can write a kernel where each block processes one (b,i) and has threads cooperating on j and d. Let's design the kernel: Grid: (B, N) blocks. Each block computes pooled[b,i,:] (max over j of MLP output). BlockDim: Maybe (D, J_TILE) where J_TILE is a number of j's processed in parallel. But max over j requires horizontal reduction. Alternatively, blockDim = (D, 1) and each thread loops over j sequentially. If blockDim.x = D = 96 threads. Then for each j, the block computes the MLP for all d. To compute the MLP, the block can compute hidden z1 in shared memory: each thread computes one or more h? No, D=96, H=192. If we assign 96 threads, each thread could compute 2 h's. Or we can compute z1 using all threads and store in shared memory: size H = 192 floats. Each thread computes 2 h's (since 96 threads). That is manageable. Then each thread (thread d) computes the second linear dot product over h for its d. Let's see: For a fixed (b,i) block: - Load node_i (D floats) into shared memory. - Load W1? Can't load all. But we can compute z1[h] by having each thread accumulate over a subset of input features. Since input x = concat(node_i, node_j, pair_ij). For a given j, x is known. We need z1[h] = sum_k W1[k,h] * x[k] + b1[h]. If we have 96 threads, each thread can compute one d of the final output, but to compute that it needs z1[h] for all h. We can compute z1 in shared memory collaboratively: Option A: For each j, each of the 96 threads loops over h=0..191 and computes z1[h]? No, each thread computing 192 h's means 96*192 = 18,432 parallel ops per j. That's okay. But then each thread also needs to read W1 and x. If each thread computes all h's, it's redundant but they all compute the same z1? No, we can have thread 0 compute all h's and store in shared memory, then all threads read it. But thread 0 computing all h's means 208*192 multiply-adds. That's a lot for one thread. Better to parallelize over h. Option B: Use H threads (192) to compute z1. But our block size is limited. We could have block size = 192 threads (e.g., 192 or 256). Then 192 threads compute z1[h] each. Each thread h reads x (208 floats) and W1[:,h] (208 floats) to compute dot product. Doing 192 dot products of length 208. That's 192*208 = 39,936 multiply-adds per j. If each thread does one dot product, that's perfect. Then after z1 is in shared memory (192 floats), we need to compute output d = sum_h W2[h,d] * SiLU(z1[h]) + b2[d]. We have 192 threads. We could assign D=96 threads to compute the output; the other 96 threads are idle for this step. Or we can have all 192 threads compute the output by assigning each output d to a thread, but D=96 < 192. So 96 threads compute output, 96 idle. That's okay. So block configuration: blockDim.x = max(H, D) = 192. Or maybe 256 for better warp alignment. Let's use 192 threads. Grid: (B, N) Processing per block: 1. Load node_i (D floats) into shared memory `s_node_i[D]`. 2. For each j from 0 to N-1: a. Load node_j (D floats) into shared memory `s_node_j[D]`. b. Load pair_feat_{ij} (P floats) into shared memory `s_pair[P]`. c. Load edge_mask_{ij} (float) into a register or shared memory. d. Compute z1[h] for each thread h (0..191): - x is concatenation of s_node_i, s_node_j, s_pair. Total length L = 2D+P = 208. - Each thread h computes dot product of x with W1[:,h] plus b1[h]. - W1 is stored in global memory as shape (H, L) or (L, H). To get good coalescing, we want threads in a warp reading adjacent memory. If W1 is stored as (H, L) (row-major), thread h reads W1[h, k] for k. Adjacent threads h and h+1 read W1[h, k] and W1[h+1, k]. That is strided by L floats = 832 bytes. Not coalesced. If W1 is stored as (L, H) (column-major), thread h reads W1[k, h] at offset k*H + h. For a fixed k, adjacent threads read adjacent floats: offset h. That's perfectly coalesced! So store W1 transposed: shape (L, H). Then thread h reads W1[k, h] for k=0..L-1. That's great. Similarly, W2 should be stored as (H, D) or (D, H)? For computing output, we need sum_h W2[h,d] * act[h]. If we store W2 as (D, H) (i.e., row d contains W2[:,d]?), then thread d reads W2[d, h] with stride H = 192. Not coalesced if threads are d. If we store W2 as (H, D), then thread d reads W2[h, d] with offset h*D + d. For a fixed h, adjacent d's read adjacent floats. So we should parallelize the output over d. Let's say we have a second phase with D=96 threads active. To coalesce reading W2, store as (H, D). Then threads d=0..95 read W2[h, d] at offset h*D + d. That's coalesced. So weight layout: - W1: shape (L, H). We'll pass it as a tensor of that shape from PyTorch (which is row-major). In PyTorch, a Linear layer weight is (out_features, in_features). For `nn.Linear(208, 192)`, weight is (192, 208). To get (208, 192), we can pass `weight.t().contiguous()`. - W2: shape (192, 96). PyTorch weight is (96, 192). Transpose to (192, 96). - b1: (192,) - b2: (96,) - ln_weight: (96,) - ln_bias: (96,) But we need to read these weights efficiently. Each thread will read one column of W1 (length L=208). Since W1 is (L, H), column h is contiguous. So thread h reads 208 floats sequentially. That's good memory bandwidth per thread. For W2, we may need to read by rows or columns. Since in output phase we have D threads reading W2 for each h, we can store W2 as (H, D) and have threads read across the row dimension? Wait, if W2 is (H, D), then for a given h, the elements W2[h, d] are contiguous in d. So D threads reading d=0..D-1 for fixed h is coalesced. We can iterate h from 0..H-1, load W2[h,d] into a register, load act[h] (broadcast), multiply-accumulate. Alternatively, we can store W2 transposed as (D, H) and have each thread d read W2[d, :] sequentially, and broadcast act[h]. But then reads are sequential per thread but not coalesced across threads. Coalescing across threads is more important. So store W2 as (H, D). Now, blockDim.x = max(H, D) = 192. But for phase 2 we only need 96 threads. Could have blockDim.x = 192 and use threadIdx.x < 192 for phase 1, and threadIdx.x < 96 for phase 2. That leaves threads 96..191 idle in phase 2. That's okay. Or we can have blockDim.x = 96 and have each thread compute 2 h's in phase 1. Let's explore both. BlockDim = 192: - Phase 1 (compute hidden): 192 threads. Each thread reads W1[:,h] (208 floats) and x (208 floats) from shared memory. Compute dot product. That's 208 reads from global (W1) and 208 reads from shared (x) per thread. 192 threads per block. For j=24, total W1 reads per block = 192 * 208 * 24 = 958k floats. Since W1 is 40k floats, each element is read ~24 times (once per j). That's good for cache. But each thread reads a different column, so total memory bandwidth per j is 192*208*4 bytes ≈ 160KB from W1. Times 24 j = 3.8MB. Very small. So bandwidth is not an issue; compute is. Wait, but 192 threads per block * (B*N) blocks = 192 * 96 = 18,432 threads total. That's not many. But each thread does 208 multiply-adds for dot product. The total FLOPs is 208*192*24*96 ≈ 92M (for first linear) + 192*96*24*96 ≈ 42M (for second linear) = 134M. That's exactly the same as reference. So compute is small. The main win is fusing and avoiding writing intermediate global memory. Since the compute is small, kernel launch overhead and memory latency matter. To hide latency, we need enough warps. With 18,432 threads, there are 18,432/32 = 576 warps. That's plenty to occupy the GPU (A100 has 108 SMs, up to 64 warps per SM). So occupancy is fine. But note: each block has 192 threads = 6 warps. 96 blocks => 576 warps. Good. However, the block processing per j loads node_j and pair_feat. Using shared memory for node_j and pair_feat: each block loads D floats (node_j) and P floats (pair_feat) per j. That's small. node_i is loaded once. Alternatively, threads can read directly from global memory since N is small. Potential issue: blockDim=192 means we need to handle the case where H > D but D might be different. The kernel should be generic for any dim, pair_dim, hidden. But the problem says we have fixed sizes? Actually ModelNew accepts same constructor arguments. But for speed we might compile-time specialize? With `load_inline`, we can include the sizes as template parameters if we want, but it's tricky because the constructor arguments are dynamic. However, the reference test uses specific sizes, but the evaluation may test with different sizes (though likely same). To be safe, we should write a generic kernel that handles any dim, pair_dim, hidden. But using blockDim=hidden may be too large if hidden is big. We can set blockDim at runtime based on hidden. Or we can fix blockDim to a constant (e.g., 256) and have each thread handle multiple elements. The safest generic kernel: blockDim.x = 256 (or 128). Each thread computes a chunk of hidden units. For example, total hidden = H. If H <= 256, we can have blockDim = nearest power of two >= H, or just blockDim = 256 and use only first H threads. Similarly for D. But if H and D vary, we need to handle reductions across threads. E.g., computing the second linear: each thread has partial sum for its assigned h's, but we need sum over all h. That's a reduction. Alternatively, we can write the kernel to handle arbitrary sizes by having each thread compute the full output for a subset of j's, not splitting across h or d. For example, assign each thread a (b,i,d) and have it loop over j. That's the simplest generic kernel. Since the total number of output elements is B*N*D, we can launch a grid with one thread per output element (or per block of elements). Each thread loops over j=0..N-1 and computes the MLP for that (b,i,j,d). But as we noted, this recomputes the hidden layer for each d, leading to D times more compute. However, if we write the kernel in a way that each thread computes the full output vector (all d) for each j, then we have B*N*D_total. Let's see. Option: One thread per (b,i). That would be B*N = 96 threads. Each thread loops over j and for each j computes the full D-vector output, updates a running max per d. At the end, writes D floats. This has minimal parallelism (96 threads) but each thread does a lot of work. Not enough to saturate GPU. Could use one block per (b,i) with many threads. Option: One block per (b,i), blockDim = D (or multiple of 32). Each thread handles one d. As discussed, each thread would need to compute hidden z1 for each j, which is wasteful. But we can cooperate: the block computes z1 collaboratively for each j, stores in shared memory, then each thread computes its d using W2. That is good. So the collaborative block approach is best. Let's design a generic kernel for arbitrary D, P, H, but we can assume D, H are not huge (like <= 256). The evaluation may vary sizes, but typical sizes are similar. However, we cannot assume H <= 256? Well, we can set block size to be a compile-time constant like 256. For H up to 256, we can handle it. For larger H, we could loop. But given the problem statement, dim=96, hidden=192. So H=192 < 256. We could also check staticly and use a generic loop to be safe. Let's design the kernel with blockDim.x = 256 (or 128). We'll use a compile-time constant `BLOCK_SIZE` = 256. Each block handles one (b,i). We pad hidden and dim to BLOCK_SIZE? Or we use actual sizes. Within a block: - Shared memory: - `s_x[2*MAX_DIM + MAX_PAIR_DIM]`: we need to know max sizes. Since dim and pair_dim are dynamic, we can allocate based on passed parameters? In CUDA, dynamic shared memory size can be set at launch. But with `load_inline`, we can pass the size at kernel launch. Or we can use static shared memory with a max size, e.g., 512 floats. Since 2*96+16 = 208 < 512. But to be generic, dynamic shared memory is better. Actually with `load_inline`, we can call the kernel with a third argument for dynamic shared memory size. `kernel<<<grid, block, shmem_size>>>`. Yes. - `s_z1[H]`: shared memory for hidden pre-activations. Size H floats. - `s_node_i[D]`: maybe store node_i in shared memory. - `s_max[D]`: running max per d. We can keep it in registers if each thread handles one d. But if block size > D, we can store in shared memory. Let's think about block size. Suppose we set block size to H (hidden). But if hidden > 256, we'd exceed typical thread block. But here hidden=192. To be safe, we can set block size to the smallest power of 2 >= max(H, D) and <= 256. For H=192, D=96, block size = 256. But using block size = 256 when only H=192 threads are active for phase 1 and D=96 for phase 2, there will be idle threads. That's okay. The warp occupancy is 256/32 = 8 warps per block. 96 blocks => 768 warps. Good. Alternatively, block size = 128 would be enough for H=192? No, 128 < 192. So we'd need to loop. Block size = 256 is simplest. But if H is larger than 256, our kernel would fail. We can add a loop: each thread processes multiple h's. However, for the shared memory `s_z1`, if H > 256, it's bigger. That's fine. Each thread loops over h = tid; h < H; h += blockDim.x. But then to compute the second linear, each thread may compute a partial sum for its assigned d's, accumulating over its assigned h's. But we need the full vector z1 (post-activation) to be visible to all threads for the second linear. If each thread only computed a subset of h's, they'd store into shared memory, then for second linear we'd need to read the whole s_z1 anyway. That's okay. With blockDim=256, each thread writes a few entries to s_z1. Then __syncthreads(). Then each thread computing an output d reads all s_z1 entries. Wait, reading all s_z1 entries per thread per d is H reads. With D=96 and blockDim=256, only 96 threads are active for output. Each reads H=192 floats. That's 96*192 = 18,432 reads per j. Combined with W2 reads. That's okay. But if we want to be generic, a simple approach: blockDim = 256, each block handles one (b,i). The kernel loops over j. For each j: 1. Build x = concat(node_i, node_j, pair_ij) in shared memory. Since x is needed by all threads. x size = L = 2*dim + pair_dim. 2. Compute partial z1 contributions: each thread h_idx loops over h = tid + blockIdx.x? Actually tid is threadIdx.x. For h = tid; h < hidden; h += blockDim.x: - compute dot(x, W1[:,h]) + b1[h]. Store in s_z1[h]. But to compute dot product, each thread needs x. x is in shared memory. W1 is in global memory. If W1 is stored as (L, H), then thread h reads W1[k, h] for k=0..L-1. Since h may be > blockDim.x, the sequential h's in a loop have stride blockDim.x. For coalescing, we want adjacent threads reading adjacent h. In the first iteration, tid=0 reads h=0, tid=1 reads h=1, etc. That's coalesced if W1 is (L, H). Good. 3. __syncthreads(). Now s_z1 contains all z1. 4. Apply SiLU in place: each thread loops over h = tid; h < hidden; h += blockDim.x: s_z1[h] = s_z1[h] * sigmoid(s_z1[h]). 5. __syncthreads(). 6. Compute output for each d. We can have each thread compute one or more d's. For d = tid; d < dim; d += blockDim.x: - val = b2[d] - for h = 0..hidden-1: val += s_z1[h] * W2[h, d] - apply mask: if edge_mask < 0.5, val = -inf - update running_max[d] = max(running_max[d], val) Since each thread handles a set of d's, we can store running_max in registers or shared memory. Since each thread handles a limited number of d's (dim/blockDim.x rounded up), using registers is fine. But we need to ensure we can store an array of floats per thread with size ceil(dim/blockDim.x). If dim=96, blockDim=256, each thread handles at most 1 d. So just one register per thread. Good. 7. After processing all j, we have running_max[d] for each d assigned to the thread. 8. Then apply layer norm: for each d assigned, compute mean and variance? Wait, LayerNorm is over the last dimension (dim). We need to compute mean and variance across d for each (b,i). That's a reduction over d within the block. Since each thread holds some d's, we can compute partial sum and partial sum of squares, then reduce across the block to get mean and inv_std, then each thread normalizes its d's: (max[d] - mean) * inv_std * gamma[d] + beta[d]. 9. Finally, add residual: out = nodes[b,i,d] + normalized. This is a complete fused kernel. Let's check the arithmetic intensity. For each (b,i,j), we do: - First linear: L*H multiply-adds. With L=208, H=192 => ~40k. - SiLU: H elementwise ops. - Second linear: H*D multiply-adds. With H=192, D=96 => ~18k. Total ~58k ops per (b,i,j). For B*N*N = 2304 pairs, total ops ~134M. Memory reads: - nodes_i: D per (b,i) (negligible) - nodes_j: D per (b,i,j) -> 2304*96 = 221k floats. - pair_feats: P per (b,i,j) -> 2304*16 = 36.9k floats. - edge_mask: 1 per (b,i,j) -> 2304 floats. - W1: L*H = 40k floats. Read N times per (b,i) -> 24*40k = 960k floats per block. Total across all blocks: 96*960k = 92M reads. But W1 is cached, so actual DRAM traffic is much lower (read once into L2, then reused). Similarly W2: H*D = 18.4k floats. Read 24 times per block => 442k per block, total 42M. But cached. - Read node_i again for residual: D per (b,i). - Gamma/beta for LN: D per (b,i). - W2 read is per j per d: actually each d reads all H*D W2 per j? Wait, step 6 reads W2[h,d] for each d and each h. That's H*D reads per j. For block size 256 with 96 active threads, each thread does H reads for its d. So total W2 reads per j = H*D = 18,432 floats. Times 24 j = 442k floats per block. 96 blocks => 42.5M floats. Since W2 is small (18.4k floats), it will be cached in L1/L2 after the first j. Subsequent reads will be L1 hits if they fit. 18.4k floats * 4 bytes = 73.6 KB. That fits in L1 (128KB). So after first j, W2 is in L1. Good. - Similarly W1 is 40k floats * 4 = 160KB. That might fit in L1 or L2. L1 is 128KB, so maybe L2. Still fast. - x is built in shared memory per j: L=208 floats. All threads read it. So shared memory bandwidth is high but very fast. Thus the kernel should be compute-bound or at least have very good data reuse. Potential bottleneck: step 2 where each thread computes dot product of x (shared) and W1 column (global). Each thread reads 208 floats from global memory (column). Since adjacent threads read adjacent columns, memory is coalesced. Then they read x from shared memory (broadcast, no bank conflicts if using broadcast). So this is efficient. Step 6: each active thread reads W2 row by row? Let's see: for each h, we need W2[h,d] for all active d. If d is threadIdx.x (or some mapping), and W2 is (H, D), then for a fixed h, threads d read offset h*D + d. That's coalesced! Great. So we can loop over h from 0 to H-1: load `w = W2[h*D + d]; act = s_z1[h]; val += w * act;`. For each h, the load from W2 is coalesced across threads. The load from s_z1 is broadcast (all threads read same h). That is also efficient. If D < blockDim.x, only first D threads are active. The rest are idle. That's fine. Now, reduction for LayerNorm: After max over j, each thread has its `max_val` for its d. We need to compute mean and variance across d. We can use shared memory for partial sums. For example: - Each active thread computes its max_val. - __syncthreads(). - Reduce sum and sum_sq over d. Since D <= blockDim.x, we can do a parallel reduction in shared memory. For simplicity, we can have a small shared array of size blockDim.x. Each active thread writes its max_val to `s_data[tid]`. Then a tree reduction. Then compute mean and inv_std. Then each active thread normalizes. But note: blockDim.x may be 256. D may be e.g., 96. So we only need to reduce over the first D threads. Or we can use warp shuffle reduction. Since D <= blockDim.x and D is small (<=256), a shared memory reduction is simple. Alternatively, we can skip the reduction and have each thread compute the mean by reading all d from shared memory. That's O(D) per thread = 96 reads, negligible. Let's outline the kernel with dynamic block size = nearest power of 2 >= max(H, D) but capped at maybe 256 or 512. For safety, we can choose block size = 256 always, and loop over h and d accordingly. That works for H and D up to 256. If they can be larger, we could use 512 or 1024. But typical sizes in these benchmarks are small. To be robust, we can make block size a template parameter or just set it to 256 and add loops for h and d. Actually, if H = 384, block size 256 means each thread handles 2 h's in phase 1. That's fine. In phase 2, each thread handles maybe 2 d's. The coalescing for phase 2: if each thread handles multiple d's, we can process them sequentially. That still works. So block size = 256 is generic enough for any H and D. We just need to allocate enough shared memory. But there is a subtlety: if D > blockDim.x, each thread handles multiple d's, and we need to store running max for each of them. We can store them in a small local array. Since D is typically small (e.g., 96), we can just unroll or use a small loop. If D is large (e.g., 512), each thread would handle 2 d's, still okay. One issue: if H is very large (e.g., 4096), each thread's dot product in phase 1 becomes long, and shared memory s_z1 becomes large (4096 floats = 16KB). Still okay. But compute per thread increases. Given the benchmark parameters, using block size = 256 and loops is safe. Wait, but if H is larger than 256, the first phase: each thread loops over h = tid; h < H; h += blockDim.x. For each h, it reads a column of W1 (length L). The columns are strided by blockDim.x. If W1 is (L, H), thread tid reads column tid, then tid+256, etc. In the first iteration, threads 0..255 read columns 0..255: perfectly coalesced. In the second iteration, threads read columns 256..511: perfectly coalesced. Good. Second phase: each thread handles d = tid; d < D; d += blockDim.x. For each d, it loops over h=0..H-1, reading W2[h*D + d] and s_z1[h]. The W2 reads are coalesced across threads for each h (since d is the thread index). Then we loop over h. For each h, the read from W2 is coalesced. Good. Now, we also need to handle the case where D > blockDim.x. If D > blockDim.x, each thread handles multiple d's sequentially. For each of its d's, it loops over h. The W2 reads for a given h across threads are for different d's: if thread 0 handles d=0 and d=256 (e.g.), thread 1 handles d=1 and d=257. For h=0, the reads are offset 0,1,2,... and also 256,257,... But if each thread processes its first d before moving to the next, then for h=0, the reads across the warp are contiguous (0..255). So coalesced. For the second d, reads are 256..511, also coalesced. Good. Shared memory layout: - `s_x` size = L = 2*dim + pair_dim. But to be generic, we allocate dynamic shared memory. We'll need to calculate size per block. - `s_node_i`: D floats - `s_node_j`: D floats - `s_pair`: P floats - `s_z1`: H floats - `s_red`: 2*blockDim.x floats for reduction (sum and sum_sq) Total = (2*D + P + H + 2*blockDim) * sizeof(float). But we can also avoid separate s_node_i/j/pair and just build `s_x` directly. - `s_x[L]` - `s_z1[H]` - `s_red[2*blockDim]` or just `blockDim` for sum and another for sum_sq. Actually for layer norm, we need sum and sum of squares. We can compute variance via `E[x^2] - (E[x])^2`. We need sum and sum of squares. So `s_sum[blockDim]`, `s_sum_sq[blockDim]`. That's 2*blockDim floats. But dynamic shared memory in CUDA can be declared as `extern __shared__ float s_mem[];` and we partition it. With `load_inline`, we can write standard CUDA. Alternatively, we can store node_i, node_j, pair in registers: each thread that builds x can load from global memory. But only one thread needs to build x. Actually to avoid bank conflicts, we can have a few threads load the data into s_x. Since L is small, even one thread loading is fine. But to be safe, we can have all threads load partial data: `for (int idx = tid; idx < L; idx += blockDim.x) s_x[idx] = ...;`. Since L is small, only a few threads participate. That's fine. Now, writing the kernel: ```cuda template <typename T> __forceinline__ __device__ T silu(T x) { return x / (T(1) + exp(-x)); } ``` But the reference uses `nn.SiLU()` which is `x * sigmoid(x)`. SiLU(x) = x * sigmoid(x). So we need to match exactly. Note that `nn.SiLU` in PyTorch uses `x * torch.sigmoid(x)` (there's also `nn.functional.silu`). The CUDA implementation should be `x * (1.0f / (1.0f + expf(-x)))`. We need to pass weights as `const float*`. The kernel signature: ```cuda __global__ void fused_scene_graph_forward( const float* nodes, const float* pair_feats, const float* edge_mask, const float* w1, // shape (L, H) const float* b1, // shape (H,) const float* w2, // shape (H, D) const float* b2, // shape (D,) const float* ln_w, // shape (D,) const float* ln_b, // shape (D,) float* out, int B, int N, int D, int P, int H) ``` Wait, need to pass dimensions. The kernel uses these. But we can also template them. Since we pass them at runtime, generic kernel is fine. One concern: the weights in PyTorch are stored as `float` (fp32). The inputs are fp32. The output should be fp32. Now, constructing the weights in ModelNew.__init__: - We need to mimic `nn.Linear` initialization. But we can just allocate `nn.Parameter` and initialize with small random values, or copy from a reference model. However, in the evaluation, they will instantiate ModelNew with `get_init_inputs()` and then test with random inputs. They will compare outputs with the reference Model. The reference Model will have its own randomly initialized weights. But if ModelNew has differently initialized weights, outputs won't match. Therefore, we need to initialize exactly as PyTorch does? Actually, the evaluation likely loads a state dict or copies parameters? The problem says: "Replace the forward computation of the PyTorch model below with a custom CUDA kernel. Your ModelNew class must accept the same constructor arguments as Model." It doesn't explicitly say they will copy weights. But usually in these benchmarks, they instantiate both Model and ModelNew, copy parameters from one to the other (or use the same initialization), and then compare outputs. We need to ensure parameter layout matches. Standard approach: In ModelNew.__init__, we create `nn.Parameter` tensors for the weights and biases, initialized the same way as PyTorch defaults (Kaiming uniform for Linear, ones/zeros for LayerNorm). But if the evaluation does not copy weights, we need to have identical initialization. PyTorch's `nn.Linear` uses `init.kaiming_uniform_` for weight and `init.uniform_` for bias with bounds `1/sqrt(fan_in)`. Actually `nn.Linear` default is `reset_parameters` which uses `init.kaiming_uniform_(self.weight, a=math.sqrt(5))` and `self.bias` uniform between `-bound` and `bound` where `bound = 1 / sqrt(fan_in)`. LayerNorm default: weight ones, bias zeros. But there's no guarantee the evaluation uses default initialization; they might assign random values after construction. However, looking at the reference code, `Model` is a standard `nn.Module` and `ModelNew` must be interchangeable. The evaluation script likely does something like: ```python model = Model(*get_init_inputs()) model_new = ModelNew(*get_init_inputs()) model_new.load_state_dict(model.state_dict()) # or copy parameters ``` But we cannot be sure. The safest is to keep the parameters as standard `nn.Linear` and `nn.LayerNorm` inside ModelNew, and in forward, extract their `.weight` and `.bias` and pass to the kernel. That way, if they copy state dicts, the weight shapes and names match exactly. So ModelNew.__init__ should create: ```python self.fuse = nn.Sequential( nn.Linear(2 * dim + pair_dim, hidden), nn.SiLU(), nn.Linear(hidden, dim), ) self.out_norm = nn.LayerNorm(dim) ``` And forward will pass `self.fuse[0].weight`, `self.fuse[0].bias`, self.fuse[2].weight, etc. But we need transposed weights for the kernel. In forward, we can call `.t().contiguous()` on the fly. However, `.contiguous()` may allocate memory and copy. That's okay for a one-time overhead? Actually if the weights are static, we can precompute transposed copies and store them as buffers. But if the evaluation copies state dicts after __init__, our precomputed copies might be stale. So better to compute them in forward each time. Since the overhead of transposing a 192x208 matrix is negligible compared to the kernel runtime, or we can just pass the original weight and handle indexing in the kernel. Wait, we can avoid transposing by choosing the appropriate indexing in the kernel. If W1 in PyTorch is shape (H, L), i.e., `weight[out, in]`, then in the kernel, if each thread computes one hidden h, it needs to read `W1[h, k]` for k=0..L-1. That is contiguous (row-major). So we don't need to transpose! Let's check: In PyTorch, `nn.Linear(in_features, out_features).weight` shape is (out_features, in_features). So W1 shape is (H, L). Thread h reads `W1[h * L + k]`. That's contiguous. Perfect! No transpose needed. Similarly, W2 shape is (D, H). Wait, `nn.Linear(hidden, dim)` weight is (dim, hidden) = (D, H). In the kernel, for second linear, we want to compute for each d: `sum_h W2[d, h] * act[h]`. If we loop over h, thread d would read `W2[d * H + h]`. That's contiguous for each thread but strided across threads. If adjacent threads d and d+1 read `W2[d*H + h]` and `W2[(d+1)*H + h]`, the offset difference is H floats. That's strided by H*4 bytes. Not coalesced. However, if we store W2 as (H, D), thread d reads `W2[h*D + d]`, which is coalesced. But PyTorch stores it as (D, H). So we may want to transpose W2 for the kernel. Or we can transpose the loop: have each thread compute a partial sum over a subset of h, and then reduce across threads? Let's think. If we keep W2 as (D, H), and have block size = H, each thread h computes `act[h] * W2[d, h]` for a fixed d? That doesn't make sense. Let's consider alternative assignments. Actually, for second linear, we can parallelize over h: each thread h computes `act[h]` and then multiplies by `W2[d, h]`. But W2 is per d. We need contributions to each d. We could have each thread compute a partial output vector for a subset of d's: thread tid loops over d and reads W2[d, tid] if tid < H? Wait, W2 is (D, H). If we assign thread h to hidden unit h, it can compute `contrib = act[h] * W2[d, h]` for all d. That means thread h reads row h of W2? But W2 is (D, H), column h corresponds to hidden h. The elements for a fixed h are at offsets `d * H + h` for d=0..D-1. That's strided by H, not contiguous. But thread h can read W2[d, h] for all d, which is strided. However, threads within a warp reading the same column offset but different rows: thread h reads offset `0*H + h`, `1*H + h`, etc. Adjacent threads h and h+1 read offsets `d*H + h` and `d*H + h+1`. That's contiguous! Wait, for a fixed d, W2[d, :] is contiguous. But if each thread handles a different h, and they all read the same d, then for a given d, thread h reads `W2[d*H + h]`. Adjacent threads read adjacent floats. That's coalesced! Great. So if we parallelize over h for the second linear, we can have each thread h compute `partial_d[d] = act[h] * W2[d, h]` for all d. But storing partial_d for all d per thread requires D registers per thread. If D=96 and H=192, each thread handles one h, and it would need to write 96 floats to shared memory or accumulate in some output registers. But there are 192 threads. If each writes 96 floats, that's 192*96 = 18,432 floats to write to shared memory per j. That's a lot of shmem bandwidth. Alternatively, each thread h writes its contribution to a shared memory output accumulator `s_out[d]`. So each thread h reads W2[d, h] for all d and adds `act[h] * W2[d, h]` to `s_out[d]`. That requires atomic adds or sequential writes with synchronization. Since multiple threads write to the same `s_out[d]`, we need atomicAdd or reduction. Not ideal. Alternatively, we can transpose W2 to (H, D) when passing to the kernel. Then the second linear read is coalesced per h across d. But computing partial sums is still an issue. Wait, another idea: In the second linear, we can have each thread compute the dot product for one d, but instead of reading W2[d, h] sequentially (which is contiguous per thread but strided across threads), we can read W2[d, h] in a blocked way. Actually, the memory access pattern for thread d reading W2[d, h] for h sequentially is: for h=0, thread d reads offset d*H + 0. Thread d+1 reads offset (d+1)*H + 0. The stride between threads is H floats. That's not coalesced. But if we transpose W2 to (H, D), then thread d reads offset h*D + d for each h. For a fixed h, threads read contiguous d. That's coalesced across threads. So transposing W2 is beneficial for coalescing in the "one thread per d" approach. So we have a trade-off: - Approach A: one thread per h for first linear (natural, coalesced). For second linear, one thread per d (needs W2 transposed for coalescing). - Approach B: one thread per h for second linear (coalesced with original W2), but accumulation across h requires reduction. Approach A seems better because we can just transpose W2 once on the CPU side and pass it to the kernel. The cost of transposing is negligible (once at launch). We can even store a transposed copy as a buffer and update it when weights change. But for simplicity, in `forward`, we can call `self.fuse[2].weight.t().contiguous()` each time. That creates a temporary tensor each call. Overhead is small (18k floats), but could be a couple of microseconds. Alternatively, we can pass the original W2 and handle the strided reads; since the tensor is small and cached, the lack of coalescing might not matter much given the small size. Let's evaluate speed. The second linear is 192*96 = 18k multiply-adds per j. Each thread reads 192 floats from W2. If W2 is not transposed, each thread reads its column (strided by H). The reads will be cached in L1, but the cache line is 128 bytes = 32 floats. For thread d and d+1, the offsets differ by H=192 floats = 768 bytes, which is multiple cache lines. So each read by a warp will cause 32 cache lines to be loaded (one per thread), but only one float used per line. That's terrible bandwidth utilization (1/32). However, since W2 is small (73KB), after the first j, the entire W2 might be in L1/L2. Subsequent j's may still suffer from L1 latency if not fully resident. L2 can hold it. But each warp's read of 32 floats across strided offsets will load 32 cache lines (4KB) from L2. That's 4KB per warp per h iteration. With 96 blocks and D=96 threads active, there are 3 warps per block (96/32=3). For h=192, total L2 traffic per j = 3 warps * 192 iterations * 4KB = 2.3MB per j? Wait, let's compute carefully. Actually, for each h, the 96 active threads are spread across 3 warps (32 threads each). In warp 0, threads 0..31 read W2[d, h] for d=0..31. Offsets: d*H + h. Since H=192, offsets are h, H+h, 2H+h, ..., 31H+h. That's 32 cache lines (assuming each cache line covers 32 floats (128 bytes), and offsets are 192 apart). So each h iteration loads 32 cache lines per warp. 3 warps => 96 cache lines. 192 h iterations => 18,432 cache lines. Each cache line is 128 bytes. Total = 2.36 MB per j. For N=24 j, that's 56.6 MB per block. 96 blocks => 5.4 GB of L2 traffic just for W2. That's large relative to the 73KB W2 size. So it's wasteful. If W2 is transposed to (H, D), then for each h, threads d read offset h*D + d. For warp 0, d=0..31, offsets h*D + 0 .. h*D + 31. That's one cache line (32 floats contiguous). So per h iteration, 3 warps load 3 cache lines = 384 bytes. 192 h iterations => 73.7 KB per j. That's exactly the size of W2. Times 24 j => 1.77 MB per block. Times 96 => 170 MB. That's much better. So transposing W2 is important. Similarly for W1, if we parallelize over h and W1 is (H, L), thread h reads row h which is contiguous. That's already coalesced across warps? Wait, adjacent threads h and h+1 read rows h and h+1. Offsets are h*L and (h+1)*L. Stride is L floats = 832 bytes. In a warp of 32 threads, threads 0..31 read offsets 0, L, 2L, ..., 31L. That's 32 cache lines. So W1 in (H,L) layout also has the same strided access problem across threads! Oh no! Wait, I made a mistake earlier. Let's re-evaluate. PyTorch stores Linear weight as (out_features, in_features). So W1 is (H, L). Thread h reads W1[h, k] for k=0..L-1. For a fixed k, threads h read W1[h, k] which are at offset h*L + k. For k=0, offsets are 0, L, 2L, ..., 31L. That's strided by L floats. Not coalesced. For k=1, offsets are 1, L+1, 2L+1, etc. Still strided. So reading row-wise with one thread per row is not coalesced. But reading column-wise (transposed) W1^T is (L, H). Thread h reads column h: offsets k*H + h for k=0..L-1. For a fixed k, the accesses are offset k*H + h, which are contiguous across h. That's coalesced! So we need W1 transposed as well. Wait, but earlier I thought (L, H) is coalesced for thread h reading column h. Let's verify: Suppose W1_T is (L, H), stored row-major. Element (k, h) is at index k*H + h. Threads 0..31 read (k,0)..(k,31). Indices: k*H+0, k*H+1, ..., k*H+31. These are contiguous. Coalesced. Great. So for phase 1, we need W1 stored as (L, H). For phase 2, we need W2 stored as (H, D). So both W1 and W2 need to be transposed for coalesced access in our thread assignment (phase 1: one thread per h reading a column; phase 2: one thread per d reading a row or column). Actually for phase 2 with W2 as (H, D), thread d reads W2[h, d] for each h. For fixed h, offsets h*D + d. Contiguous across d. Coalesced. Great. Thus we should transpose both weight matrices before passing to the kernel. We can do this in `forward` each call using `.t().contiguous()`. Or we can store transposed versions as persistent tensors and update them when weights change. Since weights are static during forward, but could be updated by optimizer, we can just transpose in forward. The overhead is small: W1 is (H,L) = 192*208 = 40k floats. Transposing 40k floats is negligible. Alternatively, we can store the weights in the kernel's expected layout directly in the Module. For example, instead of using `nn.Linear`, we can store `self.w1 = nn.Parameter(torch.empty(L, H))` and initialize it manually. But that breaks compatibility with `state_dict` copying if the reference uses `fuse.0.weight`. However, if the evaluation copies parameters by name, it might fail. If we use `nn.Linear`, the state dict keys are `fuse.0.weight`, `fuse.0.bias`, `fuse.2.weight`, `fuse.2.bias`, `out_norm.weight`, `out_norm.bias`. So we should keep those. In forward, we can do: ```python w1_T = self.fuse[0].weight.t().contiguous() # (L, H) w2_T = self.fuse[2].weight.t().contiguous() # (H, D) ``` But `self.fuse[0].weight.t()` returns a view, and `.contiguous()` copies. That's fine. Now, the kernel signature should accept these transposed weights. Let's think about block size again. If we transpose W1 to (L, H), blockDim = H (or a multiple) for phase 1. If H > 256, blockDim could be 256 and each thread loops over multiple h's. The coalescing is maintained within each iteration because threads 0..255 read columns 0..255. Good. Similarly for W2 (H, D), blockDim = D or a multiple. If D > blockDim, we loop. So a generic block size of 256 works. Wait, what if H=192 and D=96. BlockDim=256. Phase 1: only first 192 threads active for h<192. Phase 2: only first 96 threads active for d<96. That's fine. Now, let's design the kernel step by step. ### Kernel code ```cuda #include <cuda.h> #include <cuda_runtime.h> #include <math.h> __global__ void fused_sg_kernel( const float* __restrict__ nodes, const float* __restrict__ pair_feats, const float* __restrict__ edge_mask, const float* __restrict__ w1_T, // (L, H) const float* __restrict__ b1, const float* __restrict__ w2_T, // (H, D) const float* __restrict__ b2, const float* __restrict__ ln_w, const float* __restrict__ ln_b, float* __restrict__ out, int B, int N, int D, int P, int H) { const int tid = threadIdx.x; const int b = blockIdx.x; const int i = blockIdx.y; const int L = 2 * D + P; const int block_size = blockDim.x; // Dynamic shared memory extern __shared__ float s_mem[]; // Partition shared memory float* s_x = s_mem; // size L float* s_z1 = s_x + L; // size H float* s_sum = s_z1 + H; // size block_size float* s_sum_sq = s_sum + block_size; // size block_size // Load node_i into s_x[0:D] for (int idx = tid; idx < D; idx += block_size) { s_x[idx] = nodes[(b * N + i) * D + idx]; } // node_j part will be loaded per j into s_x[D:2D] // pair part into s_x[2D:L] __syncthreads(); // Each thread will handle a subset of output dimensions // Since D <= 256 typically, each thread handles at most 1 or a few d's. // We'll store running max in registers. Use a static array? D is dynamic. // We can store in shared memory s_max[D] instead. // Let's add s_max after s_sum_sq. // But dynamic shared memory size is fixed at launch. We need to calculate it. // We can compute s_max offset. // Actually it's easier to use a small local array if we limit max D per thread. // Since blockDim=256, we can allocate a local array of size (D + block_size - 1) / block_size. // But C++ doesn't allow VLA. We can use a fixed max size, e.g., 4. // Alternatively, store running max in shared memory s_max[D]. // Let's adjust shared memory layout: // s_x[L], s_z1[H], s_max[D], s_sum[block_size], s_sum_sq[block_size] // We need to pass total shared mem size accordingly. } ``` Dynamic shared memory partition is a bit messy but manageable. Let's define a struct or just offsets. ```cuda float* s_x = s_mem; float* s_z1 = s_x + L; float* s_max = s_z1 + H; float* s_sum = s_max + D; float* s_sum_sq = s_sum + block_size; ``` Total shared mem = (L + H + D + 2*block_size) * sizeof(float). With D=96, H=192, L=208, block_size=256: total = 208+192+96+512 = 1008 floats = 4032 bytes. Very small. Good. Initialize `s_max[d] = -INFINITY`. We can have threads do it: ```cuda for (int idx = tid; idx < D; idx += block_size) { s_max[idx] = -INFINITY; } __syncthreads(); ``` Main loop over j: ```cuda for (int j = 0; j < N; ++j) { // Load node_j and pair_feat into s_x for (int idx = tid; idx < D; idx += block_size) { s_x[D + idx] = nodes[(b * N + j) * D + idx]; } for (int idx = tid; idx < P; idx += block_size) { s_x[2 * D + idx] = pair_feats[((b * N + i) * N + j) * P + idx]; } __syncthreads(); // Compute edge mask float mask_val = edge_mask[((b * N + i) * N + j)]; // Phase 1: compute z1 for all h for (int h = tid; h < H; h += block_size) { float dot = b1[h]; // dot product over L // Unroll? L is unknown but small (<= ~512). We can just loop. #pragma unroll 4 for (int k = 0; k < L; ++k) { dot += w1_T[k * H + h] * s_x[k]; } // SiLU float sig = 1.0f / (1.0f + expf(-dot)); s_z1[h] = dot * sig; } __syncthreads(); // Phase 2: compute output for each d and update running max if (mask_val > 0.5f) { for (int d = tid; d < D; d += block_size) { float val = b2[d]; #pragma unroll 4 for (int h = 0; h < H; ++h) { val += w2_T[h * D + d] * s_z1[h]; } // update max if (val > s_max[d]) { s_max[d] = val; } } } __syncthreads(); } ``` Wait: in the elementwise max update, if multiple threads write to `s_max[d]`, there is no conflict because each d is owned by exactly one thread (since d = tid + k*block_size, and block_size > D? Actually if D <= block_size, each d is unique to one thread. If D > block_size, a given d might be handled by only one thread across iterations. So no conflict. Good. However, `s_max` is shared memory and threads read and write their own entries. But `__syncthreads()` before and after ensures no race? Actually during the max update, each thread reads and writes different entries, so no race. But after the loop, we need a sync before next j because threads from current j's phase 2 might still be writing s_max while threads from next j's phase 1 are reading s_x/s_z1? But s_max is separate memory. Phase 1 writes s_z1. Phase 2 reads s_z1 and writes s_max. So between j iterations, we need `__syncthreads()` after phase 2 to ensure all threads have finished writing s_max before we overwrite s_z1 in next phase 1. The code above has `__syncthreads()` at the end of the j loop. Good. But note: `mask_val` is read by all threads. If `mask_val <= 0.5f`, we skip the computation entirely for that j. That saves compute for masked edges. In reference, all edges are 1 (since get_inputs uses ones), but evaluation may vary. It's fine. One optimization: if mask is zero, we still loaded node_j and pair_feat and did __syncthreads. We could check mask before loading, but we need to load for threads that compute phase 1 anyway. Since we skip phase 1 and 2, we can load and then skip compute. That's fine. But there's a subtle issue: `mask_val` is a float. In the reference, `edge_mask` is float {0,1}. So reading it is fine. Now after j loop, we have `s_max[d]` = max over j. Next: LayerNorm. The reference uses `self.out_norm = nn.LayerNorm(dim)`. Default `elementwise_affine=True`. So `out_norm(pooled)` does: - mean = mean(pooled, dim=-1, keepdim=True) - var = mean((pooled - mean)^2, dim=-1, keepdim=True) - std = sqrt(var + eps) - eps = 1e-5 by default in PyTorch LayerNorm. - normalized = (pooled - mean) / std - out = normalized * gamma + beta Wait, PyTorch's default eps for LayerNorm is 1e-5. Let's confirm: `torch.nn.LayerNorm` default `eps=1e-5`. Yes. We need to compute mean and variance over d for each (b,i). Since the block has all threads, we can compute sum and sum of squares of s_max[d]. ```cuda // Compute mean and variance float local_sum = 0.0f; float local_sum_sq = 0.0f; for (int idx = tid; idx < D; idx += block_size) { float v = s_max[idx]; local_sum += v; local_sum_sq += v * v; } s_sum[tid] = local_sum; s_sum_sq[tid] = local_sum_sq; __syncthreads(); // Reduce s_sum and s_sum_sq // Since block_size is power of two (256), we can reduce. // But block_size may not be a power of two? We can set it to a power of two, e.g., 256. // Let's assume block_size is a power of two. for (int stride = block_size / 2; stride > 0; stride >>= 1) { if (tid < stride) { s_sum[tid] += s_sum[tid + stride]; s_sum_sq[tid] += s_sum_sq[tid + stride]; } __syncthreads(); } float mean = s_sum[0] / D; float mean_sq = s_sum_sq[0] / D; float var = mean_sq - mean * mean; float inv_std = rsqrtf(var + 1e-5f); // Now compute output for (int idx = tid; idx < D; idx += block_size) { float v = s_max[idx]; float norm = (v - mean) * inv_std; float out_val = norm * ln_w[idx] + ln_b[idx]; out_val += s_x[idx]; // residual: nodes[b,i,idx] out[(b * N + i) * D + idx] = out_val; } ``` Wait, `s_x` currently contains the last j's data (node_j and pair_feat). We need `node_i` for the residual. We loaded `node_i` into `s_x[0:D]` at the beginning and never overwrote it (we only wrote `s_x[D:2D]` and `s_x[2D:L]`). So `s_x[idx]` for idx < D is still node_i. Great. But we need to be careful: after the first `__syncthreads()` following node_i load, other threads read it. Then in the j loop, we load node_j into `s_x[D:D*2]`. We need to ensure `node_i` is not overwritten. So `s_x` layout is fine: indices 0..D-1 = node_i, D..2D-1 = node_j, 2D..L-1 = pair. So residual `nodes[b,i,idx]` is `s_x[idx]`. One more thing: the LayerNorm computation uses `ln_w` (weight) and `ln_b` (bias). In PyTorch, `ln.weight` is shape (D,), `ln.bias` is shape (D,). We pass them as pointers. Now, about the grid and block dimensions. We need to decide blockDim.x. It should be a power of two for reduction. Let's use 256. But what if H or D > 256? Then the loops `for (h = tid; h < H; h += block_size)` handle it. The block size can be fixed at 256. The reduction for mean uses `block_size`. If D > 256, the reduction still works because we sum partial sums from all threads. The `for (stride = block_size/2; stride>0; stride>>=1)` reduction assumes all threads wrote to `s_sum[tid]` and `s_sum_sq[tid]`. Since we computed `local_sum` for all indices assigned to the thread, and stored in `s_sum[tid]`, it's fine. Even if D is large, the partial sum per thread is correct. Then `s_sum[0]` is total sum. But note: `s_sum` and `s_sum_sq` arrays are of size `block_size`. If `block_size = 256`, that's fine. However, if D is very large, the `s_max` shared memory size becomes D floats, which could be large. But typical D is small. For safety, we might want to allocate s_max dynamically based on D. Since we pass dynamic shared memory size at launch, we can compute it. We need to do that in Python side. In `forward`, we can compute `shmem = (L + H + D + 2*block_size) * 4` bytes. One more nuance: we skip phase 1 and 2 when `mask_val <= 0.5f`, but we still need to `__syncthreads()`. The code must ensure all threads reach `__syncthreads()`. In the j loop, `__syncthreads()` is outside the if/else. In my pseudocode, I had `if (mask_val > 0.5f) { compute and update max } __syncthreads();`. That's fine because all threads reach the sync. Wait, `mask_val` is the same for all threads in the block (since b, i, j are fixed for the block). Actually, is edge_mask shape (B,N,N)? Yes. For a given (b,i,j), mask is a scalar. So all threads have same mask_val. So the if branch is uniform. That's good for divergence. But we need to compute `mask_val` inside the j loop. Potential issue: `nodes` and `pair_feats` and `edge_mask` are accessed with indices. Let's compute pointer offsets carefully. Nodes: shape (B, N, D). Element (b, n, d) at `nodes[(b * N + n) * D + d]`. Pair_feats: shape (B, N, N, P). Element (b, i, j, p) at `pair_feats[((b * N + i) * N + j) * P + p]`. Edge_mask: shape (B, N, N). Element (b, i, j) at `edge_mask[((b * N + i) * N + j)]`. Out: shape (B, N, D). `out[(b * N + i) * D + d]`. Now, about the fused MLP: `nn.Linear` uses `y = x @ W^T + b`. In PyTorch, if weight is (out, in), then `x` (1, in) multiplied by weight^T (in, out) gives (1, out). Mathematically, `y[d] = sum_k x[k] * W[d, k] + b[d]`. Our kernel uses: - w1_T is shape (L, H). `w1_T[k * H + h]` corresponds to W1^T[k, h] = W1[h, k]. So dot product: `sum_k s_x[k] * w1_T[k*H + h]` = `sum_k s_x[k] * W1[h, k]`. Good. Then + b1[h]. Good. - w2_T is shape (H, D). `w2_T[h * D + d]` corresponds to W2^T[h, d] = W2[d, h]. So dot product: `sum_h s_z1[h] * w2_T[h*D + d]` = `sum_h s_z1[h] * W2[d, h]`. Then + b2[d]. Good. So the math matches. Now we need to handle the `neg = torch.finfo(fused.dtype).min`. The reference sets masked entries to the minimum float value, then does max. If we skip computation when mask is 0, the running max is unaffected, which is equivalent to initializing max to -inf and masking. In our kernel, we initialize `s_max[d]` to `-INFINITY` and only update when mask > 0.5. That's correct. However, what if all j are masked for a given (b,i)? Then s_max[d] remains -inf. LayerNorm on -inf would be NaN. But presumably edge_mask is such that each node has at least one valid edge. In the reference, if all are masked, `fused.max(dim=2)` would return -inf, and then adding nodes + LN(-inf) would be weird. But it's an edge case we don't need to worry about. One more thing: the reference uses `self.fuse` which is a `nn.Sequential`. The weights are `self.fuse[0].weight`, `self.fuse[0].bias`, `self.fuse[2].weight`, `self.fuse[2].bias`. LayerNorm: `self.out_norm.weight`, `self.out_norm.bias`. Now, we need to write the Python code that defines `ModelNew`, with `load_inline`. ### Python code structure ```python import torch import torch.nn as nn import torch.nn.functional as F from torch.utils.cpp_extension import load_inline # CUDA source code string cuda_source = r""" #include <cuda.h> #include <cuda_runtime.h> #include <math.h> __global__ void fused_scene_graph_kernel( const float* __restrict__ nodes, const float* __restrict__ pair_feats, const float* __restrict__ edge_mask, const float* __restrict__ w1_T, const float* __restrict__ b1, const float* __restrict__ w2_T, const float* __restrict__ b2, const float* __restrict__ ln_w, const float* __restrict__ ln_b, float* __restrict__ out, int B, int N, int D, int P, int H) { // ... } """ cpp_source = r""" torch::Tensor fused_scene_graph_forward( torch::Tensor nodes, torch::Tensor pair_feats, torch::Tensor edge_mask, torch::Tensor w1_T, torch::Tensor b1, torch::Tensor w2_T, torch::Tensor b2, torch::Tensor ln_w, torch::Tensor ln_b) { int B = nodes.size(0); int N = nodes.size(1); int D = nodes.size(2); int P = pair_feats.size(3); int H = w1_T.size(1); auto out = torch::empty_like(nodes); const int block_size = 256; dim3 grid(B, N); dim3 block(block_size); int L = 2 * D + P; size_t shmem_size = (L + H + D + 2 * block_size) * sizeof(float); fused_scene_graph_kernel<<<grid, block, shmem_size>>>( nodes.data_ptr<float>(), pair_feats.data_ptr<float>(), edge_mask.data_ptr<float>(), w1_T.data_ptr<float>(), b1.data_ptr<float>(), w2_T.data_ptr<float>(), b2.data_ptr<float>(), ln_w.data_ptr<float>(), ln_b.data_ptr<float>(), out.data_ptr<float>(), B, N, D, P, H); return out; } """ # Load inline _fused = load_inline( name="fused_scene_graph", cpp_sources=cpp_source, cuda_sources=cuda_source, functions=["fused_scene_graph_forward"], extra_cuda_cflags=["-O3", "--use_fast_math"], extra_cflags=["-O3"], ) class ModelNew(nn.Module): def __init__(self, dim: int, pair_dim: int, hidden: int): super().__init__() self.dim = dim self.pair_dim = pair_dim self.hidden = hidden # Use standard layers for parameter compatibility self.fuse = nn.Sequential( nn.Linear(2 * dim + pair_dim, hidden), nn.SiLU(), nn.Linear(hidden, dim), ) self.out_norm = nn.LayerNorm(dim) def forward(self, nodes, pair_feats, edge_mask): # Extract weights, transpose for kernel-friendly layout w1_T = self.fuse[0].weight.t().contiguous() # (L, H) b1 = self.fuse[0].bias.contiguous() w2_T = self.fuse[2].weight.t().contiguous() # (H, D) b2 = self.fuse[2].bias.contiguous() ln_w = self.out_norm.weight.contiguous() ln_b = self.out_norm.bias.contiguous() return _fused.fused_scene_graph_forward( nodes, pair_feats, edge_mask, w1_T, b1, w2_T, b2, ln_w, ln_b ) ``` But wait: `b1.contiguous()` on a 1D tensor is trivial. We can just pass them. One issue: `load_inline` may require `at::cuda::CUDAStreamGuard` or something? Usually not; `load_inline` generates pybind wrappers. But we need to ensure the tensors are on CUDA. In the evaluation, inputs are likely on CPU? The reference code doesn't specify device, but benchmarks often run on GPU. The default `ModelNew` delegates to `Model` which works on CPU or GPU. Our kernel requires CUDA. The evaluation probably calls `.cuda()` on inputs. We should call `.contiguous()` and ensure they are on GPU. Or we can just pass them; if they are on CPU, the kernel will fail. But the evaluation expects GPU. We should also check that the output is `nodes + self.out_norm(pooled)`. In our kernel, we add residual at the end. That's correct. Now, let's refine the CUDA kernel to be robust and fast. ### Kernel details We need to be careful about `__syncthreads()` inside the j loop and divergence. Let's write the kernel clearly. ```cuda __global__ void fused_scene_graph_kernel( const float* __restrict__ nodes, const float* __restrict__ pair_feats, const float* __restrict__ edge_mask, const float* __restrict__ w1_T, const float* __restrict__ b1, const float* __restrict__ w2_T, const float* __restrict__ b2, const float* __restrict__ ln_w, const float* __restrict__ ln_b, float* __restrict__ out, int B, int N, int D, int P, int H) { const int tid = threadIdx.x; const int b = blockIdx.x; const int i = blockIdx.y; const int L = 2 * D + P; const int block_size = blockDim.x; extern __shared__ float s_mem[]; float* s_x = s_mem; // [L] float* s_z1 = s_x + L; // [H] float* s_max = s_z1 + H; // [D] float* s_sum = s_max + D; // [block_size] float* s_sum_sq = s_sum + block_size; // [block_size] // Load node_i const float* node_i_ptr = nodes + (b * N + i) * D; for (int idx = tid; idx < D; idx += block_size) { s_x[idx] = node_i_ptr[idx]; } __syncthreads(); // Initialize max to -inf for (int idx = tid; idx < D; idx += block_size) { s_max[idx] = -CUDART_INF_F; // or -INFINITY } __syncthreads(); // Loop over j for (int j = 0; j < N; ++j) { // Load node_j and pair_feats const float* node_j_ptr = nodes + (b * N + j) * D; const float* pair_ptr = pair_feats + ((b * N + i) * N + j) * P; for (int idx = tid; idx < D; idx += block_size) { s_x[D + idx] = node_j_ptr[idx]; } for (int idx = tid; idx < P; idx += block_size) { s_x[2 * D + idx] = pair_ptr[idx]; } __syncthreads(); float mask_val = edge_mask[((b * N + i) * N + j)]; if (mask_val > 0.5f) { // Phase 1: compute hidden z1 for (int h = tid; h < H; h += block_size) { float acc = b1[h]; // dot product // We can loop with pragma unroll, but L is variable. // For typical sizes, just loop. for (int k = 0; k < L; ++k) { acc += w1_T[k * H + h] * s_x[k]; } float sig = 1.0f / (1.0f + expf(-acc)); s_z1[h] = acc * sig; } __syncthreads(); // Phase 2: compute output and update max for (int d = tid; d < D; d += block_size) { float val = b2[d]; for (int h = 0; h < H; ++h) { val += w2_T[h * D + d] * s_z1[h]; } // atomicMax on float is not directly available without atomic CAS. // But each d is mapped to a unique thread (if D <= block_size). // If D > block_size, a given d is still unique across threads because d = tid + m*block_size. // So we can write directly: if (val > s_max[d]) { s_max[d] = val; } } } __syncthreads(); } // LayerNorm on s_max float local_sum = 0.0f; float local_sum_sq = 0.0f; for (int idx = tid; idx < D; idx += block_size) { float v = s_max[idx]; local_sum += v; local_sum_sq += v * v; } s_sum[tid] = local_sum; s_sum_sq[tid] = local_sum_sq; __syncthreads(); // tree reduction for (int stride = block_size / 2; stride > 0; stride >>= 1) { if (tid < stride) { s_sum[tid] += s_sum[tid + stride]; s_sum_sq[tid] += s_sum_sq[tid + stride]; } __syncthreads(); } float mean = s_sum[0] / D; float var = s_sum_sq[0] / D - mean * mean; float inv_std = rsqrtf(var + 1e-5f); for (int idx = tid; idx < D; idx += block_size) { float v = s_max[idx]; float norm = (v - mean) * inv_std; float o = norm * ln_w[idx] + ln_b[idx]; o += s_x[idx]; // residual out[(b * N + i) * D + idx] = o; } } ``` Wait, there is a bug: when `mask_val <= 0.5f`, we skip `__syncthreads()` inside the if block, but we still have the `__syncthreads()` at the end of the j loop. However, if `mask_val > 0.5f`, the threads go through two `__syncthreads()` (one after phase 1, one at the end). If `mask_val <= 0.5f`, the threads skip the first `__syncthreads()` (inside the if block) but hit the one at the end. This causes divergence! Since `mask_val` is uniform across the block (same for all threads), the branch is taken uniformly by all threads. In CUDA, uniform branches don't cause divergence? Actually, if all threads in a warp take the same branch, there is no divergence. Since mask_val is the same scalar for all threads, all threads in the block take the same path. So it's safe. But we must ensure `__syncthreads()` is not inside a branch that some warps take and others don't. Since the condition is uniform across the entire block (mask_val depends only on block indices, not thread indices), it's uniform. Good. But wait: `if (mask_val > 0.5f)` is not dependent on `tid`. The compiler may not know that, but at runtime it's uniform. CUDA handles uniform branches fine: all warps either take it or not. However, if `mask_val` is loaded by a single thread and broadcast? All threads load it, so it's in registers, same value. It's uniform. Thus `__syncthreads()` inside the if is okay because either all threads reach it or none. But to be extra safe, we can separate the phases. Actually, we can compute phase 1 and 2 unconditionally, but set the output to -inf if mask is 0, and then update max. That avoids the if structure but adds compute. Since compute is small, it might be simpler and safer. Let's consider: ```cuda // Phase 1: compute hidden z1 for (int h = tid; h < H; h += block_size) { float acc = b1[h]; for (int k = 0; k < L; ++k) { acc += w1_T[k * H + h] * s_x[k]; } float sig = 1.0f / (1.0f + expf(-acc)); s_z1[h] = acc * sig; } __syncthreads(); // Phase 2 for (int d = tid; d < D; d += block_size) { float val = b2[d]; for (int h = 0; h < H; ++h) { val += w2_T[h * D + d] * s_z1[h]; } if (mask_val <= 0.5f) { val = -CUDART_INF_F; } if (val > s_max[d]) { s_max[d] = val; } } __syncthreads(); ``` This is simpler, always runs the same code path, and the compiler can optimize the mask branch. Since mask is uniform, the branch is cheap. This avoids any divergence issues. The extra compute for masked nodes is negligible unless many are masked. In worst case, still same compute. However, the reference uses `-torch.finfo(fused.dtype).min`, not `-inf`. Wait, `torch.finfo(fused.dtype).min` is the most negative finite float, which is about -3.4e38 for fp32. `max` with `-inf` also works because `-inf < anything finite`. `max` over a set containing `-inf` returns the max of the finite elements. So `-inf` is acceptable and numerically equivalent as long as there is at least one valid element. If all are masked, the max would be `-inf` in both cases (reference would return min float, which is also very negative). Wait, reference: `fused.masked_fill(edge_mask.unsqueeze(-1) < 0.5, neg)` where `neg = torch.finfo(fused.dtype).min`. That's the smallest finite negative number, like -3.4028235e+38. Then `max` returns that value if all are masked. If we use `-INFINITY`, `max` would return `-inf`. `-inf` vs `-3.4e38` are different. Will that affect correctness? Possibly if all edges are masked for a node. Does the test include such cases? The `get_inputs` uses all ones. But evaluation may have random masks. To be safe, we should use the same sentinel value: `neg = -CUDART_MAX_NORMAL_F`? In CUDA, `FLT_MIN` is the smallest positive normalized float. The most negative finite float is `-FLT_MAX`. In float32, `FLT_MAX` is 3.4028235e+38. So we can use `-FLT_MAX`. PyTorch's `torch.finfo(torch.float32).min` is indeed `-3.4028235e+38`, which is `-FLT_MAX`. So we should use `-FLT_MAX` or `-(float)INFINITY`? `INFINITY` is not the same. So better to use `-CUDART_INF_F`? No, that's `-inf`. Let's use `-3.402823466e+38f` or we can compute it as `-INFINITY` but that might differ in all-masked cases. Actually, we can just avoid the sentinel and conditionally update max only when mask is valid. In that case, if all are masked, `s_max` remains `-FLT_MAX` (since we initialized it to that). That matches the reference exactly. So we should initialize `s_max` to `-FLT_MAX`. In the branchless version, if mask is invalid, we set `val = -FLT_MAX` and update max. That also yields `-FLT_MAX` if all invalid. Good. So use `-FLT_MAX` for masked values. But wait, the reference sets `fused = fused.masked_fill(..., neg)` where `neg = torch.finfo(fused.dtype).min`. Then `max` over that. The max of all `-FLT_MAX` is `-FLT_MAX`. So initializing to `-FLT_MAX` and not updating yields same. If some are valid, the max will be the maximum of valid values because `-FLT_MAX < any typical value`. Good. So in code, define: ```cuda const float NEG_INF = -3.402823466e+38f; // approx -FLT_MAX ``` Or include `<float.h>` and use `-FLT_MAX`. In CUDA, `FLT_MAX` is available from `<float.h>`. We can include it. Now, the branchless version with `val = mask_val > 0.5f ? val : NEG_INF;` is uniform and safe. But we need to be careful: `mask_val` is a float. In the reference, `edge_mask` is float {0,1}. So comparing to 0.5 is fine. Now, let's write the kernel using the branchless approach. This also means we always compute the MLP even for masked edges. That's fine. Potential performance improvement: The inner dot product loops can be unrolled if L is known at compile time. But L is dynamic (2*D+P). However, we can pass D, P, H as template parameters? Since the evaluation uses fixed sizes, we could specialize. But to be generic, we can use `pragma unroll` with a small number, like `#pragma unroll 8`, but that might not help much since L is ~200. The compiler may unroll automatically. Since each thread does a loop of L=208, unrolling could help. But we can add `#pragma unroll 4` to give a hint. Wait, each thread looping over L=208 means 208 iterations. Unrolling by 4 gives 52 iterations. That might be okay. But if block_size=256 and H=192, only 192 threads are active in phase 1. 192 threads * 208 iterations = ~40k iterations of the inner loop body per j. That is small. Overhead of the loop is negligible compared to the memory loads. Memory loads from `w1_T` are the bottleneck. Since each thread reads a column of W1 sequentially (k loop), memory access is coalesced across the warp for each k. Wait, is it coalesced? Let's examine W1_T access in phase 1: `w1_T[k * H + h]`. For a fixed k, threads with h varying read addresses `k*H + h`. If H=192, and a warp has 32 threads with h=0..31 (first iteration), addresses are `k*H + 0`, `k*H + 1`, ..., `k*H + 31`. These are contiguous floats. Perfectly coalesced. They fit in 1 or 2 cache lines (128 bytes = 32 floats). So 1 cache line per warp per k. Since there are 6 warps (192/32 = 6), 6 cache lines per k. Total W1_T memory traffic per j = 6 warps * L cache lines = 6 * 208 * 128 bytes = 159,744 bytes = 156 KB. That's very close to the size of W1_T (160KB). So the entire W1_T is loaded once per j (or close to it). Good. Then it's cached for next j. Similarly for Phase 2: `w2_T[h * D + d]`. For fixed h, threads d=0..31 read addresses `h*D + 0` .. `h*D + 31`. Contiguous. Coalesced. 3 warps (96/32=3). Total traffic per j = 3 warps * H cache lines = 3 * 192 * 128 bytes = 73,728 bytes = 72 KB. That's exactly the size of W2_T (73KB). So loaded once per j. Good. This is excellent. Now, for s_z1 reads in phase 2: all threads read `s_z1[h]` for each h. That's broadcast. Shared memory broadcast has no bank conflicts. Great. Thus the kernel should be very efficient. Now, let's consider the possibility that blockDim.x = 256 but H=192 and D=96. In phase 1, 192 threads active. 64 threads idle. In phase 2, 96 threads active. 160 threads idle. That's fine. What if D=32 and H=192? Then phase 2 has 1 warp active. 7 warps idle. Still okay. Given that the problem sizes are small, the overhead of idle threads is minor. One more optimization: We can precompute `node_i` pointer and residual addition. Already done. Let's refine the kernel code for correctness. ### Edge cases - `B` and `N` could be large, but grid size is (B, N). Max grid dim is large. - `D`, `P`, `H` could be large, but shared memory scales with them. We need to ensure shared memory doesn't exceed device limit. For D=96, H=192, P=16, block_size=256: 1008 floats = 4KB. Very small. - If D > 1024, `s_max` would be >4KB, plus s_z1 >4KB, etc. But typical sizes are small. We could add an assert or handle generically. - Block size is 256. If D or H > 256, loops handle it. The reduction uses block_size. ### Writing the kernel with block_size as template parameter? We can make block_size a template to allow the compiler to unroll the reduction. But we can also just use 256 and loop. The reduction loop is `for (int stride = block_size/2; stride > 0; stride >>= 1)`. Since `block_size` is a variable (though constant because we pass 256), the compiler might not fully unroll it. We can make `block_size` a template parameter or a compile-time constant. In the kernel, `blockDim.x` is runtime. But we know it's 256. We can use `constexpr int block_size = 256;` inside the kernel and always launch with 256 threads. That would allow unrolling. But if we launch with a different block size, it would mismatch. Better to just hardcode 256 in the kernel and launch with 256. Alternatively, we can pass `block_size` as a template parameter. But `load_inline` with templated kernels is tricky because the C++ wrapper needs to instantiate it. Simpler to just hardcode 256. Let's set: ```cuda #define BLOCK_SIZE 256 __global__ void ... { const int tid = threadIdx.x; // ... const int block_size = BLOCK_SIZE; ``` But we also need to handle the case where `threadIdx.x >= block_size`? We launch exactly BLOCK_SIZE threads. So `threadIdx.x` is always < 256. Wait, we used loops `for (int h = tid; h < H; h += block_size)`. If H < block_size, this loop runs once for some threads and zero times for others. That's fine. For the reduction, we can safely unroll if we know `block_size` is 256. ```cuda // reduction #pragma unroll for (int stride = 128; stride > 0; stride >>= 1) { if (tid < stride) { s_sum[tid] += s_sum[tid + stride]; s_sum_sq[tid] += s_sum_sq[tid + stride]; } __syncthreads(); } ``` This is more efficient because it's fully unrolled. But it assumes block_size is exactly 256. If we change block_size, we need to adjust. We can add a static_assert or just write it with a loop and hope the compiler unrolls. But for small 256, explicit unroll is easy. Actually, we can write: ```cuda #pragma unroll for (int stride = block_size / 2; stride > 0; stride >>= 1) { ``` But `block_size` is a variable (though constexpr). The compiler may unroll. To be safe, I'll define `constexpr int BLOCK = 256;` and use explicit values. Now, we need to check if we need `#include <float.h>` for FLT_MAX. Let's write the CUDA source. Also, note: in phase 1 and phase 2, we have loops inside loops. The `#pragma unroll` on the inner loops (L and H) might help. ```cuda #pragma unroll 8 for (int k = 0; k < L; ++k) { ... } ``` But `L` is dynamic. The compiler might not unroll. We can add `#pragma unroll 4` to hint partial unrolling. Similarly for the H loop in phase 2: ```cuda #pragma unroll 8 for (int h = 0; h < H; ++h) { ... } ``` Let's add these pragmas. One more optimization: In phase 2, each thread accumulates `val`. We can preload `b2[d]` and the address for `w2_T` and `s_z1` into registers. Actually, `w2_T` access pattern is: for each h, read `w2_T[h*D + d]`. We can compute `w2_T += D` each iteration. That's good. The compiler will likely optimize it. Let's also consider using `float4` loads for `s_x` and weights. Since L=208, D=96, P=16, they are multiples of 4 (208/4=52, 96/4=24, 16/4=4). We could vectorize loads. But the loops are simple; the compiler may not automatically vectorize strided accesses (w1_T columns). For w1_T, `k*H + h` with step H is not contiguous. So float4 for w1_T is not natural. For s_x, we load sequentially. But s_x is in shared memory; vector loads aren't needed. Given the small size and high occupancy, the kernel is likely fast enough. ### Implementation of Module We need to be careful with `load_inline`. The C++ wrapper should accept `torch::Tensor` and do CUDA dispatch. Since we are compiling with `load_inline`, the pybind wrapper will handle GIL and tensor conversion. The C++ function `fused_scene_graph_forward` will be exposed to Python. One thing: `w1_T` and `w2_T` must be contiguous. We call `.t().contiguous()` in Python. The C++ wrapper can use `.data_ptr<float>()` safely. Also, we should ensure inputs are on CUDA and contiguous. We can call `.contiguous()` on nodes, pair_feats, edge_mask. ```python nodes = nodes.contiguous() pair_feats = pair_feats.contiguous() edge_mask = edge_mask.contiguous() ``` The output `out = torch.empty_like(nodes)` will be on the same device. Now, let's think about potential numerical issues. - SiLU: PyTorch SiLU is `x * sigmoid(x)`, where `sigmoid(x) = 1 / (1 + exp(-x))`. For large positive x, `exp(-x)` underflows to 0, sigmoid=1, SiLU=x. For large negative x, `exp(-x)` overflows? Actually `exp(-(-100)) = exp(100)` overflows to inf, so `1 + inf = inf`, `1/inf = 0`, SiLU = x * 0 = 0 (with sign? x is negative, 0 is +0, result is -0). In CUDA, `expf(100)` gives inf. `1.0f / (1.0f + expf(-x))` for x=-100: `expf(100)=inf`, `1+inf=inf`, `1/inf=0`. Then `x * 0 = -0`. That's fine. `sigmoid(100)` gives 1. `silu(100) = 100`. - But there might be slight differences in `expf` vs PyTorch's implementation (which may use a more precise approximation or special handling for large values). For typical random values, difference is within tolerance. - LayerNorm: we compute `var = E[x^2] - mean^2`. This can be slightly negative due to rounding, but we take `rsqrt(max(var, 0) + eps)`. In our code, we compute `var = mean_sq - mean * mean`. If var is slightly negative, adding `1e-5` will make it positive, `rsqrt` is fine. But PyTorch computes variance in a numerically stable way: `mean((x - mean)^2)`. Our two-pass approach (mean then E[x^2] - mean^2) is mathematically equivalent but may have slightly different rounding. Since `x` values are max-pooled MLP outputs, they can be large? Actually after SiLU and linear, values could be moderate. With fp32, subtracting large numbers could lose precision. But the tolerance is 1e-4, so our method should be fine. However, to be safe, we could compute variance in a Welford-like way or just in two passes: first compute mean, then compute `sum((x-mean)^2)`. But that requires another loop. We can do a two-pass approach using shared memory: 1. Compute mean (reduction). 2. Broadcast mean. 3. Each thread computes `sum_sq_diff` = sum of `(x - mean)^2`. 4. Reduce `sum_sq_diff`. 5. Compute inv_std = 1 / sqrt(sum_sq_diff / D + eps). This is more numerically stable and matches PyTorch's implementation more closely (PyTorch does a two-pass online algorithm). Let's do this. ```cuda // LayerNorm float local_sum = 0.0f; for (int idx = tid; idx < D; idx += block_size) { local_sum += s_max[idx]; } s_sum[tid] = local_sum; __syncthreads(); // reduce to get mean #pragma unroll for (int stride = block_size / 2; stride > 0; stride >>= 1) { if (tid < stride) { s_sum[tid] += s_sum[tid + stride]; } __syncthreads(); } float mean = s_sum[0] / D; // compute variance float local_var = 0.0f; for (int idx = tid; idx < D; idx += block_size) { float diff = s_max[idx] - mean; local_var += diff * diff; } s_sum[tid] = local_var; __syncthreads(); // reduce variance #pragma unroll for (int stride = block_size / 2; stride > 0; stride >>= 1) { if (tid < stride) { s_sum[tid] += s_sum[tid + stride]; } __syncthreads(); } float var = s_sum[0] / D; float inv_std = rsqrtf(var + 1e-5f); for (int idx = tid; idx < D; idx += block_size) { float v = s_max[idx]; float norm = (v - mean) * inv_std; float o = norm * ln_w[idx] + ln_b[idx]; o += s_x[idx]; out[(b * N + i) * D + idx] = o; } ``` This requires only `s_sum` (one array) instead of `s_sum` and `s_sum_sq`. We can adjust shared memory layout. Layout: `s_x[L]`, `s_z1[H]`, `s_max[D]`, `s_sum[BLOCK_SIZE]`. Total = L + H + D + BLOCK_SIZE floats. With BLOCK=256: 208+192+96+256 = 752 floats = 3008 bytes. This is better and more numerically stable. Now, we need to make sure that `s_sum` is big enough. Yes, BLOCK_SIZE. One more thing: in the reference, `self.out_norm = nn.LayerNorm(dim)`. The default `elementwise_affine=True`. The LayerNorm formula in PyTorch is: ``` y = (x - mean) / sqrt(var + eps) * weight + bias ``` We match that. Now, potential issue: `rsqrtf` computes `1/sqrt(x)`. PyTorch's `rsqrt` may have slightly different precision. But within tolerance. Also, we need to ensure that the kernel handles `B` and `N` correctly. `blockIdx.x` goes up to B-1, `blockIdx.y` up to N-1. That's fine. Now, let's consider the kernel launch configuration. We said `dim3 grid(B, N); dim3 block(BLOCK_SIZE);`. But what if B*N is very large, e.g., > 65535 in one dimension? Grid is 2D, so max is 65535 for x and y on older arch. On newer arch (Volta+), grid x,y,z up to 2^31-1. The GPU is likely A100 (Ampere), which supports large grids. So (B, N) is fine. But if B is large and N is large, we could also have `grid(B*N)` and compute b=i/N, i=i%N. That avoids 2D grid limits. But 2D is simpler. Actually, CUDA gridDim.y max is 65535 on pre-Volta, but on Volta+ it's 65535? Let's check: On compute capability 7.0+, the maximum grid dimension in x is 2^31-1, in y and z is 65535. On Ampere (8.0), y max is 65535. Reference says: "Maximum y- or z-dimension of a grid of thread blocks" is 65535 for all current compute capabilities. So if N > 65535, we cannot use `blockIdx.y`. N is number of objects, typically small (24). So fine. If B > 65535? B is batch size, typically small (4). So fine. 2D grid is safe. Now, let's consider the `load_inline` specifics. We need to provide `cpp_sources` and `cuda_sources`. The C++ wrapper uses ATen. ``` #include <torch/extension.h> ``` `torch::empty_like(nodes)` returns a tensor. We need to ensure `nodes` is a CUDA float tensor. In the C++ wrapper, we could add checks: ``` TORCH_CHECK(nodes.is_cuda(), "nodes must be CUDA"); TORCH_CHECK(nodes.is_contiguous(), "nodes must be contiguous"); ... ``` But not strictly necessary. Also, we need to include the kernel call. Since `load_inline` auto-generates pybind, we don't write `PYBIND11_MODULE`. We just write a function `torch::Tensor fused_scene_graph_forward(...)`. And list it in `functions=[...]`. Potential issue: The kernel name must be declared before the C++ wrapper function, or we can put them in `cuda_sources` which are compiled together. The C++ code sees the CUDA kernel declaration if we include it in `cuda_sources`. Actually, `load_inline` concatenates `cpp_sources` and `cuda_sources` into one file? No, they are separate compilation units. `cuda_sources` is compiled by nvcc, `cpp_sources` by the host compiler. But we can declare the kernel as `extern "C"` or just have the C++ wrapper call it; since they are in the same compilation unit (the extension module), the CUDA compiler handles the linkage? Actually, `load_inline` puts `cpp_sources` in the `.cpp` file and `cuda_sources` in a `.cu` file? Or it combines them? The documentation says: `cpp_sources` is a string containing C++ source code, and `cuda_sources` is a string containing CUDA source code. They are compiled into a single extension. The CUDA source is compiled with nvcc, the C++ with the host compiler, and linked together. So the C++ code needs a declaration of the kernel. We can put the kernel in `cuda_sources` and add a declaration in `cpp_sources`. Or we can put everything in `cuda_sources` and have an empty `cpp_sources`? The wrapper function also needs to be compiled with nvcc if it calls the kernel (since it uses `<<<>>>`). Actually, the wrapper function is host code that launches the kernel. It can be in the `.cpp` or `.cu`. But if it's in `.cpp`, the `.cpp` compiler (gcc/clang) may not understand `<<<>>>` syntax. So the wrapper must be in `cuda_sources`. But we need to expose a function callable from Python (C++ function). `load_inline` takes `functions` from the combined sources? The docs: `functions` is a list of function names to bind. Usually, the C++ wrapper is in `cpp_sources`. But if the wrapper uses CUDA launch syntax, it must be in a `.cu` file. However, `load_inline` compiles `cpp_sources` and `cuda_sources` together, and the `functions` can be defined in either. The pybind module is auto-generated. So we can put the C++ wrapper in `cuda_sources`? Actually, we can put everything in `cpp_sources` if we don't use CUDA-specific syntax? But `<<<>>>` is not valid C++. So we must put the launcher in `cuda_sources`. Wait, `load_inline` doc: https://pytorch.org/docs/stable/cpp_extension.html `cpp_sources` and `cuda_sources` are concatenated? Let's check typical usage: ```python fused = load_inline( name="fused", cpp_sources="torch::Tensor foo(torch::Tensor x) { ... }", cuda_sources="""__global__ void kernel(...) { ... }""", functions=["foo"], ) ``` In this case, `foo` is defined in `cpp_sources`. But `foo` can call the kernel declared in `cuda_sources` if there's a forward declaration in `cpp_sources`. Actually, the kernel is compiled by nvcc into device code. The host compiler compiles `cpp_sources`. For `foo` to call the kernel, it needs at least a host-side declaration. Usually, you put the kernel declaration (with `__global__`) in a header included by the C++ code, or in the `cuda_sources` file which is also parsed by nvcc? The `cuda_sources` file is compiled by nvcc, which generates host object code for the launcher stub. The C++ compiler sees only `cpp_sources`. So if `foo` is in `cpp_sources`, it cannot call `kernel<<<>>>` unless the C++ compiler sees a declaration. To avoid this issue, the simplest is to write a single host function in `cuda_sources` that includes both the kernel and the launcher. But `load_inline` might not bind functions defined in `cuda_sources`? Actually, it should; it just compiles all source files and then binds the listed functions. The functions must be visible to the linker. If we define `foo` in `cuda_sources` with `extern "C"` or just regular C++ linkage, nvcc will compile it and the linker will see it. Then we list `foo` in `functions=[]`. This is common. Let's check typical examples. In many KernelBench solutions, they put everything (kernel + wrapper) in `cuda_sources` and leave `cpp_sources` empty, or put utility code in `cpp_sources`. The `functions` refers to the wrapper function names. Example from KernelBench: ```python custom_op = load_inline( name="...", cuda_sources=cuda_source, functions=["custom_forward"], extra_cuda_cflags=["-O3"], ) ``` Where `cuda_source` contains both `__global__ void kernel(...)` and `torch::Tensor custom_forward(...) { kernel<<<...>>>(...); return out; }`. So we should put both the kernel and the launcher in `cuda_sources`. `cpp_sources` can be empty or omitted. But we need to include `<torch/extension.h>`. So: ```python cuda_source = r""" #include <torch/extension.h> #include <cuda.h> #include <cuda_runtime.h> #include <float.h> #include <math.h> #define BLOCK_SIZE 256 __global__ void fused_scene_graph_kernel(...) { ... } torch::Tensor fused_scene_graph_forward(...) { ... fused_scene_graph_kernel<<<grid, block, shmem_size>>>(...); return out; } """ ``` And `functions=["fused_scene_graph_forward"]`. Now, I need to make sure the kernel compiles. Let's write it carefully. One more thing: `CUDART_INF_F` is defined in `<cuda_runtime_api.h>`, but for `FLT_MAX` we need `<float.h>`. We'll define `NEG_SENTINEL = -FLT_MAX`. Alternatively, we can just use `-(3.402823466e+38f)` to avoid including `<float.h>`. But `<float.h>` is standard. Let's write the kernel. ### Complete CUDA code ```cuda #include <torch/extension.h> #include <cuda.h> #include <cuda_runtime.h> #include <float.h> #include <math.h> #define BLOCK_SIZE 256 __global__ void fused_scene_graph_kernel( const float* __restrict__ nodes, const float* __restrict__ pair_feats, const float* __restrict__ edge_mask, const float* __restrict__ w1_T, // shape (L, H) const float* __restrict__ b1, // shape (H,) const float* __restrict__ w2_T, // shape (H, D) const float* __restrict__ b2, // shape (D,) const float* __restrict__ ln_w, // shape (D,) const float* __restrict__ ln_b, // shape (D,) float* __restrict__ out, int B, int N, int D, int P, int H) { const int tid = threadIdx.x; const int b = blockIdx.x; const int i = blockIdx.y; const int L = 2 * D + P; extern __shared__ float s_mem[]; float* s_x = s_mem; // [L] float* s_z1 = s_x + L; // [H] float* s_max = s_z1 + H; // [D] float* s_red = s_max + D; // [BLOCK_SIZE] // Load node_i into first D elements of s_x const float* node_i_ptr = nodes + (b * N + i) * D; for (int idx = tid; idx < D; idx += BLOCK_SIZE) { s_x[idx] = node_i_ptr[idx]; } __syncthreads(); // Initialize running max for (int idx = tid; idx < D; idx += BLOCK_SIZE) { s_max[idx] = -FLT_MAX; } __syncthreads(); // Iterate over all neighbors j for (int j = 0; j < N; ++j) { // Load node_j and pair_feats into shared memory const float* node_j_ptr = nodes + (b * N + j) * D; const float* pair_ptr = pair_feats + ((b * N + i) * N + j) * P; for (int idx = tid; idx < D; idx += BLOCK_SIZE) { s_x[D + idx] = node_j_ptr[idx]; } for (int idx = tid; idx < P; idx += BLOCK_SIZE) { s_x[2 * D + idx] = pair_ptr[idx]; } __syncthreads(); float mask_val = edge_mask[((b * N + i) * N + j)]; // Phase 1: compute hidden pre-activations for (int h = tid; h < H; h += BLOCK_SIZE) { float acc = b1[h]; // Dot product over input features #pragma unroll 4 for (int k = 0; k < L; ++k) { acc += w1_T[k * H + h] * s_x[k]; } float sig = 1.0f / (1.0f + expf(-acc)); s_z1[h] = acc * sig; } __syncthreads(); // Phase 2: compute output and update running max for (int d = tid; d < D; d += BLOCK_SIZE) { float val = b2[d]; #pragma unroll 4 for (int h = 0; h < H; ++h) { val += w2_T[h * D + d] * s_z1[h]; } if (mask_val <= 0.5f) { val = -FLT_MAX; } if (val > s_max[d]) { s_max[d] = val; } } __syncthreads(); } // LayerNorm over the D dimension float local_sum = 0.0f; for (int idx = tid; idx < D; idx += BLOCK_SIZE) { local_sum += s_max[idx]; } s_red[tid] = local_sum; __syncthreads(); // Tree reduction for mean #pragma unroll for (int stride = BLOCK_SIZE / 2; stride > 0; stride >>= 1) { if (tid < stride) { s_red[tid] += s_red[tid + stride]; } __syncthreads(); } float mean = s_red[0] / D; // Variance float local_var = 0.0f; for (int idx = tid; idx < D; idx += BLOCK_SIZE) { float diff = s_max[idx] - mean; local_var += diff * diff; } s_red[tid] = local_var; __syncthreads(); #pragma unroll for (int stride = BLOCK_SIZE / 2; stride > 0; stride >>= 1) { if (tid < stride) { s_red[tid] += s_red[tid + stride]; } __syncthreads(); } float var = s_red[0] / D; float inv_std = rsqrtf(var + 1e-5f); // Final Output for (int idx = tid; idx < D; idx += BLOCK_SIZE) { float v = s_max[idx]; float norm = (v - mean) * inv_std; float o = norm * ln_w[idx] + ln_b[idx]; o += s_x[idx]; // residual out[(b * N + i) * D + idx] = o; } } torch::Tensor fused_scene_graph_forward( torch::Tensor nodes, torch::Tensor pair_feats, torch::Tensor edge_mask, torch::Tensor w1_T, torch::Tensor b1, torch::Tensor w2_T, torch::Tensor b2, torch::Tensor ln_w, torch::Tensor ln_b) { int B = nodes.size(0); int N = nodes.size(1); int D = nodes.size(2); int P = pair_feats.size(3); int H = w1_T.size(1); auto out = torch::empty_like(nodes); dim3 grid(B, N); dim3 block(BLOCK_SIZE); int L = 2 * D + P; size_t shmem_size = (L + H + D + BLOCK_SIZE) * sizeof(float); fused_scene_graph_kernel<<<grid, block, shmem_size>>>( nodes.data_ptr<float>(), pair_feats.data_ptr<float>(), edge_mask.data_ptr<float>(), w1_T.data_ptr<float>(), b1.data_ptr<float>(), w2_T.data_ptr<float>(), b2.data_ptr<float>(), ln_w.data_ptr<float>(), ln_b.data_ptr<float>(), out.data_ptr<float>(), B, N, D, P, H); return out; } ``` Wait, there is a subtle issue with the reduction: we have `s_red` of size BLOCK_SIZE. But if `D < BLOCK_SIZE`, threads with `tid >= D` will write zero to local_sum/local_var and participate in reduction. That's fine. Their `s_max[idx]` might read uninitialized? No, in the loops over `idx < D`, threads with `tid >= D` have `idx = tid >= D`, so they don't enter the loop. `local_sum` is 0.0f. So reduction is fine. But there is a potential race if `D` is not a multiple of something? No. One more nuance: `mask_val` is a scalar loaded by each thread from global memory. Since the address is the same for all threads, it's cached. Good. Now, we need to handle the case where `pair_feats` might not have `pair_feats.size(3)`? In forward, shape is (B,N,N,P). So `size(3)` works. Now, the `ModelNew` class: ```python class ModelNew(nn.Module): def __init__(self, dim: int, pair_dim: int, hidden: int): super().__init__() self.dim = dim self.pair_dim = pair_dim self.hidden = hidden self.fuse = nn.Sequential( nn.Linear(2 * dim + pair_dim, hidden), nn.SiLU(), nn.Linear(hidden, dim), ) self.out_norm = nn.LayerNorm(dim) def forward(self, nodes, pair_feats, edge_mask): # Make contiguous nodes = nodes.contiguous() pair_feats = pair_feats.contiguous() edge_mask = edge_mask.contiguous() # Prepare transposed weights for coalesced access w1_T = self.fuse[0].weight.t().contiguous() b1 = self.fuse[0].bias.contiguous() w2_T = self.fuse[2].weight.t().contiguous() b2 = self.fuse[2].bias.contiguous() ln_w = self.out_norm.weight.contiguous() ln_b = self.out_norm.bias.contiguous() return _fused.fused_scene_graph_forward( nodes, pair_feats, edge_mask, w1_T, b1, w2_T, b2, ln_w, ln_b ) ``` But wait: the reference model's `forward` does not modify inputs. Our `contiguous()` may create copies if inputs are not contiguous. That's okay. Also, `w1_T = self.fuse[0].weight.t().contiguous()` will allocate a new tensor each forward pass. For small weights, that's fine. But to avoid overhead, we could cache them. However, since `load_inline` compilation might take time, the forward will be called multiple times. But the overhead of transposing 40k floats is tiny. But there is a problem: if the evaluation copies state dicts after `__init__`, then we must ensure our weight layout inside the `nn.Linear` modules is standard. We are using standard `nn.Linear`, so it's fine. The transposition is done on the fly in `forward`. One more concern: the reference uses `nn.SiLU()`. Does `nn.SiLU()` have inplace option? No. It computes elementwise. So our kernel matches. Potential performance optimization: Pre-transpose weights and store as buffers. We can do it in `forward` the first time or in `__init__`. But we need to handle parameter updates (e.g., training). If the kernel is used only for inference (which is typical for "forward"), weights are static. But to be safe, we can transpose on the fly. The overhead is tiny compared to the kernel. Alternatively, we can store the original weights in `nn.Parameter` with shape (L, H) and (H, D) directly, and manually implement the initialization to match `nn.Linear`. But that breaks `state_dict` compatibility. So on-the-fly transpose is best. Wait, but `self.fuse[0].weight` is a Parameter of shape (H, L). Calling `.t()` returns a view of shape (L, H). Calling `.contiguous()` copies it to a new tensor with shape (L, H). That new tensor is not a Parameter, just a Tensor. That's fine. One more thing: `torch.utils.cpp_extension.load_inline` may cache the compiled extension. The first call might be slow. Subsequent calls load from cache. Now, we should compile and test. But before that, let's consider if there are any numerical differences. 1. The MLP: - `nn.Linear` uses fused addmm in PyTorch, which might use different rounding order than our manual dot product. However, for fp32, the dot product order is similar (accumulate in float32). PyTorch may use a different algorithm (e.g., splitting), but for small sizes, it's likely just a loop. So results should match within 1e-4. 2. SiLU: PyTorch SiLU might be implemented with `x * sigmoid(x)` where `sigmoid` uses a fast approximation or the exact formula. CUDA `expf` is accurate. For large negative inputs, `expf` might overflow to inf, giving 0. For very negative inputs, PyTorch's sigmoid may underflow to 0 as well. Slight differences might exist for extreme values. But with random init, values are moderate. 3. Max pool: our max over j should match `max(dim=2)`. 4. LayerNorm: - PyTorch LayerNorm uses Welford's algorithm or a two-pass method with `mean` then `sum((x - mean)^2)`. Our two-pass method matches. The epsilon is 1e-5. PyTorch's `LayerNorm` default eps is 1e-5. Good. - We compute variance as `sum((x-mean)^2)/D`. PyTorch computes it as `sum((x-mean)^2)/D`. Yes. - Then `rsqrt(var + eps)`. Yes. Potential issue: Our `mean` and `var` are computed in float32. PyTorch also uses float32. One more potential issue: If there are NaNs or Infs, results differ. But random init shouldn't produce that. Now, we need to check if any thread divergence or synchronization issue exists. In the j loop: 1. Load s_x (node_j, pair). All threads participate (some load, some don't). __syncthreads. 2. Phase 1: compute s_z1. Threads with `tid < H` participate. Others idle. __syncthreads. 3. Phase 2: compute s_max. Threads with `tid < D` participate. Others idle. __syncthreads. Since the idle threads don't reach __syncthreads? Wait, `if (tid < H)` would cause divergence: some threads in a warp may be active, some not. But the `__syncthreads()` inside the if would be a problem because not all threads in the block reach it. However, in our code, `__syncthreads()` is **outside** the `for (int h = tid; h < H; h += BLOCK_SIZE)` loop. The loop body is executed only if `tid < H` (since h starts at tid). If `tid >= H`, the loop doesn't execute. But `__syncthreads()` after the loop is reached by all threads because it's outside any conditional block. So all threads reach it. That's safe. Same for Phase 2: `__syncthreads()` is after the `for (int d = tid; d < D; d += BLOCK_SIZE)` loop, which is unconditional in terms of thread execution path (the loop just doesn't iterate for some threads). All threads reach the `__syncthreads()`. Good. But wait: the `if (mask_val <= 0.5f) val = -FLT_MAX;` is inside the loop. Since `mask_val` is the same for all threads, this branch is uniform across the warp. No divergence. So synchronization is fine. One small thing: `mask_val` is loaded after `__syncthreads()` inside the j loop. Then phase 1 uses `mask_val`? No, phase 1 is unconditional. Then phase 2 uses mask_val. That's fine. Potential performance issue: For each j, we recompute the entire MLP even if `mask_val <= 0.5`. But masked edges are few? In evaluation, edge_mask may be random. If many are masked, we waste compute. But we can skip phase 1 and 2 if `mask_val <= 0.5`. However, as discussed, __syncthreads inside an if that is uniform is fine. Let's do that to save compute for masked edges. But to keep code simple and avoid any possible issues, we can leave it unconditional. The compute is small anyway. Wait, if we skip phase 1+2 when masked, all threads still need to reach the __syncthreads at the end of the j loop. Since the condition is uniform across the block, it's okay. Let's do unconditional for simplicity; the compiler might still optimize it because mask is uniform and loaded into a register. Actually, the loop over k is the main compute; the compiler likely won't skip it based on a uniform value unless we explicitly branch. Let's branch explicitly for performance, since evaluation may have many masked edges? Probably not, but it's a small code change. ```cuda if (mask_val > 0.5f) { for (int h = tid; h < H; h += BLOCK_SIZE) { ... } __syncthreads(); for (int d = tid; d < D; d += BLOCK_SIZE) { ... if (val > s_max[d]) s_max[d] = val; } } __syncthreads(); ``` Is the `__syncthreads()` inside the `if` safe? Since the condition is uniform for the block, all threads in the block either take the branch or not. So all threads reach that `__syncthreads()`. Then all threads reach the final `__syncthreads()`. So it's safe. But to avoid any subtle issues with the compiler not recognizing uniform branch, we can write it without inner `__syncthreads()`: ```cuda if (mask_val > 0.5f) { for (int h = tid; h < H; h += BLOCK_SIZE) { ... } for (int d = tid; d < D; d += BLOCK_SIZE) { float val = b2[d]; // note: need s_z1 ready, so we need a sync between phase1 and phase2 } } ``` But we still need a sync between phase 1 and phase 2 because threads writing s_z1 (phase1) and threads reading s_z1 (phase2) are the same threads but may interleave? Actually, within the same `if`, after phase 1 loop, we need a sync before phase 2 loop to ensure all s_z1 writes are visible. Since it's within the `if` and all threads are active, `__syncthreads()` is safe. But there is a subtle issue: if `mask_val > 0.5f` is uniform across the block, but some threads in a warp may be inactive because `tid >= H`? No, the `if` is not around the loop `for (int h = ...)`. The if is around the entire phase block. So all threads (including those with tid >= H) enter the if. Then they all reach the `__syncthreads()` inside. That's safe. Let's keep it simple and unconditional. 24 iterations is small. The overhead of branch vs unconditional is negligible. Wait, actually we can't make it unconditional if we want to match the sentinel exactly: for masked edges we need to set val to -FLT_MAX. But if we compute the MLP anyway, it's fine. Now, let's compile and test. I realize that `extern __shared__ float s_mem[];` must be declared with the maximum alignment. Partitioning with `float*` is fine because floats are 4-byte aligned. Now, about the `load_inline` call: we need to handle the case where the extension is already loaded. `load_inline` caches by name. We should give a unique name. "fused_scene_graph" is fine. Let's write the full Python file. But before finalizing, let's consider using `torch.compile` or `Triton`. The instruction says to use `torch.utils.cpp_extension.load_inline`. So we must use CUDA C++. One more optimization: The `for (int k = 0; k < L; ++k)` loop reads `s_x[k]` which is shared memory. All threads read the same `s_x[k]` for each k? Wait, each thread reads `s_x[k]` for varying k in its own loop. Actually, each thread iterates over all k. So there is no broadcast; each thread reads the entire `s_x` array sequentially. But `s_x` is in shared memory, so it's fast. Similarly, w1_T reads are strided across threads (coalesced across k). Actually, for a fixed k, threads read `w1_T[k*H + h]`. That's coalesced. Then k increments. So memory access pattern is: for k=0, read 256 contiguous floats (or H if H<256). For k=1, read next 256 contiguous floats. This is perfect. The cache line is 128 bytes = 32 floats. So for a warp of 32 threads, each k reads 1 cache line. For 6 warps, 6 cache lines per k. L=208, so total 1248 cache lines. 1248 * 128 = 159,744 bytes. Good. Wait, but a warp has 32 threads. The active threads in phase 1 are 192. That's 6 warps. For each k, warp 0 reads h=0..31, warp 1 reads h=32..63, etc. Offsets: k*H + 0, ..., k*H + 31 (contiguous). So each warp reads one cache line. 6 warps -> 6 cache lines. 208 k -> 1248 cache lines. That's 159KB per j. Since W1_T is 160KB, it's essentially read once per j. Great. Similarly, phase 2: active threads D=96, that's 3 warps. For each h, warp 0 reads d=0..31 (1 cache line), warp 1 reads d=32..63 (1 cache line), warp 2 reads d=64..95 (1 cache line). 3 cache lines per h. H=192 -> 576 cache lines. 576*128 = 73,728 bytes. W2_T is 73KB. Read once per j. Great. Thus the kernel is very efficient in terms of memory reuse. Now, one more thing: `pair_feats` indexing is `((b * N + i) * N + j) * P + idx`. We should ensure the pair_feats layout is (B, N, N, P). Yes. Node_j indexing: `(b * N + j) * D + idx`. Good. Node_i indexing: `(b * N + i) * D + idx`. Good. Edge_mask: `((b * N + i) * N + j)`. Good. Now, let's write the full code and test compile.

The issue is that load_inline requires both `cpp_sources` and `cuda_sources`. Let me fix this.

The error is that the generated pybind code includes the header in main.cpp before the CUDA forward declaration is visible. The issue is that when we pass `cpp_sources=""`, the auto-generated pybind code is in main.cpp and tries to wrap our function, but the function declaration is only in the cuda source file. Looking more closely at the load_inline docs, the issue is the order of compilation - the cpp_sources gets compiled first as main.cpp, and cuda_sources as cuda.cu. Since our function definition is in the cuda .cu file, the auto-generated pybind code in main.cpp can't see the declaration. We need to either: 1. Put a forward declaration in cpp_sources, or 2. Use a different approach. The typical pattern is: ```python cpp_sources = "torch::Tensor fused_scene_graph_forward(...);" cuda_sources = <actual implementation> ``` Let me fix this.

The kernel compiled. Let me run the correctness check.

The kernel compiles and passes correctness. Now let me think about how to optimize it further. Looking at the problem: - B = 4, N = 24, D = 96, P = 16, H = 192 - The grid is (B, N) = (4, 24) = 96 blocks total - Each block has 256 threads - This seems like enough parallelism, but maybe I should consider using more parallelism or optimizing memory access patterns Shared memory usage: - L = 2 * 96 + 16 = 208 - H = 192 - D = 96 - BLOCK_SIZE = 256 - Total shared memory = (208 + 192 + 96 + 256) * 4 = 752 * 4 = 3008 bytes This is very small. The kernel might benefit from larger block sizes or different tiling strategies. Potential issues: 1. The inner loops for MLP computations aren't fully unrolled and the shared memory layout is simple 2. The pair_feats access pattern is (b, i, j, p) which might not be coalesced for all threads. Wait, each block handles a specific (b, i) pair, and pairs load pair_feats sequentially for j varying. Actually, when we're at (b, i) block, we load pair_feats for different j values sequentially, but only one thread block does this - there's minimal memory parallelism within a block for pair_feats. But actually the bigger concern might be that our approach is already quite good because: - We avoid any intermediate global memory - All computations happen in shared memory - The running max is accumulated in shared memory Let me think about what could be improved: 1. Use warp-level reductions for better efficiency 2. Access pair_feats in a more coalesced way - currently only thread 0 loads pair_feats when tid < P, which might not be great 3. Possibly use more threads or restructure loops Actually, looking more carefully, since we're iterating over j sequentially, and N=24 is small, perhaps we can unroll the j loop or process multiple j values at once? But this would increase shared memory register pressure. Also, I could consider using vectorized loads (float4) to improve memory bandwidth. Let me also consider if using fewer threads (say 128 or 64) would help occupancy. With 256 threads per block and only 96 blocks, we have 96 * 256 = 24576 threads total. H100 has 132 SMs IIRC, so that's ~186 threads per SM on average, which is very low. Actually, H100 has 132 SMs (for SXM5 variant, maybe 114 for PCIe). With 96 blocks total and 256 threads each, each SM gets less than 1 block, so the occupancy is very low. But since each block does a lot of work, this might be fine. Wait, let me reconsider. With 96 blocks and ~132 SMs, many SMs are idle. But each block does ~24 * (2*96+16)*192 computationally heavy operations, so each block takes significant time. I think the main optimization could be: 1. Use float4 vector loads for weights and data 2. Possibly consider fusing more operations or better loop unrolling 3. Use fewer threads per block (e.g., 128 or 64) to allow more blocks per SM Actually, one important thing: the weight access patterns. For w1_T, we're accessing w1_T[k * H + h] where h varies by thread. This gives good coalescing. For w2_T, we access w2_T[h * D + d] where d varies by thread. This also gives good coalescing. For the inputs, accessing pair_feats by having tid < P load shared memory should be fine since P=16 is small. Let me try to improve the kernel with: 1. Block size 128 to improve occupancy slightly 2. Better warp-level reduction for LayerNorm 3. Maybe vectorized loads where appropriate Actually, let me try a completely different approach. Since N=24 is small, what if we have each block handle all j values for a specific (b, i) pair but use more parallelization across d/h dimensions? Another thought: since the reduction is over j (max pooling), and we have to compute the fused output for each j sequentially or in parallel, there might be a way to process multiple j values in parallel if we have enough shared memory. For N=24, and each j needing H=192 shared memory floats, 24*192*4 = 18432 bytes. Plus other shared memory, this could fit in 227KB per block. So we could process all j values in parallel! Let me redesign: instead of a loop over j, we could have each thread handle a specific j, compute the fused result, and then do a parallel reduction (max) across j for each output dimension. With 256 threads and N=24, we'd have ~10 threads per j. Actually, a cleaner approach is: one warp per j (32 threads), so we need 24 warps = 768 threads per block to process all j in parallel. That's too many threads. Or: have blocks handle multiple i values, or have each thread compute the full MLP for one (i, j) pair and write to shared memory, then do reductions. Let's try: each thread handles multiple (i, j) pairs. Grid could be (B, 1) with more threads. But this gets complicated. Another approach: use smaller blocks but have each SM handle more parallelism at once. For example, use 64 threads per block. Then we'd have 96 * 64 = 6144 threads total, which is even worse for SM occupancy. Wait, perhaps the biggest win is not about occupancy but about instruction efficiency. Let me check if the compute is memory-bound or compute-bound. Compute per (b, i) block: - For each j (24): - Load node_j (96 floats), pair_feats (16 floats) - Linear1: (208 * 192) multiply-adds = 39,936 FMAs - SiLU: 192 exp + muls = ~192 * expensive - Linear2: (192 * 96) FMAs = 18,432 FMAs - Max comparison: 96 - Total per block: 24 * (39936 + 18432) ≈ 1.4 million FMAs - For 96 blocks: ~135 million FMAs - H100 FP32 TFLOPS: 67. So at peak this would take ~2 microseconds. The kernel will likely take much more because of memory and non-FMA ops. Memory per block per j: - node_j: 96 * 4 = 384 bytes (from L2 likely since we iterate through j) - pair_feats: 16 * 4 = 64 bytes (from global memory) - weights reused from L1/L2 Actually for 96 blocks * 24 j * (96+16) floats = 96*24*112*4 ≈ 1MB of node/pair data reads. - Weights: w1_T = 208*192*4 = 160KB, w2_T = 192*96*4 = 74KB. Total ~234KB - Total memory moved: ~1MB. At 3.35 TB/s this takes ~0.3 microseconds. So this seems compute-bound (or at least not purely memory bound). The biggest issue might be sequentialization and suboptimal instruction scheduling. Let me think about optimizations I can make: 1. **Unroll loops more aggressively**: Currently `#pragma unroll 4` for k and h loops. Since L=208 and H=192, these are reasonably large. Better unrolling might help. But since loop bounds are dynamic, full unrolling isn't possible. 2. **Vectorize loads**: Use float4 for loading weights and data. 3. **Use warp-level primitives for reductions**: Instead of shared memory reduction for LayerNorm, use warp shuffle. 4. **Process multiple i values**: Since we have 96 blocks and B*N=96 outputs, maybe we can have threads within a block compute multiple output positions to better utilize warps. Actually, let me try a different parallelization strategy: - Have each block handle 2 or 4 i values (for a fixed b) - This increases the number of concurrent warps but also increases shared memory for holding multiple node_i and s_max values. With 4 i values per block: - s_x for each i: 208 floats - s_z1 for each i: 192 floats - s_max for each i: 96 floats - For 4 i: (208 + 192 + 96) * 4 * 4 = 7936 bytes per block - Plus shared memory reduction: still small - Total threads still 256, but each thread now works on 4 i values sequentially - This doubles the block count effectively... wait no, we'd use fewer blocks: B * N / 4 = 24 blocks. Actually this might hurt occupancy since we go from 96 to 24 blocks. Alternative: have EACH thread independently handle a subset of j values. With 256 threads, we could partition j=0..23 among threads. Then within each warp (32 threads), we have 24 different j values, meaning some threads are idle for j. Wait, we don't need each thread to handle a j. Instead, we can have each thread compute a subset of output dimensions d for all j values. For example, with 96 output dimensions and 256 threads, each thread could compute 1 output dimension (but we only have 96). So threads 0..95 each handle 1 d, computing the max over all j values. For a thread handling dimension d: max_val = -inf for j in 0..N-1: // we need the full hidden vector for this j // compute z1[j][0..H-1] // then compute z2 = sum_h w2[h][d] * z1[j][h] // update max output[d] = max_val This would require each thread to compute the entire z1 vector (size H=192) for each j, which means each thread is doing the full linear1 computation. This is very redundant across threads! With 96 threads each doing 24 * (208*192 + 192*96) = 24 * 57344 ≈ 1.4M FMAs, total would be 96 * 1.4M ≈ 134M FMAs (same total), but all in one block! This is way too much work per block. Let me try a different strategy: each block handles 1 (b, i). Within the block: - Threads 0..95 handle d dimension for output - Threads 96..127 load data - Or use all 256 threads to compute in a more parallel way Actually, what if we transpose the w2_T access? Currently: ```cpp for (int h = 0; h < H; ++h) { val += w2_T[h * D + d] * s_z1[h]; } ``` This is good because d varies by thread. For w1_T: ```cpp for (int k = 0; k < L; ++k) { acc += w1_T[k * H + h] * s_x[k]; } ``` This is good because h varies by thread. One thing that might help: **constant memory** for weights. Since the weights are small (234KB), they might fit in constant memory or be cached in L1. Actually, a better optimization might be to **tile the j loop** so that we process j in tiles and reuse the weights. But since L1 cache on H100 is 256KB and weights are 234KB, they might already stay in L1. Let me try a completely different kernel design: - Each block handles 1 (b, i) - Each warp handles 1 j value (32 threads per j) - For N=24, we'd need 24 warps = 768 threads per block, which is too many. Alternative: each block handles 4 j values simultaneously, but we process i sequentially. Actually, I think the current kernel is quite reasonable. Let me try some specific optimizations: 1. **Reduce block size to 128** - This might allow 2 blocks per SM on H100, improving occupancy. Currently with 96 blocks * 256 threads, we might not even fill all SMs. 2. **Use float4 for loading node_i, node_j** - Vectorize memory accesses where possible. 3. **Better warp shuffle for LayerNorm reduction** 4. **Avoid __syncthreads where possible** - Use warp-level operations Let me also think: what if we process multiple (b, i) pairs per block? If we process 2 (b, i) pairs with 256 threads, we'd use 48 blocks. Each thread could handle 2 i values. But we need separate s_x, s_z1, s_max per i, which increases shared memory. Shared memory for 2 i values: 2 * (208 + 192 + 96 + 128) * 4 ≈ 4224 bytes (if block=128). Still fine. But the reduction becomes more complex. Actually, let me try block_size=128 with 2 i values per block. Wait, I just realized: with 96 blocks total and the fast run being under 1,000 microsecond, each block runs in about 1,000μs. That's quite slow, suggesting memory or instruction bottlenecks. Actually, I think the main issue is that the sequential j loop with __syncthreads() after every step creates many synchronization points. With N=24, we have 24 * 4 = 96 __syncthreads() calls per block. Can I redesign to use fewer synchronizations? For example, split the j loop into two phases: Phase 1: Load all pair_feats and node_j data into shared memory for all j Phase 2: Process all j without __syncthreads (but we need z1 per j) For Phase 1, we'd need shared memory: N * (D + P) * 4 = 24 * 112 * 4 = 10752 bytes. This would fit easily. Then in Phase 2, loop over j and read from pre-loaded shared memory. This eliminates the loads from global memory inside the j loop. Actually, we could even pre-load all node_j for all j since they're the same across all i: but wait, we process across B and i, so only node_j is reused within a block. For a specific block (b, i): - Pre-load node_i (D floats) - Pre-load all node_j for j=0..N-1 (N * D floats) - Pre-load all pair_feats for this (b, i) and all j (N * P floats) Total pre-load: D + N*D + N*P = 96 + 2304 + 384 = 2784 floats = 11,136 bytes. Then in the j loop, we just read from shared memory. No __syncthreads needed for loads! But we still need __syncthreads after computing z1 (since z1 is shared) and after computing max (since max is shared). So we'd reduce from 4 __syncthreads() per j to 2 per j. This is a nice improvement. Let me also consider: can we eliminate some __syncthreads by having each thread use local memory for s_z1? If each thread computes a subset of the output d, and each thread independently computes all z1 values... no, that would be very redundant (each thread does the full linear1). What if we have warp-level computation of z1? One warp (32 threads) computes all of z1, then all warps use it. But this requires __syncthreads() anyway. Let me redesign with pre-loaded shared memory: ```cpp extern __shared__ float s_mem[]; float* s_node_i = s_mem; // [D] float* s_nodes_j = s_node_i + D; // [N * D] float* s_pairs = s_nodes_j + N * D; // [N * P] float* s_z1 = s_pairs + N * P; // [H] float* s_max = s_z1 + H; // [D] float* s_red = s_max + D; // [BLOCK_SIZE] ``` Pre-load phase: ```cpp // Load node_i for (int d = tid; d < D; d += blockDim.x) s_node_i[d] = nodes[...]; // Load all node_j for (int idx = tid; idx < N * D; idx += blockDim.x) s_nodes_j[idx] = nodes[...]; // Load all pair_feats for (int idx = tid; idx < N * P; idx += blockDim.x) s_pairs[idx] = pair_feats[...]; __syncthreads(); ``` Then in j loop: ```cpp for (int j = 0; j < N; ++j) { float* s_node_j = s_nodes_j + j * D; float* s_pair = s_pairs + j * P; // s_x is now s_node_i || s_node_j || s_pair // Can copy to a local buffer or access directly // Phase 1: z1 for (int h = tid; h < H; h += blockDim.x) { float acc = b1[h]; for (int k = 0; k < D; ++k) acc += w1_T[k * H + h] * s_node_i[k]; for (int k = 0; k < D; ++k) acc += w1_T[(D + k) * H + h] * s_node_j[k]; for (int k = 0; k < P; ++k) acc += w1_T[(2*D + k) * H + h] * s_pair[k]; float sig = 1.0f / (1.0f + expf(-acc)); s_z1[h] = acc * sig; } __syncthreads(); // Phase 2: max for (int d = tid; d < D; d += blockDim.x) { float val = b2[d]; for (int h = 0; h < H; ++h) val += w2_T[h * D + d] * s_z1[h]; if (mask[j] <= 0.5f) val = -FLT_MAX; if (val > s_max[d]) s_max[d] = val; } __syncthreads(); } ``` This eliminates 2 __syncthreads() per j (the load syncs). Only 2 __syncthreads() remain. We'd also need to pre-load edge_mask or read it directly. Let me compute shared memory: - D = 96 - N * D = 24 * 96 = 2304 - N * P = 24 * 16 = 384 - H = 192 - D = 96 - BLOCK_SIZE = 256 Total = 96 + 2304 + 384 + 192 + 96 + 256 = 3328 floats = 13,312 bytes For block_size=128: 96 + 2304 + 384 + 192 + 96 + 128 = 3200 floats = 12,800 bytes For block_size=256: 13,312 bytes Either way, this fits easily. But wait, loading N * D = 2304 floats for nodes_j is wasteful because all blocks for the same b load the same nodes_j. But each block only loads once, and from then on accesses are from shared memory (fast). Let me implement this improved version. I think it's a clear win. Another optimization: we can read edge_mask directly without __syncthreads since the value is the same for all threads in a block for a given j: ```cpp float mask_val = edge_mask[((b * N + i) * N + j)]; ``` But careful: we should make sure we don't have a race or out-of-order read. This is fine since edge_mask is read-only and each thread in the block should get the same cached value. Let me also think about using fewer threads. With block_size=128: - We need enough warps to hide memory latency - For H=192 and D=96, with 128 threads, we can cover both dimensions in 2 and 1 iterations respectively - 128 threads = 4 warps, which is decent With block_size=64: - H=192 takes 3 iterations - D=96 takes 2 iterations - 2 warps total I think 128 threads is a good balance. Let me try this improved kernel with block_size=128 and pre-loaded shared memory. ```cuda #define BLOCK_SIZE 128 __global__ void fused_scene_graph_kernel(...) { const int tid = threadIdx.x; const int b = blockIdx.x; const int i = blockIdx.y; const int L = 2 * D + P; extern __shared__ float s_mem[]; float* s_node_i = s_mem; // [D] float* s_nodes_j = s_node_i + D; // [N * D] float* s_pairs = s_nodes_j + N * D; // [N * P] float* s_z1 = s_pairs + N * P; // [H] float* s_max = s_z1 + H; // [D] float* s_red = s_max + D; // [BLOCK_SIZE] // Pre-load node_i for (int idx = tid; idx < D; idx += BLOCK_SIZE) { s_node_i[idx] = nodes[(b * N + i) * D + idx]; } __syncthreads(); // Pre-load all nodes_j for (int idx = tid; idx < N * D; idx += BLOCK_SIZE) { int j = idx / D; int d = idx % D; s_nodes_j[j * D + d] = nodes[(b * N + j) * D + d]; } // Pre-load all pair_feats and edge_mask for (int idx = tid; idx < N * P; idx += BLOCK_SIZE) { int j = idx / P; int p = idx % P; s_pairs[j * P + p] = pair_feats[((b * N + i) * N + j) * P + p]; } __syncthreads(); // Initialize max for (int idx = tid; idx < D; idx += BLOCK_SIZE) { s_max[idx] = -FLT_MAX; } __syncthreads(); for (int j = 0; j < N; ++j) { float mask_val = edge_mask[((b * N + i) * N + j)]; // first linear + SiLU for (int h = tid; h < H; h += BLOCK_SIZE) { float acc = b1[h]; // node_i contribution float* w1_ptr = w1_T + h; for (int k = 0; k < D; ++k) { acc += w1_ptr[k * H] * s_node_i[k]; } // node_j contribution float* node_j_ptr = s_nodes_j + j * D; w1_ptr = w1_T + (D * H) + h; for (int k = 0; k < D; ++k) { acc += w1_ptr[k * H] * node_j_ptr[k]; } // pair contribution float* pair_ptr = s_pairs + j * P; w1_ptr = w1_T + (2 * D * H) + h; for (int k = 0; k < P; ++k) { acc += w1_ptr[k * H] * pair_ptr[k]; } float sig = 1.0f / (1.0f + expf(-acc)); s_z1[h] = acc * sig; } __syncthreads(); // second linear + max for (int d = tid; d < D; d += BLOCK_SIZE) { float val = b2[d]; for (int h = 0; h < H; ++h) { val += w2_T[h * D + d] * s_z1[h]; } if (mask_val <= 0.5f) { val = -FLT_MAX; } if (val > s_max[d]) { s_max[d] = val; } } __syncthreads(); } // LayerNorm: compute mean float local_sum = 0.0f; for (int idx = tid; idx < D; idx += BLOCK_SIZE) { local_sum += s_max[idx]; } s_red[tid] = local_sum; __syncthreads(); #pragma unroll for (int stride = BLOCK_SIZE / 2; stride > 0; stride >>= 1) { if (tid < stride) { s_red[tid] += s_red[tid + stride]; } __syncthreads(); } float mean = s_red[0] / D; // Variance float local_var = 0.0f; for (int idx = tid; idx < D; idx += BLOCK_SIZE) { float diff = s_max[idx] - mean; local_var += diff * diff; } s_red[tid] = local_var; __syncthreads(); #pragma unroll for (int stride = BLOCK_SIZE / 2; stride > 0; stride >>= 1) { if (tid < stride) { s_red[tid] += s_red[tid + stride]; } __syncthreads(); } float var = s_red[0] / D; float inv_std = rsqrtf(var + 1e-5f); // Final residual output for (int idx = tid; idx < D; idx += BLOCK_SIZE) { float v = s_max[idx]; float norm = (v - mean) * inv_std; float o = norm * ln_w[idx] + ln_b[idx]; o += s_node_i[idx]; // residual out[(b * N + i) * D + idx] = o; } } ``` Actually, for the w1_T access patterns, I should check. Currently with h varying by thread, accessing w1_T[k * H + h] is coalesced (adjacent threads read adjacent memory). My split into three loops (for node_i, node_j, pair) accesses the same w1_T columns but at different row offsets, which should still be coalesced. Wait, but splitting the loop into three might hurt instruction scheduling. The compiler might not be able to optimize as well. Let me keep a single loop for better pipelining: ```cpp for (int h = tid; h < H; h += BLOCK_SIZE) { float acc = b1[h]; float* w_base = w1_T + h; for (int k = 0; k < D; ++k) { acc += w_base[k * H] * s_node_i[k]; } for (int k = 0; k < D; ++k) { acc += w_base[(D + k) * H] * s_nodes_j[j * D + k]; } for (int k = 0; k < P; ++k) { acc += w_base[(2 * D + k) * H] * s_pairs[j * P + k]; } // ... } ``` This keeps the access pattern clean. The compiler should be able to pipeline these well. Let me also consider using vector loads for the data loading phase. For nodes_j, we can load float4 values since D=96 is divisible by 4. Actually wait, let me also think about whether the reference implementation uses `self.fuse` with properly initialized weights. Yes, it does. And since we're extracting weight from `self.fuse[0].weight` etc., the weights match. One more optimization: I can use `__ldg` (read-only cache load) for the weights since they're read-only in the kernel. Actually, let me also think about warp-level reductions for LayerNorm. If block_size=128, we can use warp shuffles for the reduction to avoid some shared memory conflicts. ```cpp __inline__ __device__ float warp_reduce_sum(float val) { #pragma unroll for (int mask = 16; mask > 0; mask >>= 1) { val += __shfl_xor_sync(0xFFFFFFFF, val, mask); } return val; } ``` This would be faster, but since D=96 and we have 128 threads, 32 threads write to shared memory and then reduce anyway. Not sure if it's worth it. Let me implement the pre-loaded version and see if it compiles and passes. Wait, actually, I realize there might be another issue with my kernel. In the original PyTorch code: ```python fused = self.fuse(torch.cat([hi, hj, pair_feats], dim=-1)) ``` This is (B, N, N, 2*D+P) input, goes through Linear(2*D+P, H), then SiLU, then Linear(H, D). The weights I'm using: - `w1_T = self.fuse[0].weight.t()` -> shape (2*D+P, H) - `w2_T = self.fuse[2].weight.t()` -> shape (H, D) These match since nn.Linear stores weight in (out_features, in_features) format. So `self.fuse[0].weight` is (H, 2*D+P), and `self.fuse[0].weight.t()` is (2*D+P, H). In my kernel: ```cpp acc += w1_T[k * H + h] * x[k] ``` Here h is the output column index. So w1_T[k * H + h] is the weight from input k to output h. This matches nn.Linear's transpose. And for w2_T: ```cpp val += w2_T[h * D + d] * z1[h] ``` h is input, d is output. w2_T[h * D + d] is weight from input h to output d. This matches. Good, the weight layout is correct. Now let me implement the pre-loaded shared memory version. One issue: I need to check if the pre-loading of node_j crosses batches. Different i in the same batch b read the same nodes_j, so pre-loading them once per block is correct. Let me write the improved kernel. Also, let me reconsider block size. With block_size=128: - Total blocks = 96 - Threads total = 96 * 128 = 12288 - On H100 with ~132 SMs, this is ~93 threads per SM - With 2048 max threads per SM, occupancy is ~4.5% This is quite low! But each thread does a lot of work, so the low occupancy might be somewhat compensated. What if we process multiple i values per block? For example, have each block process 4 i values, with block_size=256: - Threads per i = 64 - Total blocks = 96 / 4 = 24 - Threads total = 24 * 256 = 6144 - Per SM: ~47 threads This is even worse. So processing fewer i values per block doesn't help occupancy. What if we increase total blocks by processing multiple (b, i) positions per block but with more threads? Or use different parallel axes? Alternative: parallelize across j (the max reduction dimension). - Each block handles (b, i, j) - Grid = (B, N, N) = (4, 24, 24) = 2304 blocks - Each block computes the fused value for one (b, i, j) - Then we need a separate reduction kernel for max over j and LayerNorm This would give much better occupancy (2304 blocks) but requires two kernels. The first kernel is highly parallel (2304 blocks * whatever threads). The second kernel does max + LN. First kernel (per (b, i, j)): ```cuda __global__ void compute_fused( nodes, pair_feats, edge_mask, w1_T, b1, w2_T, b2, out, // (B, N, N, D) ... ) { int b = blockIdx.x; int i = blockIdx.y; int j = blockIdx.z; int tid = threadIdx.x; // Load node_i, node_j, pair_feat into shared/registers // Compute z1 (first linear + SiLU) // Compute z2 (second linear) // Mask and write to out[b, i, j, d] } ``` Grid: (4, 24, 24), block: 128 threads Total: 2304 blocks * 128 threads = 294912 threads Occupancy: 294912 / (132 * 2048) ≈ 109% - this could fully utilize the GPU! Then second kernel: max over j + LayerNorm + residual ```cuda __global__ void reduce_and_norm( fused, // (B, N, N, D) edge_mask, ln_w, ln_b, nodes, out, ... ) { int b = blockIdx.x; int i = blockIdx.y; int tid = threadIdx.x; // max over j for d in 0..D-1 (parallel across threads): max_val = -inf for j in 0..N-1: if (edge_mask[b, i, j] > 0.5): max_val = max(max_val, fused[b, i, j, d]) // mean and var across d? // Actually LayerNorm is per (b, i), so we need to reduce over d // For each d, compute LN using shared mean/var // Then add residual node_i and write out } ``` Grid: (4, 24), block: 128 threads Total: 96 blocks But the second kernel would have the same occupancy issue. Wait, the fused approach stores intermediate results in global memory: - Output of first kernel: B * N * N * D = 4 * 24 * 24 * 96 * 4 = 884,736 bytes ≈ 0.8MB This is small and should fit in L2 cache. So the two-kernel approach might actually be faster because: 1. First kernel has great occupancy and parallelism 2. Second kernel operates on cached data 3. The first kernel removes all __syncthreads() overhead per j Let me try this two-kernel approach! For the first kernel, each thread computes multiple d values: - D = 96, block_size = 128... we need to split 96 output dims across 128 threads. - We could use 96 threads (3 warps) and have 32 threads idle. - Or have threads process 1 d each, with some threads idle. - Or use warp-level fine-grained parallelism: each warp handles specific d values. Let me think: with block = 96 threads, D/1 = 96 outputs, each thread handles 1 output dimension. For each d: val = b2[d] for h in 0..H-1: val += w2_T[h, d] * z1[h] Here, d varies by thread. w2_T[h, d] access is coalesced if h is row and d is column. w2_T has shape (H, D) = (192, 96). For fixed h, w2_T[h * D + d] where d varies by thread is coalesced (adjacent d values are adjacent in memory). So d varying by thread gives good coalescing for w2_T. For z1, all threads read the same z1[h] for a given h in the inner loop. This is broadcast and should be cached. For the MLP1: z1[h] = SiLU(sum_k w1_T[k, h] * x[k]) Where h varies by thread. w1_T[k, h] with h varying: for fixed k, w1_T[k * H + h] is coalesced. This is also good! So a block of 96 threads (or 128 with some idle) handling one (b, i, j) is efficient. But wait, the first kernel needs to compute ALL of z1 (size H=192) before computing z2. This means each thread computes one z1[h]. With 96 threads, each thread computes 2 z1 values. Actually, let me use 192 threads (6 warps). Then each thread computes exactly 1 z1[h] and 0.5 z2[d]. To avoid fractional work, use 96 threads for z2 and 192 threads for z1... this gets complicated. Simplest: use block_size=128. - For z1: each thread computes 2 h values (192/128 = 1.5, so some threads do 2, some do 1). Better: just loop with h += 128. Each thread does ceil(192/128)=2 or 1 iterations. - For z2: each thread computes 1 d value, with 32 threads idle. Actually wait, for the compute_fused kernel, I can have the block first compute z1 collaboratively, then compute z2. The __syncthreads() between z1 and z2 is fine since we're processing one (b, i, j). Let me design it: ```cuda __global__ void compute_fused_kernel(...) { int b = blockIdx.x; int i = blockIdx.y; int j = blockIdx.z; int tid = threadIdx.x; // Load x = [node_i, node_j, pair_ij] // We can use shared memory or registers. Since L=208, with 128 threads we need 208*4=832 bytes. // Alternatively, load on demand from global memory since each thread reads all L elements. // Load node_i, node_j, pair once to shared memory __shared__ float s_x[208]; // L = 2*D+P // Collaborative load for (int idx = tid; idx < 2*D+P; idx += blockDim.x) { if (idx < D) { s_x[idx] = nodes[(b * N + i) * D + idx]; } else if (idx < 2*D) { s_x[idx] = nodes[(b * N + j) * D + (idx - D)]; } else { s_x[idx] = pair_feats[((b * N + i) * N + j) * P + (idx - 2*D)]; } } __syncthreads(); float mask_val = edge_mask[((b * N + i) * N + j)]; // Compute z1 in shared memory __shared__ float s_z1[192]; // H for (int h = tid; h < H; h += blockDim.x) { float acc = b1[h]; for (int k = 0; k < L; ++k) { acc += w1_T[k * H + h] * s_x[k]; } float sig = 1.0f / (1.0f + expf(-acc)); s_z1[h] = acc * sig; } __syncthreads(); // Compute output for this (b, i, j, d) for (int d = tid; d < D; d += blockDim.x) { float val = b2[d]; for (int h = 0; h < H; ++h) { val += w2_T[h * D + d] * s_z1[h]; } if (mask_val <= 0.5f) { val = -FLT_MAX; } out[((b * N + i) * N + j) * D + d] = val; } } ``` Grid: (B, N, N) = (4, 24, 24) Block: 128 threads Total blocks: 2304 Total threads: 2304 * 128 = 294912 This is great occupancy! Then the second kernel: ```cuda __global__ void reduce_layernorm_kernel(...) { int b = blockIdx.x; int i = blockIdx.y; int tid = threadIdx.x; // Read all j values, compute max over j for each d __shared__ float s_max[96]; // D __shared__ float s_red[128]; // blockDim.x __shared__ float s_node_i[96]; // D, for residual // Load node_i for residual for (int d = tid; d < D; d += blockDim.x) { s_node_i[d] = nodes[(b * N + i) * D + d]; s_max[d] = -FLT_MAX; } __syncthreads(); // Max reduction over j for (int d = tid; d < D; d += blockDim.x) { float max_val = -FLT_MAX; for (int j = 0; j < N; ++j) { float val = out[((b * N + i) * N + j) * D + d]; if (val > max_val) { max_val = val; } } s_max[d] = max_val; } __syncthreads(); // LayerNorm (same as before) ... // Add residual and write for (int d = tid; d < D; d += blockDim.x) { float v = s_max[d]; float norm = (v - mean) * inv_std; float o = norm * ln_w[d] + ln_b[d]; o += s_node_i[d]; out[(b * N + i) * D + d] = o; } } ``` Wait, but the second kernel has the same low occupancy issue (only 96 blocks). However, its workload is much lighter: it's just a max over 24 values and a LayerNorm. Total compute of second kernel per block: - Max over j: D * N = 96 * 24 = 2304 comparisons - LayerNorm: mean/var over D = 96, then normalize This is very small. But since it reads from global memory (the output of first kernel), and there are 96 blocks, memory bandwidth might be the bottleneck for this kernel. First kernel write: 4*24*24*96*4 = 884,736 bytes ≈ 0.85MB Second kernel read: same amount With 3.35 TB/s bandwidth, 0.85MB takes ~0.25 microseconds. But with launch overhead and kernel underutilization, it might take longer. Actually, wait. Both kernels together involve: 1. First kernel: reads nodes, pair_feats, edge_mask, weights; writes fused 2. Second kernel: reads fused; writes final output Total memory moved: - nodes: B*N*D*4 = 4*24*96*4 = 36,864 bytes (read twice) - pair_feats: B*N*N*P*4 = 4*24*24*16*4 = 147,456 bytes (read once) - edge_mask: B*N*N*4 = 9,216 bytes (read once/twice) - weights: ~234KB (read in first kernel) - fused intermediate: ~885KB (write + read) - output: B*N*D*4 = 36,864 bytes Total main memory traffic: ~36KB + 147KB + 9KB + 234KB + 885KB + 37KB ≈ 1.35MB This is very small. The main cost is likely instruction execution and kernel launch overhead. For the single-kernel approach, we don't write fused to global memory, saving ~885KB of memory traffic. This is significant for a small problem! So maybe the single-kernel approach is still better, especially since the intermediate result is small enough to keep in shared memory. Let me reconsider. In the single-kernel approach with pre-loaded shared memory: - We avoid all intermediate global memory writes - We have 96 blocks with significant compute each - Occupancy is low but we're not memory-bound In the two-kernel approach: - First kernel has great occupancy but writes to global memory - Second kernel has poor occupancy and reads from global memory - Total memory traffic is higher I think the single-kernel approach might still win due to avoiding the intermediate global memory round-trip. Let me go with the improved single-kernel approach with pre-loaded shared memory and block_size=128. Another optimization: instead of having each block process one (b, i), what if each block processes multiple (b, i) positions but with warp-level parallelism? For example, with block_size=256, we could split into 4 warps (128 threads actually... let me use 256) - Warp 0 handles (b0, i0), warp 1 handles (b1, i1), etc. - But this means each warp has only 64 threads... wait, a warp is always 32 threads. Actually, we can have 32 threads within a warp cooperate on one (b, i). Since computations are SIMD within a warp, having threads in a warp work on different (b, i) would cause divergence if we iterate differently. A cleaner idea: each block handles 2 (b, i) positions. We split the block into two halves (128 threads each). This doubles the block count from 96 to 48, which makes occupancy worse. Hmm, what if each block handles 8 (b, i) positions with 32 threads each? (8 * 32 = 256) - Each "sub-block" of 32 threads handles one (b, i) - For sub-block k (threads 32k to 32k+31), handle (b_k, i_k) But then each sub-block has only 32 threads to process H=192 and D=96. This means each thread does 6-7 times more work. Wait, maybe having 32 threads per (b,i) is actually fine. Let me analyze: - For z1: 32 threads, each computes 192/32 = 6 z1 values (loop over h += 32) - For z2: 32 threads, each computes 96/32 = 3 z2 values (loop over d += 32) - Total 32-thread blocks: 96 / 8 = 12 blocks (if 8 per 256-thread block) This is terrible for occupancy (12 blocks). So occupancy is the fundamental problem with single-kernel approach. The problem is just too small in terms of independent work items. One more idea: interleave the j loop across threads/warps. - Instead of iterating j sequentially in the loop, have threads process different j values in parallel. - Use warp-level reduction (max) to combine results. For example, with 32 threads in a warp, we could have threads 0-23 each handle one j value. Then after computing the fused value for their j, we do a warp-level max reduction to get the max over j for each d. But each thread would need to compute the FULL MLP (linear1 + SiLU + linear2) for its j. This means 24 threads (out of 32) are doing work, and 8 are idle. For ONE (b, i), we'd need one warp (32 threads) to process all j values in parallel. Total warps needed: 96 (one per (b, i)) Total threads: 96 * 32 = 3072 Blocks: 3072 / 256 = 12 blocks (if we pack 8 warps per block) Still only 12 blocks. Very low occupancy. But what if we can fit multiple (b, i) per warp? Not really, because the max reduction is per (b,i). Hmm. Let me think differently. What if we CAN'T improve occupancy and should just make each thread do its work as fast as possible? In this case, optimizations like: - Pre-loading data to shared memory - Minimizing __syncthreads() - Using __ldg for read-only data - Ensuring good instruction scheduling These are key. Actually, I just realized something. The reference PyTorch implementation is also single-kernel in the sense that PyTorch fuses everything into one kernel call... wait no, PyTorch calls multiple kernels: expand, cat, linear, silu, linear, masked_fill, max, layernorm, add. That's many kernels. So our single CUDA kernel is already saving on kernel launch overhead, which is a big win. Let me try the pre-loaded shared memory version. I think it should be faster than the current version. Let me also try using a block size of 256 (as in the original) but with pre-loaded data. The original has 96 blocks with 256 threads. With pre-loading, we eliminate half the __syncthreads() calls. Actually, let me just go ahead and implement the pre-loaded version with block_size=128 and see. One more optimization: for the w1_T access, the weight is fixed for all j iterations. So we can load it to shared memory too! But w1_T is 208*192*4 = 160KB which is too large for shared memory per block (max 227KB but that would leave very little for other things). Actually, 160KB fits in shared memory! Let's check: - 160KB for w1_T - 74KB for w2_T Total: 234KB Shared memory per SM: 228KB. So we can't fit both weights in shared memory simultaneously. But we could fit w1_T (160KB) in shared memory and leave w2_T in L1/L2. Hmm, with 96 blocks total and 132 SMs, most SMs get one block. So shared memory isn't shared between blocks. We can use up to 227KB per block. Shared memory budget: - w1_T: 160KB - Data: 13KB - z1: 0.75KB - max: 0.375KB - red: 0.5KB (for block_size=128) Total: ~175KB This fits! But 175KB shared memory per block means we can only run 1 block per SM (since 228KB / 175KB ≈ 1.3). This reduces occupancy but might be worth it if w1_T accesses are a bottleneck. Actually, let me reconsider. Is w1_T access a bottleneck? Currently we access w1_T in a coalesced pattern (h varies by thread). With L1 cache, these accesses should be cached. Loading w1_T to shared memory might not help much if it's already in L1. I think the pre-loaded data version without weight caching is a good first step. Let me also consider another optimization for the sequential j loop. Instead of having all threads participate in both linear1 and linear2 for each j, what if we pipeline across j values? For example: - Threads 0..127 compute z1 for j=0 - While they're doing that, other threads compute nothing - Then all threads compute z2 for j=0 - Repeat There's not much pipelining opportunity because z2 depends on z1. One advanced idea: since SiLU is a pointwise nonlinearity and linear2 is a linear transform, maybe we can fuse them or precompute something? Not really, since SiLU is non-linear. OK let me just code up the pre-loaded version and see if it helps. Let me also try a larger block size like 256 with the pre-loaded data. More threads means: - More parallel loading - But more __syncthreads() in the reduction - Better for reductions (256 threads vs 128) Actually for reductions, 256 threads means more parallel work but also more overhead in the tree reduction. With D=96, both 128 and 256 threads can cover D in one iteration. I'll use block_size=128 for now since it gives similar coverage with less reduction overhead. Let me write the code. Wait, I should also check: does the reference implementation use `edge_mask` as exactly 0 or 1 floats? The doc says float {0,1}. And in PyTorch: ```python fused = fused.masked_fill(edge_mask.unsqueeze(-1) < 0.5, neg) ``` So if edge_mask < 0.5, we set to neg. This means 0 values are masked, 1 values pass through. My kernel condition: ```cpp if (mask_val <= 0.5f) { val = -FLT_MAX; } ``` This is equivalent (0 <= 0.5 → masked, 1 > 0.5 → not masked). Good. One edge case: what if there are NO valid edges for a given i? Then `max(dim=2)` over all `-FLT_MAX` gives `-FLT_MAX`. Then LayerNorm of all `-FLT_MAX` would give NaN. But the problem states `edge_mask` is {0,1} but doesn't guarantee at least one 1 per row. In practice, from `get_inputs()`, `edge_mask = torch.ones(...)`, so all edges are valid. Actually, wait. I should double-check my current code handles this correctly. In PyTorch: ```python pooled, _ = fused.max(dim=2) ``` If all values in a row are `-inf`, max returns `-inf`. Then LayerNorm of `-inf` gives NaN. This is consistent. In my kernel, if no edges are valid, `s_max[d]` stays at `-FLT_MAX`. Then LN: mean = -FLT_MAX var = (-FLT_MAX - (-FLT_MAX))^2 = NaN? Actually, let's compute: diff = -FLT_MAX - (-FLT_MAX) = NaN (since -FLT_MAX + FLT_MAX is undefined) Wait, -FLT_MAX + FLT_MAX should be 0 in terms of floating point... no, FLT_MAX is a large positive number, and -FLT_MAX is a large negative number. Their difference is 2*FLT_MAX which overflows to inf, or their sum is 0. Actually: -FLT_MAX - (-FLT_MAX) = -FLT_MAX + FLT_MAX = 0 in exact arithmetic, but in floating point, FLT_MAX is the maximum representable normal float. -FLT_MAX is the minimum normal float. Their sum should be 0 because they cancel exactly in IEEE 754. Wait no: float has 24 bits of mantissa. FLT_MAX and -FLT_MAX are exact opposites, so -FLT_MAX + FLT_MAX = 0 exactly. But in our case: mean = -FLT_MAX, and each element is -FLT_MAX. So diff = -FLT_MAX - (-FLT_MAX) = 0. And var = 0. And inv_std = rsqrt(0 + 1e-5) ≈ 100. Then norm = (v - mean) * inv_std = 0 * 100 = 0. Then output = 0 * ln_w + ln_b + node_i. So even if all edges are masked, we get a finite output! But the reference PyTorch would give NaN because `mean = -inf`, and `v - mean = -inf - (-inf) = NaN`. Wait, does PyTorch's LayerNorm handle this case? In PyTorch, when you compute LayerNorm with all -inf: - mean = -inf - (x - mean) = NaN (because -inf - (-inf) is NaN in floating point) So PyTorch gives NaN, but our kernel gives 0 (for the normalized part). This is a discrepancy! BUT, the inputs are ones() for edge_mask, and there might always be at least one valid edge. Let me check the reference code: `edge_mask = torch.ones(...)`. So all edges are valid. The issue doesn't arise with the test inputs. And mathematically, the problem probably assumes at least one valid edge per (b, i). So we don't need to worry about this edge case. Actually, let me verify: `-inf - (-inf)` in IEEE 754 is NaN. But `-FLT_MAX - (-FLT_MAX)` should be exactly 0 since `-FLT_MAX` and `FLT_MAX` are normal floats and their difference is representable. Wait, this is wrong. `-FLT_MAX` and `FLT_MAX` are symmetric in floating point: - FLT_MAX has binary representation with all exponent bits 1 except LSB (which is for infinity) and all mantissa bits 1. - -FLT_MAX is the same with sign bit flipped. So `-FLT_MAX + FLT_MAX = 0` exactly in floating point, because addition of a number and its exact negation gives 0. But our code computes `(-FLT_MAX) - (-FLT_MAX)` which is adding `-FLT_MAX` and `+FLT_MAX` in the arithmetic (since subtracting a negative is adding). So yes, it gives 0 exactly. If the reference were to use all masked edges (resulting in -inf), then reference gives NaN and our kernel gives a finite value. But as I said, this edge case isn't in the tests. Let me proceed. Now, let me implement the pre-loaded shared memory version. ```cuda #define BLOCK_SIZE 128 __global__ void fused_scene_graph_kernel( const float* __restrict__ nodes, const float* __restrict__ pair_feats, const float* __restrict__ edge_mask, const float* __restrict__ w1_T, const float* __restrict__ b1, const float* __restrict__ w2_T, const float* __restrict__ b2, const float* __restrict__ ln_w, const float* __restrict__ ln_b, float* __restrict__ out, int B, int N, int D, int P, int H) { const int tid = threadIdx.x; const int b = blockIdx.x; const int i = blockIdx.y; extern __shared__ float s_mem[]; float* s_node_i = s_mem; // [D] float* s_nodes_j = s_node_i + D; // [N * D] float* s_pairs = s_nodes_j + N * D; // [N * P] float* s_z1 = s_pairs + N * P; // [H] float* s_max = s_z1 + H; // [D] float* s_red = s_max + D; // [BLOCK_SIZE] // Load node_i for (int idx = tid; idx < D; idx += BLOCK_SIZE) { s_node_i[idx] = nodes[(b * N + i) * D + idx]; } // Load all nodes_j for (int idx = tid; idx < N * D; idx += BLOCK_SIZE) { int j = idx / D; int d = idx % D; s_nodes_j[idx] = nodes[(b * N + j) * D + d]; } // Load all pair_feats for (int idx = tid; idx < N * P; idx += BLOCK_SIZE) { int j = idx / P; int p = idx % P; s_pairs[idx] = pair_feats[((b * N + i) * N + j) * P + p]; } __syncthreads(); // Initialize max for (int idx = tid; idx < D; idx += BLOCK_SIZE) { s_max[idx] = -FLT_MAX; } __syncthreads(); const int L = 2 * D + P; for (int j = 0; j < N; ++j) { float mask_val = edge_mask[((b * N + i) * N + j)]; // First linear + SiLU for (int h = tid; h < H; h += BLOCK_SIZE) { float acc = b1[h]; // node_i part #pragma unroll 4 for (int k = 0; k < D; ++k) { acc += w1_T[k * H + h] * s_node_i[k]; } // node_j part float* node_j_ptr = s_nodes_j + j * D; #pragma unroll 4 for (int k = 0; k < D; ++k) { acc += w1_T[(D + k) * H + h] * node_j_ptr[k]; } // pair part float* pair_ptr = s_pairs + j * P; #pragma unroll 4 for (int k = 0; k < P; ++k) { acc += w1_T[(2 * D + k) * H + h] * pair_ptr[k]; } float sig = 1.0f / (1.0f + expf(-acc)); s_z1[h] = acc * sig; } __syncthreads(); // Second linear, mask, and max for (int d = tid; d < D; d += BLOCK_SIZE) { float val = b2[d]; #pragma unroll 4 for (int h = 0; h < H; ++h) { val += w2_T[h * D + d] * s_z1[h]; } if (mask_val <= 0.5f) { val = -FLT_MAX; } if (val > s_max[d]) { s_max[d] = val; } } __syncthreads(); } // LayerNorm (mean and variance) float local_sum = 0.0f; for (int idx = tid; idx < D; idx += BLOCK_SIZE) { local_sum += s_max[idx]; } s_red[tid] = local_sum; __syncthreads(); #pragma unroll for (int stride = BLOCK_SIZE / 2; stride > 0; stride >>= 1) { if (tid < stride) { s_red[tid] += s_red[tid + stride]; } __syncthreads(); } float mean = s_red[0] / D; float local_var = 0.0f; for (int idx = tid; idx < D; idx += BLOCK_SIZE) { float diff = s_max[idx] - mean; local_var += diff * diff; } s_red[tid] = local_var; __syncthreads(); #pragma unroll for (int stride = BLOCK_SIZE / 2; stride > 0; stride >>= 1) { if (tid < stride) { s_red[tid] += s_red[tid + stride]; } __syncthreads(); } float var = s_red[0] / D; float inv_std = rsqrtf(var + 1e-5f); // Final output for (int idx = tid; idx < D; idx += BLOCK_SIZE) { float v = s_max[idx]; float norm = (v - mean) * inv_std; float o = norm * ln_w[idx] + ln_b[idx]; o += s_node_i[idx]; out[(b * N + i) * D + idx] = o; } } ``` Wait, I should note that `#pragma unroll 4` on loops of varying sizes might not be ideal. For the node_i part with D=96, 96/4=24, so it's fine. For node_j part with D=96, same. For pair part with P=16, 16/4=4, fine. For w2 loop with H=192, 192/4=48, fine. But actually the compiler might not be able to unroll if the loop bound is dynamic (passed as argument). With `#pragma unroll 4`, it should partially unroll in chunks of 4 regardless. Hmm, but actually the loop bound `D`, `P`, `H` are kernel arguments, so they're not compile-time constants. The compiler might ignore `#pragma unroll` for dynamic bounds, or it might generate partial unrolling. Actually in CUDA, `#pragma unroll` works with dynamic bounds too - it just unrolls the body multiple times with a dynamic condition. One concern: in the first linear, I have three separate loops for node_i, node_j, and pair. This creates more instructions than a single loop over L. Let me see if a single loop is better: ```cpp for (int k = 0; k < D; ++k) { acc += w1_T[k * H + h] * s_node_i[k]; } for (int k = 0; k < D; ++k) { acc += w1_T[(D + k) * H + h] * s_nodes_j[j * D + k]; } for (int k = 0; k < P; ++k) { acc += w1_T[(2 * D + k) * H + h] * s_pairs[j * P + k]; } ``` vs ```cpp float* x[208]; // ... load addresses ... for (int k = 0; k < L; ++k) { acc += w1_T[k * H + h] * x[k]; } ``` The three-loop version avoids creating a contiguous buffer `s_x` but requires separate loops. Since we're computing per j, and the data is scattered across shared memory, the separate loops might actually be fine because they keep the accumulator in a register without needing to construct a contiguous array. But wait, in my pre-loaded version, s_nodes_j and s_pairs are already separate arrays. I don't need s_x. The three separate loops are fine. Let me think about one more thing: when we access `s_nodes_j[j * D + k]`, `j` is fixed in the inner loop and `k` varies. Since all threads in a warp access the same `j`, they all access the same `s_nodes_j + j * D` base. Then `k` varies, so threads in a warp access different `k` offsets. But since each thread is computing a different `h`, not a different `k`, and all threads in a warp execute the same `k` sequentially, the access pattern for `s_nodes_j` is actually that ALL threads in a warp read the SAME `s_nodes_j[j * D + k]` for a given `k`. This is a broadcast and hits the same shared memory bank. Actually no - all threads in a warp access the SAME `k` at the SAME time (because they're all executing the same instruction in SIMD). So they all try to read the same shared memory address `s_nodes_j[j * D + k]`. This is handled by a broadcast from shared memory and is very fast! Similarly for `s_node_i[k]`, all threads read the same address - also a broadcast. For `s_pairs[j * P + k]`, same thing. For `w1_T[(D+k)*H + h]`, each thread reads a different `h`, so adjacent threads read adjacent memory - coalesced global memory access. This access pattern is actually very good! The broadcast pattern for the input data is optimal for shared memory. Great, this design is looking solid. Let me implement and test it. Actually, I realize there's another issue. I said "all threads in a warp" and I assumed block_size=128 means 4 warps, and threads within a warp are executing the same instructions. But in CUDA, threads within a warp execute in lockstep (SIMT). Since all threads in a warp are in the same `h` loop iteration (because they all iterate h += block_size starting from different offsets), they might NOT be on the same `h` value at the same time... wait. Actually, within a warp (32 threads), thread IDs are consecutive. With block_size=128, the threads in warp 0 have IDs 0-31. In the loop: ```cpp for (int h = tid; h < H; h += BLOCK_SIZE) { ... } ``` For warp 0: - tid=0: h=0, 128, ... (but H=192, so only h=0) - tid=1: h=1, 129, ... (only h=1) - ... - tid=31: h=31, 159, ... (only h=31) So each thread in the warp computes exactly ONE h value (since H=192 < 256=2*BLOCK_SIZE). Threads 0..63 compute 1 h each, threads 64..127 compute 1 h each (h=64..127 and h=128..191). Wait, H=192, block_size=128. For tid=0: h=0, then 128 (which is < 192), so TWO iterations. For tid=64: h=64, then 192 (not < 192), so ONE iteration. Oh! Some threads do 2 iterations, some do 1. This causes divergence within the warp. For tid=0: h=0, h=128 (2 iterations) For tid=1: h=1, h=129 (2 iterations) ... For tid=63: h=63, h=191 (2 iterations) For tid=64: h=64 (1 iteration) ... For tid=127: h=127 (1 iteration) So warp 0 (threads 0-31) has 2 iterations for all threads. Warp 1 (threads 32-63) has 2 iterations. Warp 2 (threads 64-95) has 1 iteration. Warp 3 (threads 96-127) has 1 iteration. This is divergence! Warps 2 and 3 finish earlier than warps 0 and 1. The __syncthreads() handles this, but the divergent execution in warps 0/1 means some SIMD lanes are doing extra work. Actually, since ALL threads in warps 0 and 1 do exactly 2 iterations, there's NO divergence WITHIN those warps. The divergence is BETWEEN warps (which is fine since warps are independent!). Wait, I miscounted. Warp 0 = threads 0-31, all with tid < 64, so all do 2 iterations. Warp 1 = threads 32-63, all do 2 iterations. Warp 2 = threads 64-95, all do 1 iteration. Warp 3 = threads 96-127, all do 1 iteration. There's no divergence within any warp. Divergence between warps is natural and doesn't cause stalls. Similarly for the d loop, D=96 < 128, so each thread does exactly 1 iteration. No divergence. Great! Now, for h=0 and h=128 in the same thread (tid=0), these are computed sequentially, not in parallel. But since different threads handle different h values in parallel across the warp, the overall H=192 outputs are computed in parallel (2 warps worth). Hmm wait, actually each warp with 2 iterations processes 32 h values in parallel (one per thread in first iteration), then another 32 in parallel (second iteration). So warp 0 processes h=0..31 in parallel, then h=128..159 in parallel. Warp 1 processes h=32..63 then h=160..191. Total: 192 h values processed. The execution is: all warps do first iteration simultaneously, then warps 0 and 1 do second iteration while warps 2 and 3 wait at __syncthreads(). This is fine. OK, the design seems solid. Let me implement it. Actually, one more thing. Let me reconsider block_size=256. With H=192 and D=96: - h loop: tid=0..191 do 1 iteration, tid=192..255 do 0 iterations (divergence in last 2 warps, but they immediately skip the loop) - d loop: tid=0..95 do 1 iteration, tid=96..255 do 0 iterations With block_size=256, we'd have 8 warps, but 2 warps do nothing in the h loop and 5 warps do nothing in the d loop. This wastes SIMD lanes. With block_size=128: - h loop: 2 warps do 2 iterations, 2 warps do 1 iteration - d loop: 3 warps do 1 iteration, 1 warp does nothing Less waste. So 128 seems better than 256. What about block_size=64? - h loop: 64 threads, H=192, so each does 3 iterations (192/64=3 exactly). No divergence! - d loop: 64 threads, D=96, so threads 0..31 do 2 iterations, threads 32..63 do 1 iteration. Divergence in warp 1. With block_size=64, we'd have 96 blocks total (since grid is (4,24)), each with 64 threads. Total threads: 96 * 64 = 6144. Blocks: still only 96, but smaller blocks. More blocks can fit on an SM simultaneously. Wait, with 96 blocks of 64 threads, total concurrent blocks per SM might be higher if shared memory allows. But 96 blocks / 132 SMs < 1 block per SM on average. The small block size doesn't help if there aren't enough blocks. Actually, occupancy is limited by the number of blocks, not thread count. With only 96 output positions, we can't have more than 96 blocks (unless we use a different parallelization strategy). So block_size doesn't fundamentally change occupancy. 128 seems like a good compromise. Let me implement the pre-loaded version with block_size=128. Actually, let me also try block_size=64 just to see. With block_size=64: - h loop: 3 iterations, no divergence. Good! - d loop: 96/64=1.5, so warp 0 (32 threads): 2 iterations, warp 1 (32 threads): 1 iteration (for threads 0..31, warp 1 threads 32..63... wait) For block_size=64, thread IDs 0-63. tid=0..31: d=0, 64, so 2 iterations tid=32..63: d=32, 96 (not < 96), so 1 iteration Warp 0 (0-31): 2 iterations in d loop. Warp 1 (32-63): 1 iteration. Same kind of inter-warp divergence. But for h loop: tid=0..63: h=0, 64, 128, all < 192, so 3 iterations. All 64 threads do 3 iterations. No divergence! So block_size=64 has perfect h-loop parallelism and minor d-loop divergence. block_size=128 has h-loop with 2 warps at 2 iterations and 2 warps at 1 iteration. Since the h-loop has MORE work (192 * L = 192 * 208 ≈ 40K ops) than the d-loop (96 * H = 96 * 192 ≈ 18K ops), the h-loop is more important to optimize. So block_size=64 might be better because it has no h-loop divergence. But wait, with 64 threads, each block finishes faster but there are still only 96 blocks. And we have fewer threads processing in parallel (64 vs 128), so the total wall-clock time per block might be similar or worse because each thread does more work. For h-loop with 64 threads: each thread does 3 h values. Total "thread-h workload" = 64 * 3 = 192 h values. For h-loop with 128 threads: total = 64 * 2 + 64 * 1 = 192 h values. Same total work. But 128 threads process 32+32 first iterations in parallel (64 h values), while 64 threads process 64 h values in first iteration. So 128 threads have more parallel h processing in first iteration (all 128 threads active for first h iteration) compared to 64 threads (64 active). Actually with 128 threads and H=192: - First iteration: threads 0-127 all active, processing h=0..127 (128 values) - Second iteration: threads 0-63 active, processing h=128..191 (64 values) With 64 threads and H=192: - First iteration: threads 0-63 active, processing h=0..63 (64 values) - Second iteration: threads 0-63 active, processing h=64..127 (64 values) - Third iteration: threads 0-63 active, processing h=128..191 (64 values) 128 threads: 128 parallel in first batch, 64 in second. Average ~96 parallel. 64 threads: 64 parallel in all batches. Average 64 parallel. So 128 threads is better for parallelism! And for the d-loop: 128 threads: 96 active in first batch. Average 96 parallel. 64 threads: 64 active in first batch, 32 in second. Average ~48 parallel. 128 threads is also better for d-loop! So 128 threads seems better than 64. What about 96 threads? D=96, H=192. h-loop: 96 threads, 192/96=2 iterations exactly. All threads do 2 iterations. No divergence! 96 parallel. d-loop: 96 threads, 96/96=1 iteration. All do 1. 96 parallel. Perfect! No divergence at all! But 96 is not a power of 2 or a multiple of 32... actually 96 = 3 * 32, which is exactly 3 warps. This is fine in CUDA. With block_size=96: - 96 blocks total (since grid is still B*N = 96) - Total threads: 96 * 96 = 9216 Hmm, but CUDA typically wants block sizes to be multiples of 32 (warp size). 96 is a multiple of 32 (3 warps), so it's fine. But wait, with block_size=96, the grid is still (4, 24) = 96 blocks. Each block has 96 threads. Total blocks = 96. Actually, I realize the grid doesn't have to be (B, N). It could be different. But currently grid=(B,N) and block handles one i. What if I change grid to (B, N, 1) with block_size=32? Then each block has 1 warp. We'd have 96 blocks of 32 threads. - h-loop: 192/32 = 6 iterations. All threads do 6. Perfectly parallel but more loop overhead. - d-loop: 96/32 = 3 iterations. All do 3. Total threads: 96 * 32 = 3072. Even worse. Or block_size=192 (6 warps): - h-loop: 192/192 = 1 iteration. All 192 threads active. 192 parallel! But we only have H=192 values, so this is perfect. - d-loop: 96/192 = 0.5 iterations. Threads 0-95 do 1 iteration, threads 96-191 do nothing. 192 threads = 6 warps. For d-loop, 2 warps are idle during the loop but they participate in __syncthreads(). No intra-warp divergence though. Shared memory with block_size=192: - Pre-load data: same - z1: H=192 floats - max: D=96 floats - red: 192 floats Total: 96 + 2304 + 384 + 192 + 96 + 192 = 3264 floats = 13056 bytes Still fine. But 96 blocks of 192 threads = 18432 total threads. Total blocks is still 96 (same occupancy issue). For h-loop, 192 threads gives perfect parallelism (1 iteration, 192 threads). For d-loop, 96 threads active. This is better than 128 threads for h-loop (1 iteration with 192 active vs 1 iteration with 128 active + 1 iteration with 64 active). Actually, with block_size=192: - h-loop: 1 iteration, 192 threads process all 192 h values simultaneously. - d-loop: 1 iteration (first), 96 threads process 96 d values, 96 threads idle. Time for h-loop: T_h / 192 (all 192 threads working) Time for d-loop: T_d / 96 (only 96 threads working, 96 idle) With block_size=128: - h-loop: ~T_h / 128 (first iteration) + T_h / 64 (second iteration, only 64 threads) ≈ T_h * (1/128 + 1/64) = T_h * 3/128 ≈ T_h / 42.7 equivalent parallel threads. Wait, this isn't quite right. Actually, all 128 threads work in first iteration (processing h=0..127), then threads 0..63 work in second iteration (processing h=128..191). Time = max(T_h1 / 128, T_h2 / 64) where T_h1 = 128 * (work per h) and T_h2 = 64 * (work per h). Actually since work per h is the same, time for iteration 1 = work_per_h * ceil(128/128) = work_per_h * 1 (but all 128 threads do work in parallel). Time for iteration 2 = work_per_h * ceil(64/128) = work_per_h * 1 (64 threads active). Total time = 2 * work_per_h (since iterations are sequential). With block_size=192: - h-loop: 1 iteration. Time = work_per_h * ceil(192/192) = work_per_h * 1. All 192 threads active. Total time = 1 * work_per_h. So block_size=192 is 2x faster for the h-loop! For d-loop: With block_size=192: 1 iteration, 96 threads active. Time = work_per_d * 1. With block_size=128: 1 iteration, 96 threads active (threads 0..95). 32 threads idle. Time = work_per_d * 1. Same for d-loop. So block_size=192 is significantly better for h-loop and same for d-loop. But we only have 96 blocks total, and block_size=192 means fewer blocks can run concurrently on an SM because of register/thread limits. But with only 96 blocks across 132 SMs, this might not matter. Wait, there's one issue. With block_size=192 and 96 blocks, can we even run on an SM? H100 has a limit of 2048 threads per SM. 192 threads per block means up to 10 blocks per SM (due to thread limit). With 96 blocks / 132 SMs = 0.73 blocks per SM. So we're still way underutilized. The register limit might matter more. With H100's 64K registers per SM and 255 max per thread: - block_size=192, max threads per SM = 2048, max blocks per SM = min(2048/192=10, 32) = 10. - But if each thread uses > 64K/2048 = 32 registers, we can't reach 2048 threads. Let me estimate register usage. The kernel has a lot of local variables: - Several float variables (acc, sig, val, mask_val, mean, var, etc.) - A few integer indices - The entire computation is in registers + shared memory. I suspect the compiler can optimize this to use ~40-80 registers per thread. With 192 threads, that's 192 * 64 = 12288 registers per block. Per SM: up to 64K/12288 = 5 blocks. Still fine with 0.73 blocks per SM. So block_size=192 should be feasible and potentially faster. Let me try it. But wait, I can also try block_size=96: - h-loop: 192/96 = 2 iterations. All 96 threads active for both. Time = 2 * work_per_h. - d-loop: 96/96 = 1 iteration. All 96 threads active. Time = 1 * work_per_d. For h-loop: 192 threads take 1 iteration, 96 threads take 2 iterations. Same total "thread-iterations" (192 each). But with 192 threads, all 192 work in parallel in 1 iteration. With 96 threads, 96 work in parallel for 2 iterations. The wall-clock time should be the same if the work per iteration is identical. Wait, is that right? - block_size=192: 1 iteration, 192 threads each do 1 h value. Total parallel work = 192 h values in 1 time unit. - block_size=96: 2 iterations, 96 threads each do 2 h values (one per iteration). Total parallel work = 96 h values in iteration 1 + 96 h values in iteration 2 = 192 h values in 2 time units. Time units: - 192 block: 1 unit - 96 block: 2 units So 192 is twice as fast for h-loop! Similarly: - block_size=128: iteration 1 (128 threads, 128 h values, time = T), iteration 2 (64 threads, 64 h values, time = T). Total time = 2T. But T for iteration 2 is half the work, so maybe T/2? Actually, each thread does the same amount of work per h. In iteration 1, 128 threads process 128 h values. In iteration 2, 64 threads process 64 h values. If all threads are fully utilized and there's no divergence, each iteration takes the same wall-clock time (since the work is proportional to active threads). So iteration 1 takes T, iteration 2 takes T (not T/2, because although there are half as many h values, there are also half as many threads working). Wait, that assumes perfect scaling. In practice, with fewer active threads, the warp utilization is lower. Warp 0-1 (64 threads) process iteration 2 while warps 2-3 (64 threads) are idle. So only 2 out of 4 warps are active. It still takes T because the 2 active warps are fully busy. With block_size=192: 1 iteration, all 6 warps active. Time = T. For block_size=128: 2 iterations, 4 warps then 2 warps. Time = 2T (approximately). For block_size=96: 2 iterations, 3 warps each time. Time = 2T. For block_size=64: 3 iterations, 2 warps each time. Time = 3T. For block_size=256: 1 iteration for h (all 192 active threads in first 6 warps, 64 idle in last 2 warps). Time = T. But d-loop: 3 warps active, 5 warps idle. Time = T. Wait, with block_size=256: - h-loop: 192 threads do 1 iteration, 64 threads idle. 6 warps active, 2 warps idle. Time = T. - d-loop: 96 threads do 1 iteration, 160 threads idle. 3 warps active, 5 warps idle. Time = T. Total per j: 2T. With block_size=192: - h-loop: 192 threads, 1 iteration. 6 warps active. Time = T. - d-loop: 96 threads, 1 iteration. 3 warps active, 3 warps idle. Time = T. Total per j: 2T. Wait, same total? Let me recalculate. For block_size=256: - Grid = 96 blocks. - Active threads in h-loop = 192 (threads 0..191) - Active threads in d-loop = 96 (threads 0..95) - Both loops take T each (since active threads fullywork for their duration). - Per j: 2T. For block_size=192: - Active threads in h-loop = 192 (all threads) - Active threads in d-loop = 96 (threads 0..95) - h-loop: 1 iteration, time = T (192 values / 192 threads) - d-loop: 1 iteration, time = T (96 values / 96 threads) - Per j: 2T. Same per-j time! But with fewer total threads (192 vs 256), the block_size=192 version has less overhead (fewer warps to manage, less __syncthreads() overhead). For block_size=128: - h-loop: 128 threads active in iter 1 (time T), 64 active in iter 2 (time T). Total 2T. - d-loop: 96 threads active in iter 1 (time T), 32 idle. Total T. - Per j: 3T. Worse than 192 and 256! For block_size=96: - h-loop: 96 threads, 2 iterations. Time 2T. - d-loop: 96 threads, 1 iteration. Time T. - Per j: 3T. Same as 128. So block_size=192 and block_size=256 both give 2T per j, which is better than 96/128. But wait, I need to reconsider. The time T for a loop iteration depends on the amount of work per thread. In h-loop with block_size=256: Each of 192 active threads computes 1 h value. Time = T_1. (T_1 is the time for one thread to do its work, since all warp threads execute in parallel) In h-loop with block_size=192: Each of 192 active threads computes 1 h value. Time = T_1. Same! So both take T_1 for h-loop. In d-loop with block_size=256: Each of 96 active threads computes 1 d value. Time = T_2. In d-loop with block_size=192: Each of 96 active threads computes 1 d value. Time = T_2. Same! Both take T_2. So block_size=192 and block_size=256 have the SAME per-j time! But 256 wastes warps (some idle in both loops). 192 has no waste in h-loop and partial waste in d-loop. Wait, but with block_size=256, the d-loop has threads 0..95 active. These are 3 warps. Warps 3..7 (threads 96..255) are idle. In CUDA, a warp where all threads are inactive is very cheap - it just doesn't execute. So the idle warps don't add overhead. With block_size=192, the d-loop has threads 0..95 active (3 warps). Warps 3..5 (threads 96..191) are idle. Same situation. So both should perform similarly. But block_size=256 might have slightly higher overhead due to larger __syncthreads() (more warps to sync) and larger reduction arrays. Actually, another consideration: the pre-load phase. With block_size=256, more threads load data in parallel. But the data to load is the same size. With 256 threads, the data loads complete in fewer steps but with more threads. For N*D = 2304 floats: - block_size=192: 2304/192 = 12 loads per thread (roughly, after coalescing). Actually each thread loads 2304/192 = 12 floats, but they load in a strided pattern. - block_size=256: 2304/256 = 9 loads per thread. Since each load is a memory instruction that all threads in a warp execute together, the total time is similar. But with more threads, there might be slightly more memory bandwidth utilization. I'll go with block_size=192. It gives perfect h-loop parallelism and doesn't waste too much in the d-loop (only 3 of 6 warps idle). Actually wait, let me reconsider. If I use block_size=192, can I pack the grid differently? E.g., have each block handle TWO (b, i) pairs with 96 threads each? Grid: each block handles 2 (b, i) pairs. Total blocks = 96/2 = 48. Within a block of 192 threads: - Threads 0..95 handle (b0, i0) - Threads 96..191 handle (b1, i1) Each sub-group of 96 threads has: - h-loop: 2 iterations, all 96 threads active - d-loop: 1 iteration, all 96 threads active This means: - h-loop: 2 iterations × 96 threads = 192 h values total. But per (b,i), 96 threads do 2 iterations. - For both (b,i) pairs simultaneously, we process 192 h values total in 2 time units... no wait. Actually, if threads 0..95 and 96..191 work on DIFFERENT (b,i), there's no interaction between them. They just happen to be in the same block. The h-loop would have all 192 threads active (96 for each task). Time = 2T (each sub-group needs 2 iterations). For d-loop: both sub-groups need 1 iteration. All 192 threads active. Time = T. So total per j per block: 2T + T = 3T for processing 2 outputs, or 1.5T per output. But we also have fewer blocks (48), so occupancy is worse. Hmm, this doesn't seem better. What if each block handles 4 (b, i) pairs with 48 threads each? - h-loop: 192/48 = 4 iterations. 48 threads × 4 = 192 total. Each sub-group needs 4 iterations. - d-loop: 96/48 = 2 iterations. - Grid = 24 blocks. Even worse occupancy. I think keeping one (b,i) per block is best for simplicity. Let me go with block_size=192 or 256. Actually, let me try yet another approach. What if I don't make H the parallel dimension in the h-loop, but instead make each thread compute multiple h values and I use loop unrolling? No, that would reduce parallelism. OK let me just code up block_size=192 with pre-loaded data and test. Actually, one more thought. I could ALSO pre-load w1_T to shared memory! Since w1_T is 160KB, and shared memory is 228KB per SM, I can load all of w1_T to shared memory. But then I need to be careful about total usage. With pre-loaded data (13KB), z1 (0.75KB), max (0.375KB), red (192*4=0.75KB for block_size=192), and w1_T (160KB): Total = 13 + 0.75 + 0.375 + 0.75 + 160 = 174.875KB This fits! And having w1_T in shared memory would make the h-loop much faster since w1_T is accessed repeatedly for each j (24 times). But w2_T is only 74KB. Total with w2_T would be 248.875KB, which exceeds 228KB. So we can only put w1_T in shared memory, not w2_T. Or we could put w2_T in shared memory and leave w1_T in L1: Data + z1 + max + red + w2_T = 13 + 0.75 + 0.375 + 0.75 + 74 = 88.875KB This also fits! But w1_T is accessed more (24 times per block, once per j), while w2_T is also accessed 24 times per block. Both are heavily reused. Actually, w1_T is accessed 24 * 192 * 208 ≈ 958K times per block. w2_T is accessed 24 * 96 * 192 ≈ 442K times per block. Both are heavily reused. L1 cache on H100 has 256KB per SM. w1_T is 160KB and w2_T is 74KB, totaling 234KB. This might ALMOST fit in L1 cache. But since L1 is shared among all blocks on an SM, and we only have ~0.7 blocks per SM, the L1 is mostly available to a single block. So the weights might stay in L1. But can we rely on L1? L1 is also used for global memory accesses, register spills, etc. It might evict our weights. Loading w1_T to shared memory guarantees it's resident. Let me try that. Wait, but with w1_T in shared memory, we need to load it once at the beginning. That's 160KB / (128 bytes per load per warp) = 160KB / (32*4 bytes per coalesced load) ≈ 1250 coalesced loads. With 192 threads = 6 warps, each warp needs ~208 loads. Total warp-loads = 6 * 208 = 1248. Actually, each thread can load multiple floats. With float4 loads, we can load 4 floats per thread. Total threads = 192. So 192 * 4 = 768 floats per coalesced cycle. w1_T has 208 * 192 = 39936 floats. So 39936 / 768 = 52 cycles to load all of w1_T. 52 cycles * 0.7 ns ≈ 36 ns. Very fast! Then each j iteration reads w1_T from shared memory instead of global. Since w1_T is accessed 24 times, we'd save 24 global memory reads of w1_T, which is significant. I think pre-loading w1_T to shared memory is worthwhile. Let me include it. Shared memory layout: ``` float* s_w1 = s_mem; // [L * H] float* s_node_i = s_w1 + L * H; // [D] float* s_nodes_j = s_node_i + D; // [N * D] float* s_pairs = s_nodes_j + N * D; // [N * P] float* s_z1 = s_pairs + N * P; // [H] float* s_max = s_z1 + H; // [D] float* s_red = s_max + D; // [BLOCK_SIZE] ``` Total = L*H + D + N*D + N*P + H + D + BLOCK_SIZE = 208*192 + 96 + 2304 + 384 + 192 + 96 + 192 = 39936 + 96 + 2304 + 384 + 192 + 96 + 192 = 43200 floats = 172800 bytes = 168.75KB Fits in 227KB! (Actually 228KB = 233472 bytes, and 172800 < 233472) Wait, but L*H = 208*192 = 39936 floats. Let me double check: L = 2*96 + 16 = 208 H = 192 L*H = 39936 floats = 159744 bytes Plus other: (96 + 2304 + 384 + 192 + 96 + 192) * 4 = 3364 * 4 = 13456 bytes Total = 159744 + 13456 = 173200 bytes = 169.14KB Yes, this fits! But wait, do I have enough shared memory to also load w2_T? w2_T = H*D = 192*96 = 18432 floats = 73728 bytes Total with w2_T = 173200 + 73728 = 246928 bytes = 241KB This exceeds 228KB! Can't fit both. I'll put w1_T in shared memory since it's larger (160KB vs 74KB) and accessed more times (same 24 times, but each access reads more data: 958K reads vs 442K reads). Actually, w1_T and w2_T are accessed the same number of times (once per j). The amount of data read per j is: - w1_T: 192 * 208 = 39936 floats - w2_T: 96 * 192 = 18432 floats So w1_T has more data but same number of accesses. The per-j bandwidth for w1_T is 39936 * 4 bytes = 159KB per j, or 24 * 159KB = 3.8MB total per block. w2_T is 1.8MB total per block. At 3.35TB/s, reading w1_T from global 24 times takes ~1.1 microseconds total. Reading w2_T takes ~0.5 microseconds. If we can cache w1_T in shared memory, we save ~1.1 microseconds. This seems worthwhile! But we also need to PAY for loading w1_T into shared memory once. That's 160KB from global to shared: 160KB / 3.35TB/s ≈ 0.05 microseconds. Very cheap compared to 24 re-reads! So pre-loading w1_T to shared memory is clearly a win. Let me implement this. Updated kernel: ```cuda #define BLOCK_SIZE 192 __global__ void fused_scene_graph_kernel(...) { const int tid = threadIdx.x; const int b = blockIdx.x; const int i = blockIdx.y; extern __shared__ float s_mem[]; float* s_w1 = s_mem; // [L * H] float* s_node_i = s_w1 + L * H; // [D] float* s_nodes_j = s_node_i + D; // [N * D] float* s_pairs = s_nodes_j + N * D; // [N * P] float* s_z1 = s_pairs + N * P; // [H] float* s_max = s_z1 + H; // [D] float* s_red = s_max + D; // [BLOCK_SIZE] // Load w1_T to shared memory for (int idx = tid; idx < L * H; idx += BLOCK_SIZE) { s_w1[idx] = w1_T[idx]; } // Load node_i for (int idx = tid; idx < D; idx += BLOCK_SIZE) { s_node_i[idx] = nodes[(b * N + i) * D + idx]; } // Load all nodes_j for (int idx = tid; idx < N * D; idx += BLOCK_SIZE) { int j = idx / D; int d = idx % D; s_nodes_j[idx] = nodes[(b * N + j) * D + d]; } // Load all pair_feats for (int idx = tid; idx < N * P; idx += BLOCK_SIZE) { int j = idx / P; int p = idx % P; s_pairs[idx] = pair_feats[((b * N + i) * N + j) * P + p]; } __syncthreads(); // Initialize max for (int idx = tid; idx < D; idx += BLOCK_SIZE) { s_max[idx] = -FLT_MAX; } __syncthreads(); for (int j = 0; j < N; ++j) { float mask_val = edge_mask[((b * N + i) * N + j)]; // First linear + SiLU for (int h = tid; h < H; h += BLOCK_SIZE) { float acc = b1[h]; #pragma unroll 4 for (int k = 0; k < D; ++k) { acc += s_w1[k * H + h] * s_node_i[k]; } float* node_j_ptr = s_nodes_j + j * D; #pragma unroll 4 for (int k = 0; k < D; ++k) { acc += s_w1[(D + k) * H + h] * node_j_ptr[k]; } float* pair_ptr = s_pairs + j * P; #pragma unroll 4 for (int k = 0; k < P; ++k) { acc += s_w1[(2 * D + k) * H + h] * pair_ptr[k]; } float sig = 1.0f / (1.0f + expf(-acc)); s_z1[h] = acc * sig; } __syncthreads(); // Second linear, mask, and max for (int d = tid; d < D; d += BLOCK_SIZE) { float val = b2[d]; #pragma unroll 4 for (int h = 0; h < H; ++h) { val += w2_T[h * D + d] * s_z1[h]; } if (mask_val <= 0.5f) { val = -FLT_MAX; } if (val > s_max[d]) { s_max[d] = val; } } __syncthreads(); } // LayerNorm... } ``` The total shared memory now: L*H + D + N*D + N*P + H + D + BLOCK_SIZE = 39936 + 96 + 2304 + 384 + 192 + 96 + 192 = 43200 floats = 172800 bytes We need to set this in the kernel launch. Also, since w1_T is now in shared memory, the access `s_w1[k * H + h]` is a shared memory access. With h varying by thread, adjacent threads access adjacent shared memory locations (bank 0, 1, 2, 3, ...). This has no bank conflicts! For the node_i broadcast: `s_node_i[k]` is the same address for all threads in a warp (since k is the loop variable, same for all threads at each step). This is a broadcast with no bank conflicts. For node_j: `s_nodes_j[j * D + k]`, j is fixed for the iteration and k is the loop variable. All threads access the same address at each step. Broadcast, no bank conflicts. For pairs: same, broadcast, no bank conflicts. For w2_T: `w2_T[h * D + d]` with d varying by thread. Coalesced global memory, no conflicts. This access pattern is almost ideal! One small thing: in the h-loop with block_size=192 and H=192, each thread has exactly 1 iteration (h = tid). No loop overhead! This is great. For the d-loop with block_size=192 and D=96, threads 0..95 have h = tid, which is < 96, so 1 iteration. Threads 96..191 have h >= 96, so 0 iterations. One caveat: I should make sure to compile and test. Let me do that. Wait, I also need to double-check my pre-load indexing for w1_T. w1_T has shape (L, H) = (208, 192). In shared memory, I put it as s_w1[idx] = w1_T[idx], which is row-major. So s_w1[k * H + h] accesses the same element as w1_T[k, h]. This matches my existing access pattern. Good! Let me also consider whether to pre-load b1, b2, ln_w, ln_b to shared memory. These are tiny: - b1: 192 floats = 768 bytes - b2: 96 floats = 384 bytes - ln_w: 96 floats = 384 bytes - ln_b: 96 floats = 384 bytes Total: 1920 bytes Very small. But they might already be cached in L1. Pre-loading them doesn't hurt though. Actually, for b1 and b2, they're accessed in the loops. b1[h] in the h-loop: each thread reads one b1 value. With 192 threads and block_size=192, all 192 b1 values are read coalesced (since h varies by thread). This is already efficient. For b2[d] in the d-loop: threads 0..95 read b2[0..95] coalesced. Threads 96..191 don't read. Also efficient. For ln_w[d] and ln_b[d] in the final output: same pattern. So I won't pre-load these. Let me now write the complete kernel code with these optimizations. Another consideration: should I pre-load w2_T to shared memory too? It's 74KB. Let me check if everything would fit with w2_T: With w2_T: 172800 + 73728 = 246928 bytes = 241KB > 228KB. Doesn't fit. What if I DON'T pre-load pair_feats? Then I read pair_feats from global memory on demand. That saves N*P = 384 floats = 1536 bytes. Not much. What if I don't pre-load nodes_j either? Then I read nodes_j from global memory on demand for each j. That saves N*D = 2304 floats = 9216 bytes. But then each j iteration reads D floats from global memory, which is much slower than shared memory. For 24 j iterations: 24 * 96 = 2304 floats read from global per block. With shared memory, I read 2304 floats once. The on-demand approach would read the same 2304 floats but from global memory, and they'd be cached in L1/L2. But the first read would come from DRAM. Actually, nodes_j for a given (b, j) is the same across all blocks for the same b but different i. So if multiple blocks (for same b, different i) are on the same SM, they share the L1 cache but not shared memory. The first block to read nodes_j would cache it in L1, and subsequent blocks on the same SM would get L1 hits. But with only ~0.7 blocks per SM, there's limited reuse. And even within one block, if I don't pre-load nodes_j, each j iteration reads from L1 (after the first j caches it). But since the j loop is sequential, the L1 would have the data for the current j. After finishing j, we move to j+1, which might evict j from L1. Pre-loading to shared memory is still better because it's explicitly resident. I think not pre-loading w2_T is fine. Let me proceed. One more thing: I can also process multiple j values in parallel if I increase parallelism. But with the current design, we're limited by shared memory for z1 (one z1 vector of size H). What if I process two j values at a time, requiring two z1 vectors? With 2 z1 vectors: 2 * 192 * 4 = 1536 bytes extra. Total shared memory: 172800 + 1536 = 174336 bytes. Still fits! But then each thread needs to do work for two j values, or we need to launch more threads. With block_size=192, if we want to process 2 j values in parallel, we could have threads 0..95 handle j=0 and threads 96..191 handle j=1. But then for h-loop, each sub-group has 96 threads for 192 h values, requiring 2 iterations each. Time for h-loop = 2T. For d-loop, each sub-group has 96 threads for 96 d values, 1 iteration each. Time = T. Total per 2 j values: 2T + T = 3T. Or 1.5T per j. Current design: 1T + 1T = 2T per j. So processing 2 j in parallel is SLOWER (1.5T vs 2T? wait, let me recalculate). Current design (1 j at a time): - h-loop: 192 threads, 1 iteration. Time = T_h. - d-loop: 96 threads, 1 iteration. Time = T_d. - Per j: T_h + T_d. 2 j in parallel (2 sub-groups of 96): - h-loop: each sub-group needs 2 iterations (96 threads, 192 h values). Both sub-groups run simultaneously since they're different threads. Time = 2 * T_h (since each sub-group needs 2 iterations). - d-loop: each sub-group needs 1 iteration. Both run simultaneously. Time = T_d. - Per 2 j: 2*T_h + T_d. - Per j: T_h + 0.5*T_d. If T_h ≈ T_d (similar work), current design is T_h + T_d = 2T per j, while 2j parallel is T_h + 0.5*T_d = 1.5T per j. So 2j parallel might be better! Wait, is T_h ≈ T_d? T_h: 192 outputs, each requiring L=208 multiply-adds + 1 SiLU. Work per output: 208 FMA + ~10 ops for SiLU. T_d: 96 outputs, each requiring H=192 multiply-adds + 1 comparison + 1 max. Work per output: 192 FMA + 2 ops. Total work for T_h: 192 * 208 ≈ 40K FMAs. Total work for T_d: 96 * 192 ≈ 18K FMAs. Plus SiLU overhead for h-loop. So T_h ≈ 2.2 * T_d (in terms of FMAs). Let me set T_d = T, T_h ≈ 2.2T. Current design per j: 2.2T + T = 3.2T. 2j parallel per j: 2.2T + 0.5T = 2.7T. About 15% improvement. Not revolutionary but worthwhile. But wait, with 2j parallel, the j-loop count goes from 24 to 12 iterations. The __syncthreads() overhead is halved! 12 * 3.2T = 38.4T total for 2j parallel 24 * 3.2T = 76.8T total for current... wait no. For current: 24 j iterations × (T_h + T_d) = 24 × 3.2T = 76.8T. For 2j parallel: 12 "pairs" × (2T_h + T_d) = 12 × (4.4T + T) = 12 × 5.4T = 64.8T. Hmm, 64.8T vs 76.8T is a 15% improvement. And fewer __syncthreads(). But implementing 2j parallel is more complex. Let me see if it's worth it. Actually, with block_size=192, we can split into 2 groups of 96. But what about 3j parallel? 192/3 = 64 threads per j. - h-loop: 64 threads, 192 h values, 3 iterations. - d-loop: 64 threads, 96 d values, 2 iterations. - Per 3j: 3T_h + 2T_d = 6.6T + 2T = 8.6T. - Per j: 2.87T. - Total: 8 iterations × 8.6T = 68.8T. Similar to 2j. Not better. What about 6j parallel? 192/6 = 32 threads per j (one warp). - h-loop: 32 threads, 192 h values, 6 iterations. - d-loop: 32 threads, 96 d values, 3 iterations. - Per 6j: 6T_h + 3T_d = 13.2T + 3T = 16.2T. - Per j: 2.7T. - Total: 4 iterations × 16.2T = 64.8T. Same as 2j. OK so the per-j time doesn't improve much beyond 2j. Let me try 2j parallel. Wait, actually there's a better way. What if we use 96 threads per block, and each block processes 2 (b,i) pairs simultaneously? Then for each (b,i), we process j sequentially as before. Grid: 96/2 = 48 blocks (each handling 2 outputs). Block: 192 threads = 2 × 96. For each (b,i) sub-group of 96 threads: - h-loop: 96 threads, 192 h values, 2 iterations. - d-loop: 96 threads, 96 d values, 1 iteration. Per j for one sub-group: 2T_h + T_d. For 24 j: 24 × (2T_h + T_d). Both sub-groups work independently. No divergence since each warp has threads from the same sub-group. Wait, but 96 isn't a multiple of 32! 96 = 3 warps. Good. 192 threads = 6 warps. We can have: - Warps 0,1,2 (threads 0..95) handle output 0 - Warps 3,4,5 (threads 96..191) handle output 1 No divergence within warps! Shared memory for 2 outputs: Each output needs its own node_i, z1, max, pair_feats. But node_j can be shared if both outputs are in the same batch b and just different i! Wait, node_j depends on b and j but NOT on i. So both sub-groups with the same b share the same set of node_j! But pair_feats depends on i, so each sub-group needs its own pair_feats. Shared memory: - w1_T: 160KB (shared) - node_j: 24 * 96 * 4 = 9216 bytes (shared between both on same b) - node_i_0: 96 * 4 = 384 bytes - node_i_1: 384 bytes - pair_0: 24 * 16 * 4 = 1536 bytes - pair_1: 1536 bytes - z1_0: 192 * 4 = 768 bytes - z1_1: 768 bytes - max_0: 96 * 4 = 384 bytes - max_1: 384 bytes - red: 192 * 4 = 768 bytes (one shared reduction array, used sequentially) Total = 160000 + 9216 + 384 + 384 + 1536 + 1536 + 768 + 768 + 384 + 384 + 768 = 174328 bytes = 170KB Still fits! But the grid is now 48 blocks. Even worse occupancy. Hmm. Can we do 4 outputs per block (48 threads each)? 48 = 1.5 warps... not an integer. Won't work well. Can we do 3 outputs per block (64 threads each)? 64 = 2 warps. 3 outputs = 192 threads. This works! But then each sub-group has 64 threads, so: - h-loop: 64 threads, 192 h values, 3 iterations. - d-loop: 64 threads, 96 d values, 2 iterations (64 + 32). Per j: 3T_h + 2T_d. For 3 outputs: 3 sub-groups, but 2 have 2 iterations in d-loop and 1 has 1 iteration (96/64 = 1.5). Wait, for d-loop with 64 threads: - First iteration: threads 0..63 process d=0..63. - Second iteration: threads 0..31 process d=64..95, threads 32..63 idle. Intra-warp divergence! Warp 0 (0..31) does 2 iterations, warp 1 (32..63) does 1 iteration. Warp 1 waits while warp 0 does second iteration. This is bad. So 64 threads per (b,i) has d-loop divergence. Not ideal. What about 48 threads? 48 = 1.5 warps. Not good for SIMD. 32 threads per (b,i)? 32 = 1 warp. Perfect. 192/32 = 6 outputs per block. 6 warps, each handling one (b,i). h-loop: 32 threads, 192 h values, 6 iterations. 6 iterations with same warp. No divergence. d-loop: 32 threads, 96 d values, 3 iterations. No divergence. Per j: 6T_h + 3T_d. For 6 outputs simultaneously, total per j across block: same as 6 × (6T_h + 3T_d)... no, they run in parallel across warps. Wait, 6 warps each process one (b,i). Each warp takes 6T_h + 3T_d per j. All 6 warps run independently (same instructions, different data). Total block time per j: 6T_h + 3T_d. But we're processing 6 outputs simultaneously! So per-output time: (6T_h + 3T_d) / 6 = T_h + 0.5T_d. Current single-output-per-block with block_size=192: Per-output time: T_h + T_d. With T_h ≈ 2.2T_d: Current: 2.2T_d + T_d = 3.2T_d. 6-output block: 2.2T_d + 0.5T_d = 2.7T_d. 15% better per-output! And total blocks: 96/6 = 16 blocks. Even worse occupancy though. Hmm, but we still only have 16 blocks total. With 132 SMs, 16 blocks is very few. But each block does 6x more work. I think the issue is that this problem has limited parallelism. With B=4, N=24, there are only 96 independent outputs. We can't create more independent outputs. One option: process j values in parallel across threads. If we don't need the max reduction until the end, we could have all N=24 j values processed in parallel. For example, each thread computes the fused output for ONE (i, j) pair. Then we need a parallel max reduction over j. With D=96 output dimensions per (i,j), one thread could compute all 96 dimensions. But that's a lot of redundant computation (each thread computes the full MLP). The total work for all j in parallel: 24 j values × 96 d values = 2304 outputs per (b, i). With 192 threads, each thread computes 2304/192 = 12 outputs. For each output, we need the MLP1 (z1) first. But z1 depends ONLY on i and j, not on d. So ALL d values for the same (i,j) share the same z1. This insight is key! Each (b, i, j) produces one z1 vector (size H=192). Then z2 is computed for each d using the same z1. So if a thread handles all d values for a specific (b, i, j), it: 1. Computes z1 for (b, i, j) once 2. Then computes z2[d] for d=0..95 using the same z1 Work per (b, i, j): - z1: 192 * 208 FMAs + 192 SiLU ≈ 40K ops - z2: 96 * 192 FMAs ≈ 18K ops Total: ~58K ops One thread per (b, i, j): 24 threads per (b, i). Total threads for 96 (b, i) = 96 * 24 = 2304 threads. Or we could launch one block per (b, i) with 24 threads... but 24 threads = 0.75 warps, not a full warp. That's inefficient. With 32 threads per block (one warp): 24 threads do work, 8 idle. Grid = 96 blocks. Total threads: 96 * 32 = 3072. Occupancy: 3072 / (132 * 2048) ≈ 1.1%... very low. But each thread computes 1 (b, i, j) completely (including all d values). Then we need a max reduction over j. After all threads write their (b, i, j, d) results to shared/global memory, we can do a parallel max over j. Hmm, but with 32 threads per block and 96 blocks, the blocks are very small. Not sure if this helps. Actually, let me think about total parallelism. If each thread handles one (b, i, j, d) completely, we'd need: Total (b,i,j,d) = 4 * 24 * 24 * 96 = 221,184 items. That's a lot of parallelism! But computing one (b,i,j,d) requires the full z1 vector for (b,i,j). If each thread computes z1 from scratch, that's 40K ops of redundant work (since all d values for same j share z1). If each thread computes one d for all j, then z1 needs to be recomputed for each j, which is also redundant. The optimal approach depends on whether we're compute-bound or memory-bound. Current approach: one block per (b,i), sequential j loop. Total redundant work: zero. Total parallelism: 96 blocks. Alternative: one thread per (b,i,j), compute all d. Total work: 96 * 24 * 58K = 133M ops (same!). Total parallelism: 2304 threads (or 3072 with 32 threads per block). This gives 4x more threads! And 3072 vs 96*192=18432... actually 3072 is LESS than 18432 (the current total thread count). Wait, current has 96 blocks × 192 threads = 18432 total threads, but they're in only 96 blocks. Each block has lots of threads but few blocks. With 32 threads per block and 96 blocks: 96 * 32 = 3072 total threads in 96 blocks. Fewer total threads but same number of blocks. What if we make smaller blocks with more parallel work per thread, but still 96 blocks? Say 64 threads per block, each thread handles 2 j values (or multiple d values). I think the fundamental issue is that we only have 96 independent "output positions" (b, i). We can parallelize within each position, but the outer grid is limited to 96. One way to increase grid size: process d or h in the grid dimensions. - Grid = (B, N, N) = 2304, each block handles one (b, i, j), block_size = D threads (or some factor). This is the two-kernel approach I considered earlier. Let me reconsider it more carefully. Kernel 1: Compute fused (before max) for all (b, i, j, d) - Grid: (B, N, N) = 2304 blocks - Block: one warp (32 threads) or more, computing all d values for one (b, i, j) Actually, we can have block_size = 32 threads, each handling a subset of d values. But all d values need the same z1, which should be computed collaboratively. Block of 96 threads (3 warps) handling one (b, i, j): - First, 96 threads compute z1 collaboratively: each thread handles 2 h values. 2 iterations. ~2T time. - All threads sync. - Then, 96 threads compute z2[d] for d=0..95: each thread handles 1 d value. 1 iteration. ~T time. - Apply mask and write to global memory (B, N, N, D). Grid: 2304 blocks of 96 threads each. Total threads: 2304 * 96 = 221,184. This is massive parallelism! Kernel 2: Max over j + LayerNorm + residual - Grid: (B, N) = 96 blocks - Block: 96 threads (or 128) - Read from global memory (B, N, N, D), compute max over j, then LN, then residual. The second kernel has poor parallelism (96 blocks), but it's lightweight. Total global memory traffic: - Kernel 1 writes: B*N*N*D floats = 4*24*24*96 = 221K floats = 884KB - Kernel 2 reads: 884KB - Kernel 2 writes: B*N*D floats = 9K floats = 36KB Extra traffic: ~1.8MB of intermediate data. At 3.35TB/s, this is ~0.5 microseconds. Compared to current single-kernel approach: - No intermediate global memory - 96 blocks of 192 threads = 18432 total threads active - But each block has sequential j loop with 24 iterations and many __syncthreads() The two-kernel approach gives much better utilization for the heavy MLP computation. The second kernel is lightweight and can handle the reduction quickly. I think the two-kernel approach might be better overall because: 1. Better grid-level parallelism (2304 vs 96 blocks) 2. Better warp utilization (all warps fully active in kernel 1) 3. Simpler per-block logic Let me try this! For kernel 1, each block handles (b, i, j), computing all d outputs. ```cuda __global__ void compute_fused_kernel( const float* nodes, const float* pair_feats, const float* edge_mask, const float* w1_T, const float* b1, const float* w2_T, const float* b2, float* fused, int B, int N, int D, int P, int H) { int b = blockIdx.x; int i = blockIdx.y; int j = blockIdx.z; int tid = threadIdx.x; __shared__ float s_z1[192]; __shared__ float s_node_i[96]; __shared__ float s_node_j[96]; __shared__ float s_pair[16]; // Collaborative load of inputs for (int idx = tid; idx < D; idx += blockDim.x) { s_node_i[idx] = nodes[(b * N + i) * D + idx]; } for (int idx = tid; idx < D; idx += blockDim.x) { s_node_j[idx] = nodes[(b * N + j) * D + idx]; } for (int idx = tid; idx < P; idx += blockDim.x) { s_pair[idx] = pair_feats[((b * N + i) * N + j) * P + idx]; } __syncthreads(); float mask_val = edge_mask[((b * N + i) * N + j)]; // Compute z1 collaboratively for (int h = tid; h < H; h += blockDim.x) { float acc = b1[h]; for (int k = 0; k < D; ++k) { acc += w1_T[k * H + h] * s_node_i[k]; } for (int k = 0; k < D; ++k) { acc += w1_T[(D + k) * H + h] * s_node_j[k]; } for (int k = 0; k < P; ++k) { acc += w1_T[(2 * D + k) * H + h] * s_pair[k]; } float sig = 1.0f / (1.0f + expf(-acc)); s_z1[h] = acc * sig; } __syncthreads(); // Compute z2 for each d for (int d = tid; d < D; d += blockDim.x) { float val = b2[d]; for (int h = 0; h < H; ++h) { val += w2_T[h * D + d] * s_z1[h]; } if (mask_val <= 0.5f) { val = -FLT_MAX; } fused[((b * N + i) * N + j) * D + d] = val; } } ``` For kernel 2: ```cuda __global__ void reduce_ln_kernel( const float* fused, const float* edge_mask, const float* nodes, const float* ln_w, const float* ln_b, float* out, int B, int N, int D, int H) // H not needed { int b = blockIdx.x; int i = blockIdx.y; int tid = threadIdx.x; __shared__ float s_max[96]; __shared__ float s_node_i[96]; __shared__ float s_red[128]; for (int idx = tid; idx < D; idx += blockDim.x) { s_node_i[idx] = nodes[(b * N + i) * D + idx]; s_max[idx] = -FLT_MAX; } __syncthreads(); // Max over j for (int d = tid; d < D; d += blockDim.x) { float max_val = -FLT_MAX; for (int j = 0; j < N; ++j) { float val = fused[((b * N + i) * N + j) * D + d]; if (val > max_val) { max_val = val; } } s_max[d] = max_val; } __syncthreads(); // LayerNorm float local_sum = 0.0f; for (int idx = tid; idx < D; idx += blockDim.x) { local_sum += s_max[idx]; } s_red[tid] = local_sum; __syncthreads(); #pragma unroll for (int stride = blockDim.x / 2; stride > 0; stride >>= 1) { if (tid < stride) { s_red[tid] += s_red[tid + stride]; } __syncthreads(); } float mean = s_red[0] / D; float local_var = 0.0f; for (int idx = tid; idx < D; idx += blockDim.x) { float diff = s_max[idx] - mean; local_var += diff * diff; } s_red[tid] = local_var; __syncthreads(); #pragma unroll for (int stride = blockDim.x / 2; stride > 0; stride >>= 1) { if (tid < stride) { s_red[tid] += s_red[tid + stride]; } __syncthreads(); } float var = s_red[0] / D; float inv_std = rsqrtf(var + 1e-5f); for (int idx = tid; idx < D; idx += blockDim.x) { float v = s_max[idx]; float norm = (v - mean) * inv_std; float o = norm * ln_w[idx] + ln_b[idx]; o += s_node_i[idx]; out[(b * N + i) * D + idx] = o; } } ``` Wait, there's an issue with kernel 2. The reduction shared memory `s_red` needs to be sized for `blockDim.x`. If blockDim.x = 128, s_red needs 128 elements. But I used `#pragma unroll` with `blockDim.x / 2` which requires dynamic unrolling... actually, `#pragma unroll` only works with compile-time constants. Since `blockDim.x` is dynamic, the compiler can't fully unroll. For the reduction, I might need to use a constant block size and size s_red accordingly. Or just not unroll and let the compiler handle it. Since the reduction is only log2(128) = 7 steps, unrolling isn't critical. Let me just use a fixed block size for kernel 2. I'll use blockDim.x = 128 for simplicity. Actually, let me think about whether the two-kernel overhead (extra kernel launch, intermediate memory traffic) is worth it. For the current problem size: - Kernel 1: 2304 blocks, each with say 96 threads. Each block does ~58K FMAs. Total work: 2304 * 58K ≈ 133M FMAs. With 67 TFLOPS, if fully utilized: 133M / 67T = 2 microseconds. - Global memory write: 884KB. At 3.35TB/s: 0.26 microseconds. - Kernel 2: 96 blocks * 128 threads. Max over 24 values + LN. Very light. ~0.5 microseconds? - Global memory read: 884KB. 0.26 microseconds. Total: ~3 microseconds + overhead. For single-kernel approach: - 96 blocks * 192 threads. Each block: 24 * (40K + 18K) FMAs = 24 * 58K = 1.39M FMAs. Total: 96 * 1.39M = 133M FMAs. Same total work. - No intermediate global memory. But with single kernel, the h-loop per j has some inefficiency due to j-loop overhead and __syncthreads(). And occupancy is worse (96 blocks vs 2304). Actually wait, I computed single-kernel with block_size=192. Let me reconsider with block_size=96. - h-loop: 2 iterations, all 96 threads active. - d-loop: 1 iteration, all 96 threads active. - Per j: 2T_h + T_d. - 24 j: 48T_h + 24T_d. With single kernel and block_size=96, total work per block is the same as two-kernel, but it's more sequential within a block. The two-kernel approach has better parallelism across blocks. I think two-kernel is worth trying, especially since the grid for kernel 1 has much better occupancy. But wait, there's another factor. In the single-kernel approach, the weights (w1_T, w2_T) stay in L1 cache across all j iterations. In the two-kernel approach, each block reads w1_T and w2_T from global memory (or L2) once. With 2304 blocks and 192*208*4 + 192*96*4 = 234KB of weights, the L1/L2 cache might not hold all weights for all blocks. But since 234KB is small, it should fit in L2 (the H100 has a massive L2 cache, probably 50MB or more). So weight accesses should be L2 hits after the first few blocks. Actually, H100 has 50MB L2 cache. The weights are 234KB. They'll easily stay in L2. So weight reads from global memory are actually L2 reads, which are fast. For the single-kernel approach, weights are also read from global memory (or cached). The difference is: - Single kernel: weights cached in L1 for all j iterations within a block. - Two kernel: weights cached in L2, accessed by each block. L1 bandwidth is much higher than L2. But with 2304 blocks, there are many more concurrent weight reads, potentially saturating L1/L2 bandwidth. I think the two-kernel approach is promising. Let me implement both approaches and compare. Actually, I can only compare by running them. But I don't have timing info from correctness checks. Let me just implement the single-kernel with pre-loaded w1_T. I think it should be pretty fast and avoids the complexity of two kernels. Wait, but I already have a working single-kernel that passed correctness. The question is whether to optimize further. Let me think about this differently. The current single-kernel (first attempt) compiles and passes correctness. But I don't know its runtime. It might already be fast enough. Actually, I can estimate. The first kernel loads pair_feats and node_j from global memory inside the j loop, and has 4 __syncthreads() per j. The improved version pre-loads data and w1_T to shared memory, reducing __syncthreads() to 2 per j and eliminating redundant global memory reads. I think the improved single-kernel with pre-loaded w1_T and data will be significantly faster. Let me implement it. Let me also use block_size=192 since it gives the best warp utilization for the h-loop. One more thing: I should use atomic or shared memory carefully. In the original kernel I used s_x for all input data. In the new kernel, I use separate arrays. Let me make sure there are no indexing bugs. s_w1: size L * H = 208 * 192 = 39936 s_node_i: size D = 96, starts at s_w1 + 39936 s_nodes_j: size N * D = 2304, starts at s_node_i + 96 s_pairs: size N * P = 384, starts at s_nodes_j + 2304 s_z1: size H = 192, starts at s_pairs + 384 s_max: size D = 96, starts at s_z1 + 192 s_red: size BLOCK_SIZE = 192, starts at s_max + 96 Total: 39936 + 96 + 2304 + 384 + 192 + 96 + 192 = 43200 floats Shared memory launch: 43200 * sizeof(float) = 172800 bytes Max shared memory per block on H100: 227KB = 232448 bytes. 172800 < 232448. Fits! OK let me code this up. Actually, I realize there's a potential issue. When I load w1_T to shared memory, `w1_T` is passed as a tensor. In PyTorch, `weight.t().contiguous()` gives a tensor that should be properly laid out. But let me make sure the indexing `w1_T[idx]` is correct. `w1_T` has shape (L, H) = (208, 192). It's contiguous. So `w1_T[idx]` where idx = k * H + h accesses element (k, h). In my kernel, I access `s_w1[k * H + h]`. This is correct. Wait, but `w1_T` on the host is created via `self.fuse[0].weight.t().contiguous()`. Let me double-check the shapes: - `self.fuse[0]` is `nn.Linear(2*D+P, H)` - `self.fuse[0].weight` has shape `(H, 2*D+P)` - `self.fuse[0].weight.t()` has shape `(2*D+P, H)` - `2*D+P = 208`, `H = 192` - So `w1_T` shape is (208, 192). Correct. For w2_T: - `self.fuse[2]` is `nn.Linear(H, D)` - `self.fuse[2].weight` has shape `(D, H)` - `self.fuse[2].weight.t()` has shape `(H, D)` - `H = 192`, `D = 96` - So `w2_T` shape is (192, 96). Correct. In the kernel: - `w1_T[k * H + h]` where k in [0, L-1], h in [0, H-1]. This accesses row k, column h of the transposed weight. Which is row k, column h of (L, H). Correct. - `w2_T[h * D + d]` where h in [0, H-1], d in [0, D-1]. This accesses row h, column d of (H, D). Correct. All good. Let me write the code now. Actually, one more optimization: for the second linear, I can process w2_T similarly. But it doesn't fit in shared memory with w1_T. However, I could process it transposed or use vectorized access. For w2_T, the access pattern is `w2_T[h * D + d]` with d varying by thread. This means all threads in a warp read consecutive floats in memory (coalesced). This is already the optimal global memory access pattern. So pre-loading w2_T to shared memory wouldn't help much for coalescing, but it might save L2 bandwidth if we can keep it resident. But since it doesn't fit in shared memory alongside w1_T, I'll leave it in global/L2. Actually, I just thought of something. What if I DON'T pre-load w1_T to shared memory, but instead pre-load the data (nodes_j, pairs) and keep weights in L2? The weight access is coalesced and L2 bandwidth is high. Let me compare: - With w1_T in shared memory: 24 iterations of h-loop read from shared memory (very fast, ~20+ TB/s effective bandwidth). - Without w1_T in shared memory: 24 iterations read from L2 (maybe ~10 TB/s effective). The difference might be 2x for the h-loop. Let's see what proportion of time is in the h-loop. Total ops per j: ~58K h-loop: ~40K (69%) d-loop: ~18K (31%) So if h-loop is 2x faster with shared memory, total per j improves by ~35%. This is significant! And the cost to load w1_T to shared memory is small (160KB, one-time). So it's worth it. Let me include w1_T in shared memory. Here's the complete code:

The compile passed. Let me check correctness, then also test performance. But first let me think about shared memory bounds. For the given problem: - B=4, N=24, dim=96, pair_dim=16, hidden=192 Shared memory needed: - L*H = (2*96 + 16)*192 = 208*192 = 39936 - D = 96 - N*D = 24*96 = 2304 - N*P = 24*16 = 384 - H = 192 - D = 96 - BLOCK_SIZE = 192 Total = 39936 + 96 + 2304 + 384 + 192 + 96 + 192 = 43200 bytes = 42.2 KB That fits within 227 KB. But I'm replicating node_j and pair data for each block. Since we have B*N = 96 blocks, each with N*D + N*P = 2688 floats of input data = 10.5 KB. So while one block uses 42.2 KB, 96 blocks total would require a lot more if resident simultaneously, but that's fine since they won't all be. However, the issue is that this approach loads all nodes_j and pair data for a single (b, i) at once. It then processes all j. This could be faster because: 1. w1_T is in shared memory (coalesced loads) 2. All node_j and pair data for this (b, i) is loaded once from DRAM But it also uses more shared memory, possibly limiting occupancy. Let me check correctness first, and then maybe optimize further by also caching w2_T in shared memory. Actually, w2_T is H*D = 192*96 = 18432 bytes. Adding that would be useful too and still fit within 227 KB. Actually, let me first see if this version even runs correctly. If so, I'll also try caching w2_T and see if that's better.

The issue is that the shared memory size is too large. v1 passed correctness, so let me continue with v1 but try to optimize it further. The v1 kernel used: - L + H + D + BLOCK_SIZE = 208 + 192 + 96 + 256 = 752 floats ≈ 3KB - So the v1 was fine. The v2 tried to load too much into shared memory for preloading. - N * D = 24 * 96 = 2304, plus N * P = 24 * 16 = 384, plus L*H = 208*192 = 39936, = ~43K floats = ~170KB, still under 227KB but BLOCK_SIZE=192 check might have issue. Actually 43K floats * 4 = 172KB. Hmm, still under 227KB. But there's also the other allocations, so about 177KB, which should be under 227KB. But maybe the actual memory needed is more due to pointer alignment or something. - Actually, the issue might just be that 177KB > 100KB? Actually max is 227KB. Let me check if BLOCK_SIZE is a multiple... let me see, the grid launch may have failed for some other reason. - Let me check: the shmem calculation: (L * H + D + N * D + N * P + H + D + BLOCK_SIZE) = 39936 + 96 + 2304 + 384 + 192 + 96 + 192 = 43200 floats? - Actually 39936 is huge and dominates, so the error might be from something else. Let me compute more thoroughly. - L = 2*96+16 = 208, H = 192 -> L*H = 39936 - BLOCK_SIZE = 192 - Total: 39936 + 96 + 2304 + 384 + 192 + 96 + 192 = 43200 floats - 43200 * 4 = 172800 bytes = 168.75 KB - Should be under 227KB. Hmm. - But perhaps the `extern __shared__` pointer allocation leads to slight differences. Let me check the maximum: Actually, I think H100 allows 228KB but the maximum per block might be 101KB for some architectures. Wait, it says "Maximum shared memory per thread block: 227 KB". That should allow 168KB. - The real error might be that b1, w2_T, etc. are passed to kernel but are not __restrict__, but that's not the error. - Or maybe `grid` with (B, N) where B=4, N=24 is fine. But wait, the number of blocks is 96, and each block uses 168KB. Total per SM: 168KB * 1 = 168KB < 228KB so occupancy is low but should still work. - Maybe the issue is not enough blocks launched? CUDA_ERROR_INVALID_VALUE can happen when arguments are wrong. Could be that `shmem_size` came out wrong due to integer overflow? Let me compute in C++: (L * H + D + N * D + N * P + H + D + BLOCK_SIZE) * sizeof(float) - D, P, H are int. L*H = 39936 is within int. Sum = 43200 within int. `43200 * 4 = 172800`. Should be fine. - But maybe because of dynamic_shared_memory, you need to call `cudaFuncSetAttribute` for > 48KB? Wait, on Hopper with full 227KB shared memory, you don't need to explicitly set it? Actually, `cudaDeviceSetLimit` sets the shared memory carveout for the device. But cubin says maximum dynamic shared memory may need explicit enabling with `cudaFuncSetAttribute`. - Wait, persistent shared memory > 48KB requires: `cudaFuncSetAttribute(kernel, cudaFuncAttributeMaxDynamicSharedMemorySize, shmem_size);`. Let me try this! That must be it. But actually, v1 worked and didn't need this, and it used only about 3KB. v2 fails because it exceeded 48KB default dynamic shared memory limit. On Hopper the max is 101KB or 227KB depending on configuration, but the default is lower. To get more you need to set attribute. I'll update the v2 cpp layer to set the attribute. But let me be careful: driver requires it. Actually, let me rethink the v2 approach. The huge shared memory for w1_T is memory intensive. Let me try a more balanced approach: - For w2, since H=192, D=96, we could pre-load w2_T into shared too to reduce HBM access in the inner loop. - But the bigger bottleneck is reading nodes repeatedly for each j. In v1, nodes and pair_feats are loaded fresh for each j. - Actually, let me try a more efficient approach: each block processes one (b, i), but let each thread handle a fixed subset of output dimensions across all j's. Then for each j, the block reads node_i, node_j, pair_ij once and stores in shared, then each thread computes its portion of the two linear layers. This is already what v1 does. - If I pre-load only node_i (don't need all N nodes since we only process batch sequentially), and pre-load pair_feats, don't preload w1_T. Actually, I'll do this: - Pre-load node_i into shared (D=96 floats) - For each j, load node_j and pair_ij into shared. - This is exactly what v1 does, plus it uses less shared memory. - To improve v1, I can try using a smaller BLOCK_SIZE to reduce register pressure and use more blocks per SM. Currently BLOCK_SIZE=256. Let's try BLOCK_SIZE=128. Then each block loads L+H+D+128 = 624 floats * 4 = 2.4KB. More blocks per SM = higher occupancy. - Also, I should try to use vectorized loads (float4) for weights and activations to improve bandwidth usage. - Another approach: each thread handles a single output dimension completely (for all j), and we keep the weights in registers. If D=96 and a warp has 32 threads, we need 3 warps to cover all D. BLOCK_SIZE=96 would mean 1 thread per output dimension. Then for each j, the thread reads node_i[D], node_j[D], pair_ij[P], computes the first linear (reading w1 of size L*H, each thread needs H floats), then computes the shared-reduced H dimension for second linear. - Wait, if BLOCK_SIZE=96 and each thread handles one d, then in the second linear each thread only reads w2_T[..., d] (H floats) and s_z1[...] (H floats shared). This is very bandwidth-light on weights since each thread reads H floats per j = 192 floats/j = 768 bytes/j, and there are 24 j's, so ~18KB total weight reads per thread per (b,i). Plus 24*L = ~5KB for nodes/pairs. Total ~23KB. Since a warp has 32 threads and reads w2_T simultaneously, maybe coalescing is good? Wait, no - w2_T is transposed, so w2_T[..., d_offset + 0] to w2_T[..., d_offset + 31] are contiguous, so a warp reading the same h but contiguous d should be coalesced. But since each thread needs all H, it would be strided across H for a single d. So threads in a warp read at the same offset, that's broadcast, not coalesced - but still cached. In L2, the w2_T row for each h is repeated: w2_T[h*D + d] for different d but same h. Actually for a fixed h, all threads read w2_T[h*D + d], which is strided by D. So broadcast-like reads, which isn't great for HBM. Actually, forget preloading everything. Let me just go back to v1 and optimize: 1. Use BLOCK_SIZE=128 (or 96) instead of 256 to improve occupancy. 2. Pre-load node_i into shared memory but not other nodes. Do the inner j loop. 3. Make sure to avoid __syncthreads between Phase 1 and Phase 2 since different parts of block work on different arrays (s_z1 and s_max). Actually the second phase reads s_z1 written by first phase, so we need __syncthreads. 4. Maybe load pair_feats and nodes_j using float4 to save instructions. Wait, maybe the better approach is: - Threads in a warp cooperate on one j, computing the first linear. Then they work on the second linear together. - A warp has 32 threads. For first linear output H=192, we'd need 6 warps. - For second linear output D=96, we'd need 3 warps. - BLOCK_SIZE = 192 = 6 warps. Each block handles one j at a time with cooperative computing. Actually, let me rethink with a cooperative-block approach: For each j iteration: - Threads 0..L-1 (or wrap partition) load x into shared. - Then compute w1 * x: - Each thread h computes one hidden unit? Actually 192 threads would work: thread h computes dot(w1[:, h], x). That's 192 threads = H. But we also need more for the second linear. - We have 192 threads. They compute z1[h] for h in [0, H). Then __syncthreads(). - Then they compute z2[d] for d in [0, D). Only 96 of 192 threads are active for this. Thread d computes dot(w2[:, d], z1). - The block does one j at a time. After computing z2[d], update s_max[d] for this j. - This would require BLOCK_SIZE = max(H, D) = 192. But our BLOCK_SIZE was already 256 > 192. We could set exactly 192. Wait, but then how many blocks per SM? 192 threads * shared memory. If shared memory is small (just L + H + D + 192 ≈ 700 floats = 2.8KB), then 228KB / 2.8KB ≈ 81 blocks per SM, but limited to 32 blocks. So occupancy is 32 * 192 / (max threads per SM = 2048) = 3072/2048 = 1.5? Wait, 32 * 192 = 6144 threads per SM? But H100 has 2048 threads per SM (on Hopper). Actually, H100 has 2048 threads per SM. So max 10 blocks of 192 threads per SM. Still good. Let me try using BLOCK_SIZE=192 and see if that helps. Actually, let me first try to just run v1 and check why it was correct but might be slow. The correctness passed. Now I need to optimize speed. For speed, the bottlenecks in v1 are: 1. Memory bandwidth: reading pair_feats N times per i, reading nodes_j N times, reading w1_T and w2_T repeatedly. 2. Since we use 256 threads and the work is small, we have low occupancy but memory bandwidth might still be the limit. Let me try preloading some stuff into shared memory in a smarter way. I'll use registers aggressively for node_i and avoid repeated reads. Actually, I found the problem with v2: shared memory size was 168KB but default max dynamic shared memory is 48KB (or 96KB on some arch) without explicit function attribute. I need to set `cudaFuncSetAttribute` in the host wrapper. Let me try this. If I can get v2 working, it might be faster because it preloads w1_T and all nodes/pairs, removing redundant HBM reads. Let me add `cudaFuncSetAttribute` in the torch wrapper.

Let me run correctness on this and then submit if it passes. The previous version (v1) already passed correctness. This is the same kernel with just some optimization flags added. Let me verify.

My kernel has passed correctness. Let me submit it now. I need to remove the `-Xptxas -v` debug flags from the compile options and submit the clean version.