Let me analyze the problem. We need to optimize a kernel that does: 1. GEMM (Linear): x @ W^T + b 2. Swish activation: x * sigmoid(x) 3. Divide by 2.0 4. Clamp between -1 and 1 5. Tanh 6. Clamp between -1 and 1 The dimensions are: batch_size=1024, in_features=8192, out_features=8192. This is a large GEMM: 1024 x 8192 x 8192. For an H100, we have 3.35 TB/s memory bandwidth and 67 TFLOPS FP32 (non-tensor core) or much higher with tensor cores. Let me calculate the compute intensity: - GEMM FLOPs: 2 * 1024 * 8192 * 8192 = ~137 GFLOPs - For FP32 at 67 TFLOPS, this would take ~2ms - Memory bandwidth for reading input + writing output: - Input: 1024 * 8192 * 4 bytes = 32 MB - Output: 1024 * 8192 * 4 bytes = 32 MB - At 3.35 TB/s, memory-bound time would be ~19 microseconds But we also need to consider weight loading for the GEMM. The weights are 8192 * 8192 * 4 = 256 MB. With 3.35 TB/s, loading the weights takes ~76 microseconds. Since this is a compute-bound GEMM, we should use cuBLAS for the GEMM part and then fuse all the elementwise operations into a single kernel. This is the standard optimization approach. But wait - can we fuse GEMM + activation? Using Cutlass would be ideal but it's complex with load_inline. Let me think about what's possible. Actually, the best approach with cpp_extension.load_inline would be: 1. Use cublasSgemm for the GEMM 2. Fuse swish, divide, clamp, tanh, clamp into a single elementwise kernel Or we can try to use Cutlass if available, but it might be tricky with inline loading. Let me first try a simple approach: 1. Use torch.matmul for the linear part (let PyTorch handle GEMM) 2. Fused elementwise kernel for the rest Wait, but we need to write our own ModelNew class. We can still use PyTorch's matmul but write a custom fused kernel for the activations. However, the evaluation compares against the PyTorch reference that does elementwise operations separately. A fused kernel should be faster. But actually, the main savings would come from fusing the gemm + activation if possible. Let me think if we can do this with cublasLt which supports epilogues. But cublasLt epilogues don't support all these operations. Actually, on H100, we might be able to write a custom cutlass kernel or use CuTe to fuse GEMM + activations. But that's complex. Let me first try the simpler approach: 1. Do GEMM with torch.matmul or a custom cublas call 2. Fused elementwise kernel for the rest But the reference already uses nn.Linear which calls an optimized GEMM. The overhead of the separate elementwise operations might not be that huge compared to GEMM. Let me think... The GEMM is 137 GFLOPs. Each elementwise op is: - Swish: sigmoid + multiply = ~2 FLOPs per element - Divide: 1 FLOP - Clamp: min/max operations - Tanh: ~10 FLOPs (using fast approximations) - Clamp: min/max Total elementwise FLOPs: ~10 FLOPs per element Number of elements: 1024 * 8192 = 8.4M Total elementwise FLOPs: ~84 MFLOPs So the GEMM dominates at 137 GFLOPs vs 84 MFLOPs. However, each elementwise kernel launch has kernel launch overhead, and each separate kernel writes to and reads from global memory. 1024 * 8192 * 4 bytes = 32 MB per tensor - After GEMM: write 32MB - Swish: read 32MB, write 32MB - Divide: read 32MB, write 32MB - Clamp: read 32MB, write 32MB - Tanh: read 32MB, write 32MB - Clamp: read 32MB, write 32MB Total memory traffic for elementwise part: 32 * 11 = 352 MB At 3.35 TB/s, that's ~105 microseconds. With fused kernel: read 32MB, write 32MB = 64MB = ~19 microseconds. So the fused approach saves ~86 microseconds of memory bandwidth. But the GEMM itself takes ~2ms. So the speedup would be about 4-5% from fusing. Is there a better approach? Can we fuse GEMM + elementwise? Using CUTLASS, we could write a custom GEMM kernel with a custom epilogue that applies all the elementwise ops. Let me think about whether this is doable with load_inline... Yes, CUTLASS has a streamlined header-only library that can be included. However, compiling CUTLASS kernels with load_inline can take a very long time due to heavy template instantiation. Alternatively, we could write a custom CUDA kernel that does a tiles of GEMM using shared memory and then applies the activations. This is essentially a manual CUTLASS-like approach. For 8192x8192 * 8192x1024, this is a very large GEMM. We could tile it: - Each thread block computes a tile of the output - Load A and B tiles into shared memory - Compute partial sums in registers - Apply elementwise ops and write output Let me think about whether this would outperform cublas + fused elementwise... cuBLAS is extremely optimized and uses tensor cores, warp shuffles, and other advanced features. A manually-written GEMM is unlikely to beat cuBLAS on H100 for this size. However, we can try using CUTLASS to fuse the epilogue. Let me check if the evaluation environment has CUTLASS available... Actually, let me try using cublasLt which supports CUBLASLT_EPILOGUE_RELU, GELU, etc. But it doesn't support swish + divide + clamp + tanh + clamp. So cublasLt epilogues won't work directly. What about writing a kernel that calls cublas for the GEMM into a workspace, and then our fused kernel does the rest? That's the straightforward approach. But wait - for 1024 x 8192 x 8192, the matrix multiplication M=1024, N=8192, K=8192. This might not saturate cuBLAS fully due to the relatively small M dimension. Let me think about this differently. Maybe we can use: 1. cublas for the GEMM 2. A stream-ordered approach where we overlap computation Actually, the most practical optimized solution would be to write a fused GEMM + epilogue kernel ourselves. This might be called via Cutlass. But Cutlass compilation is very slow. Another approach: use WGMMA (warp group matrix multiply accumulate) on Hopper. H100 has support for WGMMA in PTX. We could write a kernel that uses wgmma instructions directly. This is complex but could give the best performance. Actually, let me reconsider. For M=1024, N=8192, K=8192, a single batch GEMM, cuBLAS is very likely to call into a highly optimized kernel that uses tensor cores. The overhead of separate elementwise kernels is relatively small compared to the GEMM time. But let me check: maybe the reference is slow because PyTorch doesn't use cublasLt or because the elementwise ops are memory bandwidth bound. Actually, I realize something: can we use torch.compile or JIT? No, because the reference is Model and we need to write ModelNew. Let me think about what tensor cores can do. The GEMM at 137 GFLOPs using tensor cores at ~1000 TFLOPS for FP32 accumulation from BF16/FP16 inputs, or even 67 TFLOPS for pure FP32. At 67 TFLOPS FP32, that's 2ms for the GEMM. The memory bandwidth for 32MB output at 3.35 TB/s is ~10 microseconds. But tensor cores operate at much higher throughput, so GEMM can be much faster than 2ms. Actually, for M=1024, N=8192, K=8192, a good tensor core kernel might achieve close to peak. Let me calculate more carefully: - FP32 cuBLAS peak: 67 TFLOPS - Time = 137 GFLOPs / 67 TFLOPS = 2.04 ms But with TF32 tensor cores (cuBLAS automatically uses this), effective TFLOPS could be higher. Wait, actually TF32 tensor cores are ~495 TFLOPS. If cuBLAS uses TF32, time = 137 / 495 = 0.27 ms = 270 microseconds. At 270 microseconds for GEMM, removing 6 elementwise kernel launches and memory round-trips is still meaningful. The best approach might be to try CUTLASS for GEMM + custom epilogue. But the compile time might be prohibitive. Actually, let me think about a different approach. What if we register our custom kernel as a custom op or just write a simple kernel that does: 1. Use Aten/cuBLAS for the GEMM 2. Fused elementwise kernel This should give some speedup. Let me start with this and see. Wait, I should also consider that the problem has get_init_inputs() returning [in_features, out_features], which means ModelNew(in_features, out_features) should initialize properly, but the reference doesn't pass bias. Let me check... Actually get_init_inputs returns [in_features, out_features] but the Model constructor expects bias=True by default. So ModelNew should accept in_features, out_features, and optionally bias (defaulting to True for compatibility). Now, let me write the kernel. For maximum performance, I'll try to write a fused kernel that includes both the GEMM and elementwise ops. But first, let me try a simpler version. Actually, we can use cublas for the GEMM and then a custom elementwise fusion kernel. Let me write this first. But wait - can we do better? What if we use cublaslt and a custom epilogue... Actually, there's a way to approximate swish and tanh with RELU or GELU in cublasLt, but it's not exact. Let me think about the numerical requirements: fp32 with atol=rtol=1e-4. This means we need fairly accurate results. For swish, tanh, etc., we need proper implementations. Actually, I just realized: the best approach might be to write a kernel that calls cublasSgemm or cublasGemmEx into a temporary buffer, and then our fused elementwise kernel transforms it. But the temporary buffer adds memory traffic. Can we compute the GEMM directly into the output and then apply the fused kernel in-place? Yes, we can use cublas to write to the output buffer, then apply our fused elementwise kernel in-place. This is optimal for the GEMM + elementwise separation approach. Wait, but is it better than nn.Linear + elementwise? nn.Linear also calls cublas and writes to a temporary buffer. The difference is whether we do elementwise ops in separate kernels or one fused kernel. Let me write the code: ```python import torch import torch.utils.cpp_extension cuda_code = ''' #include <torch/extension.h> #include <cuda.h> #include <cuda_runtime.h> #include <cublas_v2.h> #include <cmath> // Fused elementwise kernel __global__ void fused_ops_kernel(float* __restrict__ x, int n) { int idx = blockIdx.x * blockDim.x + threadIdx.x; if (idx < n) { float v = x[idx]; // Swish: x * sigmoid(x) v = v / (1.0f + expf(-v)); // Divide by 2 v *= 0.5f; // Clamp [-1, 1] v = fminf(fmaxf(v, -1.0f), 1.0f); // Tanh v = tanhf(v); // Clamp [-1, 1] v = fminf(fmaxf(v, -1.0f), 1.0f); x[idx] = v; } } void forward_cuda(torch::Tensor x, torch::Tensor weight, torch::Tensor bias, torch::Tensor out) { // x: (M, K), weight: (N, K), bias: (N), out: (M, N) auto M = x.size(0); auto K = x.size(1); auto N = weight.size(0); cublasHandle_t handle; cublasCreate(&handle); float alpha = 1.0f, beta = 0.0f; cublasSgemm(handle, CUBLAS_OP_T, CUBLAS_OP_N, N, M, K, &alpha, weight.data_ptr<float>(), K, x.data_ptr<float>(), K, &beta, out.data_ptr<float>(), N); // Add bias if (bias.defined() && bias.numel() > 0) { // Could do a separate kernel or use cublas to add bias } cublasDestroy(handle); int n = M * N; const int threads = 256; const int blocks = (n + threads - 1) / threads; fused_ops_kernel<<<blocks, threads>>>(out.data_ptr<float>(), n); } PYBIND11_MODULE(TORCH_EXTENSION_NAME, m) { m.def("forward_cuda", &forward_cuda, "Fused forward"); } ''' ``` Wait, this has a few issues: 1. Creating/destroying cublas handle every call is expensive 2. Bias isn't handled properly 3. I was told not to write my own PYBIND11_MODULE Let me fix these. For cublas handle, I should cache it. One way is to create it once. But in load_inline, I can use a static variable inside the function. Also, let me check: can I use torch.matmul in Python and just write the fused elementwise kernel in CUDA? Yes, that's cleaner. ```python import torch import torch.nn as nn import torch.utils.cpp_extension cuda_code = ''' #include <torch/extension.h> #include <cuda.h> #include <cuda_runtime.h> __global__ void fused_ops_kernel(float* __restrict__ x, int64_t n) { int64_t idx = (int64_t)blockIdx.x * blockDim.x + threadIdx.x; if (idx < n) { float v = x[idx]; // Swish: x * sigmoid(x) v = v / (1.0f + expf(-v)); // Divide by 2 v *= 0.5f; // Clamp [-1, 1] v = fminf(fmaxf(v, -1.0f), 1.0f); // Tanh v = tanhf(v); // Clamp [-1, 1] v = fminf(fmaxf(v, -1.0f), 1.0f); x[idx] = v; } } torch::Tensor forward_cuda(torch::Tensor x) { int64_t n = x.numel(); const int threads = 256; const int blocks = (n + threads - 1) / threads; fused_ops_kernel<<<blocks, threads>>>(x.data_ptr<float>(), n); return x; } ''' # Compile fused_ops = torch.utils.cpp_extension.load_inline( name="fused_ops", cpp_sources="", cuda_sources=cuda_code, functions=["forward_cuda"], extra_cuda_cflags=["-O3", "--use_fast_math"], verbose=False, ) class ModelNew(nn.Module): def __init__(self, in_features, out_features, bias=True): super().__init__() self.gemm = nn.Linear(in_features, out_features, bias=bias) def forward(self, x): x = self.gemm(x) x = fused_ops.forward_cuda(x) return x ``` This is much cleaner. But wait - can we do even better? What about doing GEMM + fused elementwise in a single kernel without intermediate write? For that, we'd need to write our own GEMM kernel or use CUTLASS. Let me think about whether a manually written simple GEMM could be competitive... For M=1024, N=8192, K=8192, if we tile with 128x128 blocks, we'd need 8x64 = 512 blocks. Each block does 128x128x8192 MACs = 268M operations. That's too much for a single block. Better tiling: 64x128 tiles, 16x64 = 1024 blocks. Each block does 64*128*8192 = 67M ops. Still a lot but manageable with multi-stage pipelining. Actually, for H100, we should use WGMMA. But writing WGMMA PTX is extremely complex. What if we try CUTLASS? The CUTLASS 3.x (for Hopper) streamlined API might be usable. However, compilation times can be very long. Let me try a different approach. Can we use the `cutlass` Python package or include cutlass headers in load_inline? The issue is compile time. Actually, let me first test the simple cublas + fused elementwise approach and see what speedup we get. If it's not enough, we can try something more advanced. But hold on - there's a key insight. The problem states `bias=True` in the reference. But `get_init_inputs()` only returns `[in_features, out_features]`. This means the Model will be initialized with the default `bias=True`. Our ModelNew should accept the same arguments. Also, I need to think: the reference does x = self.gemm(x) and then applies operations. Can we somehow avoid needing to call a separate nn.Linear? We could store the weight and bias as parameters and call them the same way, or we could implement our own matmul + bias using cublas. Actually, using nn.Linear is fine because it will use the optimized cublas/cublasLt GEMM kernels automatically. The speedup comes from fusing the elementwise ops. But is there significant kernel launch overhead from 6 separate elementwise operations? Launch overhead is typically 5-10 microseconds. For 6 kernels that's 30-60 microseconds. And memory roundtrips add more. For a GEMM that might take 100-500 microseconds (using tensor cores), saving 30-60 microseconds + memory bandwidth is significant. Wait, let me reconsider. M=1024, N=8192, K=8192: - With FP32 non-tensor core (67 TFLOPS): 2ms - With TF32 tensor cores (495 TFLOPS without sparsity): ~0.27ms - With BF16/FP16 tensor cores: even faster if accumulation is FP32 PyTorch's Linear will use whatever cuBLAS picks. For this size, it's likely to use tensor cores with TF32. If GEMM takes ~0.27ms, and 6 separate elementwise kernels take maybe 0.05ms each (launch + execution), that's 0.3ms for elementwise, which is actually MORE than the GEMM time! So fusing them into one kernel taking ~0.02ms would give a significant speedup. But wait, are the elementwise kernels really that slow? Each one processes 8.4M elements. At 3.35 TB/s bandwidth: - Read + write: 64MB per kernel - Time per kernel: 64MB / 3.35TB/s = 19 microseconds So 6 kernels = ~114 microseconds + ~30-60 microseconds launch overhead = ~150 microseconds total. One fused kernel = 64MB / 3.35TB/s = 19 microseconds + ~5 microseconds launch = ~24 microseconds. Difference: ~126 microseconds. If GEMM takes 270 microseconds, total reference = 420 microseconds, fused = 294 microseconds. Speedup: 1.43x That's decent but maybe not enough. Can we do better? What if we try to fuse the bias addition with the elementwise ops too? In my simple approach, nn.Linear does GEMM + bias (fused in cuBLAS), and then our kernel does the elementwise ops. So bias is already fused into GEMM by cuBLAS. Actually, I want to check: does PyTorch's nn.Linear fuse bias into the GEMM? Yes, it calls `torch.nn.functional.linear` which uses `addmm` or `matmul + add`, and PyTorch/cuBLAS typically fuses these. What if we write our own custom op that handles the entire forward pass, doing GEMM with cublas and then the fused elementwise kernel? We'd need to manage the cublas handle. Actually, using nn.Linear in forward() is perfectly fine - it calls into PyTorch's optimized GEMM. The key optimization is the fused elementwise kernel. But maybe we can get more speedup by doing the GEMM ourselves with cublas and doing the fusion in a custom way. Let me think... Actually, there's another idea: can we precompute the transformation on the weights and bias so that x @ W^T + b followed by swish, /2, clamp, tanh, clamp can be simplified or approximated? Not really, because swish and tanh are nonlinear and depend on the specific values of x. What about using tile-level fusion? We load a tile of input, compute the GEMM tile, apply activations, and write output. This would reduce the intermediate memory traffic to zero. To do this effectively, we'd need CUTLASS. Let me see if I can write a cutlass-based solution with load_inline. Actually, I just realized something important. The values after GEMM before going into the elementwise ops might be large in magnitude. But after swish and clamping, they're bounded. Not sure if this helps with anything. Let me try a more advanced approach. I'll use cublas for the GEMM but launch it asynchronously, then have our fused kernel run on the same stream. The current approach already does this implicitly since CUDA ops on the same stream are serialized. Wait, I should check if there's an even simpler optimization. The sequence of operations is: 1. Swish(x) 2. / 2 3. clamp(-1, 1) 4. tanh 5. clamp(-1, 1) Notice that since tanh is applied after clamp(-1, 1), and tanh maps [-1, 1] to approximately [-0.76, 0.76], the final clamp(-1, 1) is actually a no-op because tanh([-1, 1]) is within [-1, 1]. Wait, is that true? tanh(-1) ≈ -0.7616 and tanh(1) ≈ 0.7616. Yes! So the final clamp is strictly a no-op. But is it numerically equivalent? Yes, because tanh([-1, 1]) ⊆ [-1, 1], so clamp(tanh(v), -1, 1) = tanh(v) for all v ∈ [-1, 1]. And before tanh, we clamp to [-1, 1]. So the final clamp is indeed redundant. Actually wait, there's floating point precision to consider. tanh(v) for v very close to 1 might round to slightly more than 1? No, tanh(1) = 0.76159... which is well within [-1, 1]. So we can remove the final clamp! But the reference includes it. Our result should be numerically identical because the final clamp doesn't change any values. So removing it is safe from a correctness standpoint, but doesn't change much in terms of performance since it's in the fused kernel anyway. Actually, in a fused kernel, removing one fminf/fmaxf pair doesn't save much. OK let me write a more optimized fused kernel using vectorized loads/stores for better memory bandwidth. This is important for elementwise kernels. We can process 4 float elements per thread using float4. ```cpp __global__ void fused_ops_kernel(float* __restrict__ x, int64_t n) { int64_t idx = ((int64_t)blockIdx.x * blockDim.x + threadIdx.x) * 4; if (idx + 3 < n) { float4 v4 = reinterpret_cast<float4*>(x)[idx / 4]; // Process 4 elements #pragma unroll for (int i = 0; i < 4; i++) { float v = ((float*)&v4)[i]; v = v / (1.0f + expf(-v)); v *= 0.5f; v = fminf(fmaxf(v, -1.0f), 1.0f); v = tanhf(v); v = fminf(fmaxf(v, -1.0f), 1.0f); ((float*)&v4)[i] = v; } reinterpret_cast<float4*>(x)[idx / 4] = v4; } else { for (int64_t i = idx; i < n; i++) { float v = x[i]; v = v / (1.0f + expf(-v)); v *= 0.5f; v = fminf(fmaxf(v, -1.0f), 1.0f); v = tanhf(v); v = fminf(fmaxf(v, -1.0f), 1.0f); x[i] = v; } } } ``` This should give much better memory bandwidth utilization. Also, I can use `__ldg` for read-only loads, but since we're reading and writing the same array, we can't use __ldg unless we make a copy. But the array is large, and we're doing in-place operations, so we just need good coalesced access. For the grid configuration, with float4, each thread processes 4 elements. For 8.4M elements, we'd need ~2.1M threads, or at 256 threads/block, ~8K blocks. That's enough to saturate the GPU. Now, one more optimization: can we use Grid-stride loops? This gives better occupancy. But with 8K blocks at 256 threads = ~2M threads, and H100 has many SMs, this should be fine. Actually, let me think about whether we should keep using nn.Linear or call cublas directly. Advantages of nn.Linear: - Simple, uses PyTorch's extremely optimized backend - Handles all the dimensions, transposes, etc. - Uses cublasLt under the hood Disadvantages: - We rely on PyTorch to schedule the GEMM Advantages of custom cublas: - We could potentially use cublasGemmEx with FP16/BF16 weights for faster computation - But this might violate fp32 precision requirements Actually, the problem says "produce numerically equivalent outputs at fp32 precision". So we need fp32-accurate results. Using BF16/FP16 for the GEMM internal computation might be acceptable if the result is within 1e-4, but it's risky. Better to stick with FP32. Wait, PyTorch might use TF32 tensor cores by default. Are TF32 results within 1e-4 of true FP32? Typically TF32 has ~1e-4 relative precision, so yes, usually within tolerance. But PyTorch's default for matmul on Ampere/Hopper is to use TF32. And the reference uses nn.Linear, so it will get whatever PyTorch gives it. Our ModelNew using nn.Linear will use the same GEMM, then fused elementwise. So the elementwise fusion is the main optimization. Let me write this. Actually, let me also consider: could we write a custom Triton kernel? No, because we only have load_inline available, not Triton. OK, let me write the complete code. But first, I need to think about whether we can fuse the bias addition into our elementwise kernel too, to save even more... Actually, nn.Linear already fuses bias into the GEMM. Wait, is that true? Let me check. PyTorch's `nn.Linear` calls `F.linear(input, self.weight, self.bias)`. `F.linear` does: ```python if input.dim() == 2 and bias is not None: # fused op is marginally faster ret = torch.addmm(bias, input, weight.t()) else: output = input.matmul(weight.t()) if bias is not None: output += bias return output ``` Yes, for 2D input it uses `torch.addmm`, which fuses GEMM + bias addition. This is calling into cublasLt with epilogue fusion. So bias is already fused. Now, what about writing the entire thing in a custom kernel that uses cublas for GEMM and then our fused elementwise kernel on the same buffer? We'd allocate output, pass it to cublas, then call our kernel. Let me compare: 1. `nn.Linear` returns tensor → our fused kernel processes it in-place 2. We allocate output → call cublas → call fused kernel in-place Option 1 is simpler and should be equivalent. nn.Linear returns a freshly allocated tensor that was written by cublas. Our fused kernel reads and writes this tensor in-place. Is there any overhead from nn.Linear that we avoid by rolling our own? Not really - nn.Linear just does addmm or matmul + bias. OK let me write the code and test it. Wait, I just realized - could we do warp-level GEMM manually in the kernel? For example, each thread processes one output element, doing the dot product with the input and weight vectors. This avoids storing the intermediate GEMM output entirely! For M=1024, N=8192, each thread computes one output element: - Each thread does 8192 multiply-adds = ~8K FLOPs - Total threads: 1024 * 8192 = 8.4M threads - At 256 threads/block: ~33K blocks - Compute per thread: 8192 FMAs = ~16K FLOPs. At 67 TFLOPS FP32, each thread takes... wait, this doesn't work that way. Actually, if each thread computes a dot product serially: - 8.4M threads * 8192 FMAs each = 137 GFLOPs total - But each thread needs to load its corresponding column of weights (8192 floats = 32KB) and the input row (8192 floats = 32KB). Wait, if thread (i,j) computes output[i,j], it needs: - Input row i (shared by all threads in the same output row) - Weight column j (unique to thread, or shared by threads computing the same output column) For 256 threads in a block, if they all compute different elements in the same row, they share the input row but each needs a different weight column. That doesn't help much with shared memory. If threads compute elements in a tile, they can cooperatively load input and weight tiles. Actually, this brings me back to: can we write a reasonable GEMM + epilogue kernel? For M=1024, we could have each block compute a tile of outputs, say 64x64 or 128x64. This is a standard tiling approach. But writing a high-performance GEMM is very complex. Let me think about this more carefully. For M=1024: - We have 1024 rows of output - Each row has 8192 elements - If a block computes a 64x64 tile, we need 16 * 128 = 2048 blocks - Each block does 64*64*8192 = ~33M MACs = ~66M FLOPs - At 67 TFLOPS, each SM would need ~1 microsecond. But we have many SMs. - A block with 256 threads would do 33M / 256 = ~130K FMAs per thread. That's quite a lot per thread. If each FMA takes 1 cycle and GPU runs at ~1.5 GHz, that's ~87 microseconds per thread. But with memory loads, it would be much more. Actually, the standard approach is to tile K dimension. So a 64x64 output tile with K tile of 16: - We need 8192/16 = 512 iterations over K - Each iteration loads 64*16 + 64*16 = 2048 floats = 8KB to shared memory - Each thread does 16*16 = 256 MACs per iteration - 512 iterations * 256 threads = good occupancy with enough blocks This is definitely doable, but writing it correctly and competitively vs cuBLAS is hard. On H100, the real question is whether we should try to use tensor cores. Without tensor cores, a manual GEMM won't beat cuBLAS's tensor core kernels by a huge margin. And manual tensor core programming is extremely complex. So I think the best practical approach is still: 1. Use PyTorch's nn.Linear (which calls optimized cuBLAS) 2. Fused elementwise kernel with vectorized memory access Let me also verify if we can do the GEMM + elementwise fusion by reusing output tensor. Actually, I just had another idea. What if we implement our own batched GEMM using a custom kernel specifically for M=1024, N=8192, K=8192 where each thread computes 8 or 16 output elements? We could load weight tiles into shared memory and stream through the K dimension. Because the output is relatively small (32MB) and weights are large (256MB), the bottleneck is weight loading. In our manual kernel: - We iterate over M in tiles, and within each tile, we iterate over N in tiles - For each output tile, we load the corresponding A tile from input and B tiles from weights - Stream through K tiles - After computing the output tile, apply the activation and write to output This is exactly what CUTLASS does. The memory traffic for output tiles is just the final write (32MB). For weights, we still need to load all 256MB. For cuBLAS + fused elementwise: - cuBLAS loads weights and input, writes output (32MB) - Fused kernel reads and writes output (64MB) - Total memory traffic: input (32MB) + weights (256MB) + output write (32MB) + output read/write for elementwise (64MB) = 384MB For manual fused GEMM+epilogue: - Load input (32MB) + weights (256MB) + write output (32MB) = 320MB Difference is only 64MB (output read/write for elementwise). At 3.35 TB/s, that's 19 microseconds. So manually fusing GEMM + epilogue saves only 19 microseconds of memory bandwidth! But it avoids the separate kernel launch. So total savings maybe 25-30 microseconds. If cuBLAS GEMM takes 270 microseconds, the speedup from manual fusion is minimal (10% or less). Wait, but there's another factor. cuBLAS with addmm writes output, then our kernel reads it. The CPU overhead of launching two kernels might be non-trivial. Hmm, but is there a way to overlap cuBLAS and elementwise? Not in a serialized stream. OK, let me also think about using streams/pipelines. Actually no, cuBLAS and our custom kernel run on the same stream by default. Another idea: since M=1024 is relatively small compared to N=K=8192, maybe the GEMM is not as compute-bound as I calculated, because it might be harder to achieve peak utilization with small M. For cuBLAS, M=1024 is actually reasonably large. But let me check: cublas Lt or cublas might choose different algorithms. Actually, I think the easiest and most reliable approach is: 1. nn.Linear for GEMM + bias 2. Vectorized fused elementwise kernel And then if I want to push further, I can explore using cublas directly with a persistent cublas handle, and maybe experimenting with different algorithms. Wait, actually there's something else. For H100, we can use float4 vectorized loads. But does the GEMM output align to 16 bytes? PyTorch tensors are typically 4-byte aligned, but for a tensor of shape (1024, 8192), each row has 8192*4 = 32768 bytes, which is divisible by 16. So float4 alignment is perfect within rows. Let me write the final code. Actually, one more thought: can we use grid-stride loops in the elementwise kernel for better occupancy? With n=8,388,608 elements and processing 4 per thread, we need n/4 = 2,097,152 threads. At 256 threads/block, that's 8192 blocks. H100 has 132 SMs (I think? Or 144?). Either way, 8192 blocks gives ~60+ blocks per SM, which is plenty. So we don't need grid-stride loops. OK, I'll write the code now. Let me be careful about: 1. Not including PYBIND11_MODULE 2. Compiling with `--use_fast_math` for faster expf and tanhf 3. Using float4 vectorized access for the main loop 4. Proper error handling? The evaluation probably doesn't care about errors as long as output is correct. Wait, actually `--use_fast_math` might affect numerical accuracy. The tolerance is 1e-4. Let me think about whether fast_math's expf and tanhf are accurate enough. CUDA's `expf` with fast math should be accurate to within a few ULPs, well within 1e-4 for reasonable ranges. `tanhf` with fast math uses a faster approximation. For values in [-1, 1] (after clamping), tanhf should be very accurate. But swish is applied before clamping, so inputs to expf could be large. For large negative inputs, expf(-x) → 0, and v / (1 + 0) → v. For large positive inputs, expf(-x) → 0, and v * expf(-x) should → 0... wait, swish = x * sigmoid(x) = x / (1 + exp(-x)). For large negative x, exp(-x) is huge, so sigmoid(x) → 0, and swish → 0. For large positive x, exp(-x) → 0, sigmoid(x) → 1, and swish → x. The issue is for large positive x, if expf(-x) underflows to 0, we get v / 1 = v, which is correct. For large negative x, if expf(-x) overflows, we have a problem. But in CUDA, expf of large values returns inf, and v / (1 + inf) = 0, which is correct for swish. So `--use_fast_math` should be fine. Let me use it. One thing to be careful about: the custom CUDA kernel should be defined in `cpp_sources` or `cuda_sources`? With `load_inline`, I should put CUDA code in `cuda_sources` and C++ wrapper code in `cpp_sources` (or empty if I define wrappers in cuda_sources, which is fine since nvcc can compile mixed code). Actually for `load_inline`, both are compiled together. Let me just put everything in `cuda_sources`. Let me draft the kernel: ```python kernel_code = """ import torch import torch.nn as nn import torch.utils.cpp_extension cuda_source = ''' #include <torch/extension.h> #include <cuda.h> #include <cuda_runtime.h> __forceinline__ __device__ float swish(float x) { return x / (1.0f + expf(-x)); } __forceinline__ __device__ float activate(float x) { x = swish(x); x *= 0.5f; x = fminf(fmaxf(x, -1.0f), 1.0f); x = tanhf(x); x = fminf(fmaxf(x, -1.0f), 1.0f); return x; } __global__ void fused_forward_kernel(float* __restrict__ x, int64_t n) { int64_t idx = ((int64_t)blockIdx.x * blockDim.x + threadIdx.x) * 4; if (idx + 3 < n) { float4 v4 = reinterpret_cast<float4*>(x)[idx / 4]; v4.x = activate(v4.x); v4.y = activate(v4.y); v4.z = activate(v4.z); v4.w = activate(v4.w); reinterpret_cast<float4*>(x)[idx / 4] = v4; } else { #pragma unroll for (int64_t i = idx; i < n; ++i) { x[i] = activate(x[i]); } } } torch::Tensor fused_forward(torch::Tensor x) { int64_t n = x.numel(); const int threads = 256; const int blocks = (n + threads * 4 - 1) / (threads * 4); fused_forward_kernel<<<blocks, threads>>>(x.data_ptr<float>(), n); return x; } ''' """ ``` Wait, I need to think about the `fmaxf`/`fminf` with `--use_fast_math`. With fast math, these should compile to straightforward `max`/`min` instructions. Also, let me double check: the second loop `for (int64_t i = idx; i < n; ++i)` - if idx = n-3, we'd process 3 elements. But wait, the `else` branch is entered when `idx + 3 >= n`, meaning we don't have a full float4 to process. But each thread is supposed to process exactly its 4 elements. The issue is if `idx + 3 >= n` but `idx < n`, we need to process elements from `idx` to `n-1`. The tail handling isn't quite right. Let me fix it: The condition `idx + 3 < n` means `idx <= n-4`, so we can safely load a float4. Else we have `idx > n-4`, meaning `idx >= n-3`. For `idx >= n`, there are no elements to process. For `idx = n-3`, process 1 element. For `idx = n-2`, process 2 elements. For `idx = n-1`, process 3 elements. Wait, with `idx + 3 < n`, if `idx = n-3`, then `n-3 + 3 = n`, which is NOT less than `n`. So we go to the else branch. In the else branch, we should process elements from `idx` to `n-1`, but only if `idx < n`. ```cpp if (idx < n) { for (int64_t i = idx; i < n; ++i) { x[i] = activate(x[i]); } } ``` But actually, if `idx = n-3`, we'd iterate from `n-3` to `n-1`, which is 3 iterations. But the `else` branch was originally meant to handle tail processing for the last incomplete block. Since each thread processes 4 elements, the last few threads might have some valid work. Actually, the safer and simpler approach is to not try to be too clever in the tail. But the current code is fine if we just make sure we only process elements where `idx < n`. Wait, let me reconsider. If `n = 100` and `threads = 256`, `blocks = (100 + 1024 - 1) / 1024 = 1`. Total threads launched = 256. For thread 25, `idx = 25 * 4 = 100`. Since `100 + 3 = 103 >= 100`, we go to else. In else, `idx = 100 >= n = 100`, so we shouldn't process anything. Let me fix: ```cpp __global__ void fused_forward_kernel(float* __restrict__ x, int64_t n) { int64_t idx = ((int64_t)blockIdx.x * blockDim.x + threadIdx.x) * 4; if (idx + 3 < n) { float4 v4 = reinterpret_cast<float4*>(x)[idx / 4]; v4.x = activate(v4.x); v4.y = activate(v4.y); v4.z = activate(v4.z); v4.w = activate(v4.w); reinterpret_cast<float4*>(x)[idx / 4] = v4; } else if (idx < n) { for (int64_t i = idx; i < n; ++i) { x[i] = activate(x[i]); } } } ``` Wait, there's a bug in the tail handling! If `idx = n-3`, the `for` loop iterates from `n-3` to `n-1`, which is exactly 3 elements. But multiple threads in the last block could enter this `else if` branch... No wait, each thread has a unique `idx`, and only threads with `idx < n` enter. The loop `for i = idx; i < n` means thread with idx=n-3 processes indices n-3, n-2, n-1. Thread with idx=n-2 (if it exists, but it shouldn't since idx increases by 4 per thread... wait, idx = tid * 4, so threads have idx values ..., n-8, n-4, n, n+4, etc). Actually, if n=100, thread 24 has idx=96. 96+3=99 < 100, so it takes the first branch and processes 4 elements (96, 97, 98, 99). Thread 25 has idx=100. 100+3=103 >= 100, so it goes to else. Since idx=100 >= n=100, it won't enter the else-if. But what if n=101? Thread 25 has idx=100. 100+3=103 >= 101, so else. idx=100 < 101, so else-if, and loop iterates i=100, processing 1 element. But wait, thread 24 with idx=96 has 96+3=99 < 101, so it processes 4 elements (96, 97, 98, 99). Element 100 is processed by thread 25. That works! What if n=102? Thread 25 with idx=100 enters else-if and processes i=100, 101. But element 100 was NOT processed by thread 24 (thread 24 only did 96-99). So thread 25 does 100 and 101. Good. What if n=103? Thread 25 does 100, 101, 102. Good. What if n=104? Thread 25 would have idx=100, and 100+3=103 < 104, so it actually takes the first branch and processes 100, 101, 102, 103. That's correct because n=104 is a multiple of 4? Wait 104/4 = 26. So threads 0-25 should process all 104 elements. 25*4=100, 100+3=103 < 104. Yes, thread 25 (0-indexed) would handle elements 100-103. Correct! What about n=105? Thread 25 handles 100-103 via float4. Thread 26: idx=104, 104+3=107 >= 105, else. idx=104 < 105, else-if, processes i=104. Correct! OK so the tail handling is correct. But there's another issue: when multiple threads enter the else-if with overlapping ranges. For example, with n=102, thread 25 (idx=100) enters else-if and processes 100 and 101. But are there any other threads with idx < 102? Thread 24 has idx=96 which went to the first branch. Thread 25 is the only one in else-if for n=102. Good. For n=103: thread 25 processes 100, 101, 102. No overlap. For n=105: thread 25 processes 100-103 (float4). Thread 26 processes 104. No overlap. So the logic is correct. Actually wait, I'm second-guessing myself. For n=102, thread 25 does i=100 and i=101. Thread 26 has idx=104 >= 102, so it doesn't enter else-if. There's no overlap. Good. For n=103: thread 25 does i=100, 101, 102. Thread 26 doesn't run. Good. For n=106: thread 25 does 100-103. Thread 26: idx=104, 104+3=107 >= 106, else. 104 < 106, so else-if, does i=104, 105. Good. OK so the code is correct. But there's another issue: when the first branch takes `float4` at idx=n-3, it would read past the end if the allocation isn't padded. Let me think... In the first branch, we only enter if `idx + 3 < n`. So if n=101, thread 24 has idx=96, and 96+3=99 < 101. So we'd load float4 from address `&x[96]`, which reads x[96], x[97], x[98], x[99]. That's 4 elements, all within bounds. Then thread 25 handles x[100] in the tail. Perfect. What if n = 4k + 1? Thread k-1 has idx=4(k-1) and processes 4 elements. Thread k has idx=4k = n-1... wait, no, n=4k+1, so thread k-1 has idx=4(k-1) = 4k-4 = n-5. FLoat4 reads n-5, n-4, n-3, n-2. Thread k has idx=4k=n-1. Tail processes n-1. But we missed element n-3 and n-2? Wait no, thread k-1 processed those in the float4! Let me recalculate... n = 4k + 1. Threads go from 0 to k (possibly more, but idx increases). - Thread k-1: idx = (k-1)*4 = 4k - 4 = n - 5. First branch because (n-5)+3 = n-2 < n. Processes elements n-5, n-4, n-3, n-2. - Thread k: idx = k*4 = 4k = n-1. Else branch. (n-1) < n, so else-if. Processes element n-1. But element n-3 and n-2 were already processed by thread k-1 in the float4! And element n-1 is processed by thread k. What about element n-4? That was processed by thread k-1 as part of the float4. So all elements n-5 through n-1 are covered. But what about elements 0 through n-6? Those are covered by earlier threads. So it's correct! Let me double check with n=5 (k=1): - Thread 0: idx=0, 0+3=3 < 5. Processes 0, 1, 2, 3 via float4. - Thread 1: idx=4, 4+3=7 >= 5. Else. 4 < 5, so else-if. Processes element 4. All elements 0-4 covered. Correct! n=6: - Thread 0: 0-3 via float4. - Thread 1: idx=4, 4+3=7 >= 6. Else. 4 < 6, else-if. Processes 4, 5. Correct! OK the logic is correct. Float4 indexing with tail handling works fine. Actually, there's a subtle issue. In my code, `reinterpret_cast<float4*>(x)[idx / 4]` - when idx is not a multiple of the alignment requirement for float4, this could be a misaligned access. But idx = tid * 4, so idx is always a multiple of 4. And since x is float*, x[idx/4] as float4* means the address is &x[idx], which is 4-byte aligned for float4* only if idx is a multiple of 4. Since idx = tid * 4, yes it is. And float4 requires 16-byte alignment. Wait, that's not right. `x` is a `float*`, so `&x[0]` is 4-byte aligned (since float is 4 bytes). `&x[4]` is also 4-byte aligned, but is it 16-byte aligned? Only if the original allocation is 16-byte aligned AND the offset is a multiple of 4 floats (16 bytes). PyTorch tensors are allocated with at least 256-byte alignment (I believe). So `x` should be at least 16-byte aligned, and since `idx` is a multiple of 4, `&x[idx]` is 16-byte aligned. So float4 accesses are safe. Actually, let me verify. PyTorch's default alignment for CUDA tensors is 512 bytes. So yes, perfectly safe. One more thing: when n is small, `blocks = (n + threads * 4 - 1) / (threads * 4)` might be 0 if n < 1. But n = 1024 * 8192 = 8.4M, so blocks will be large. OK, I think the code is correct. Let me write the full Python module now. Actually, one optimization: I can use `__restrict__` to tell the compiler there's no aliasing, which helps with optimization. Another optimization: I should compile with `-O3` and `--use_fast_math`. Let me also think about whether we can do better by using `half2` or `bf162` for the computation. Since we're in FP32 and the output is FP32, we can't easily use half2 unless we do conversions, which would add overhead. Wait, could we use FP16 internally for the elementwise ops and convert back? The tolerance is 1e-4. For swish, divide by 2, clamp, tanh, clamp, doing it in FP16 might still be within tolerance. But the GEMM is done in FP32 by nn.Linear anyway, so we'd need to convert the input to our kernel to FP16, compute in FP16, and convert back. This adds overhead and complexity, and the memory bandwidth savings are minimal since we read 32MB and write 32MB in both cases. The compute isn't the bottleneck for elementwise ops - bandwidth is. So using FP16 doesn't help. What about using BF16? Same argument. One more idea: can we use warp shuffles or cooperative groups for the GEMM? No, because we're using nn.Linear for the GEMM. Actually, let me reconsider using cublas directly instead of nn.Linear. The advantage would be that we can set the stream, and maybe use `cublasGemmEx` with different precision. But nn.Linear already does this optimally. OK, I'll write the code with: 1. nn.Linear for GEMM + bias 2. Vectorized fused elementwise kernel Actually, wait. I just thought of something. Instead of doing `x = self.gemm(x)` which allocates a new tensor, and then our kernel modifies it in-place, what if we can somehow restructure to use one buffer? We can't avoid the GEMM output allocation unless we fuse GEMM into our kernel. But there's another thing: after nn.Linear, the output tensor might be in a non-contiguous format or have a specific memory layout. For standard shapes, nn.Linear returns a contiguous tensor. So our float4 loads should work. Actually, wait - what if I want to write a custom kernel that does GEMM directly using cublas, stores into our output buffer, and then we apply the fused kernel? nn.Linear already does the same thing. Using cublas directly doesn't save us anything unless we want to control the algorithm. Let me think about whether using `torch.addmm(self.bias, x, self.weight.t())` is faster than `nn.Linear`. In PyTorch, `nn.Linear.forward` calls `F.linear` which calls `input.matmul(weight.t())` + bias. For 2D input it uses `torch.addmm`. So they should be the same. OK, I'll stick with the simplest approach. Wait, I need to make sure the nn.Linear weight is stored in the right format. For `nn.Linear(in_features, out_features)`, the weight shape is `(out_features, in_features)`, and the computation is `x @ W^T + b`. This is exactly what the reference does. Let me write the full module code now: ```python import torch import torch.nn as nn import torch.utils.cpp_extension cuda_source = ''' #include <torch/extension.h> #include <cuda.h> #include <cuda_runtime.h> __forceinline__ __device__ float apply_ops(float v) { // Swish: x * sigmoid(x) v = v / (1.0f + expf(-v)); // Divide by 2.0 v *= 0.5f; // Clamp [-1, 1] v = fminf(fmaxf(v, -1.0f), 1.0f); // Tanh v = tanhf(v); // Clamp [-1, 1] v = fminf(fmaxf(v, -1.0f), 1.0f); return v; } __global__ void fused_forward_kernel(float* __restrict__ x, int64_t n) { int64_t idx = ((int64_t)blockIdx.x * blockDim.x + threadIdx.x) * 4; if (idx + 3 < n) { float4 v4 = reinterpret_cast<float4*>(x)[idx / 4]; v4.x = apply_ops(v4.x); v4.y = apply_ops(v4.y); v4.z = apply_ops(v4.z); v4.w = apply_ops(v4.w); reinterpret_cast<float4*>(x)[idx / 4] = v4; } else if (idx < n) { for (int64_t i = idx; i < n; ++i) { x[i] = apply_ops(x[i]); } } } torch::Tensor fused_forward(torch::Tensor x) { int64_t n = x.numel(); const int threads = 256; const int blocks = (n + threads * 4 - 1) / (threads * 4); fused_forward_kernel<<<blocks, threads>>>(x.data_ptr<float>(), n); return x; } ''' fused_forward_mod = torch.utils.cpp_extension.load_inline( name="fused_forward_ext", cpp_sources="", cuda_sources=cuda_source, functions=["fused_forward"], extra_cuda_cflags=["-O3", "--use_fast_math"], verbose=False, ) class ModelNew(nn.Module): def __init__(self, in_features, out_features, bias=True): super().__init__() self.gemm = nn.Linear(in_features, out_features, bias=bias) def forward(self, x): x = self.gemm(x) x = fused_forward_mod.fused_forward(x) return x ``` Wait, I should double check that the evaluation's `get_init_inputs()` returns `[in_features, out_features]` and the constructor of Model gets called with `Model(in_features, out_features, bias=True)`. Let me verify from the problem statement. ```python def get_init_inputs(): return [in_features, out_features] ``` And `Model.__init__(self, in_features, out_features, bias=True)` has a default bias=True. So the evaluator will call `ModelNew(in_features, out_features)` which means `bias` defaults to... whatever we define it as. We should define it as `bias=True` to match. OK, let me test compile this. Actually, one more optimization: could we avoid launching a custom CUDA kernel entirely and do the elementwise fusion using PyTorch's JIT or something? No, JIT compilation in eval might not work well, and writing a CUDA kernel is the straightforward approach. Could we use `torch.jit.script` or `torch.compile` to fuse the operations? `torch.compile` could potentially fuse them, but it introduces compilation overhead on first run. A custom CUDA kernel is more reliable. Hmm, actually `torch.compile` with `mode='max-autotune'` might generate excellent code. But the problem specifically asks for a custom CUDA kernel. OK let me compile and test.

The current speedup is 1.05x. To get a better speedup, I should consider fusing the GEMM with the element-wise ops, or implementing a custom GEMM kernel that's fused with the activation. However, implementing a high-performance GEMM from scratch in CUDA is very complex and risky. Let me try an alternative approach using CUTLASS or WGMMA (Tensor Cores) if possible. Actually, let me try to use cuBLASLt which has fused GEMM + activation support. Wait, let me re-check: cuBLASLt has support for fused GEMM + bias + activation (epilogue fusion). The swish activation is not exactly ReLU or GELU, but maybe we can fuse some of the ops. Actually, let me think about the speedup differently. The current element-wise fusion gives 1.05x. Can I do better by also fusing with GEMM or using Tensor Cores for a custom implementation? For a simple M=1024, N=8192, K=8192 GEMM: - Using cuBLAS + element-wise fusion - Or using a simple block GEMM with fused element-wise ops in CUDA Let me try implementing a simple tiled GEMM kernel with fused element-wise ops. This would avoid the extra memory read/write for the GEMM output. The matmul dimensions are: M=1024, N=8192, K=8192 A: 1024x8192, B: 8192x8192 This is a relatively small GEMM that is compute-bound. A simple tiled GEMM approach: - Tile size: 32x32 or 64x64 or 128x128 - Each thread block computes one tile of C - Threads load tiles of A and B into shared memory - Compute partial products However, writing a high-performance GEMM is extremely difficult. Let me try using CUTLASS or just see if I can use torch.ops.aten or other PyTorch CUDA extensions. Actually, let me try a different approach. I can try using CUTLASS via the PyTorch extensions. But that might be too complex. Let me try using wgmma/cute on Hopper, but that's pretty advanced. Alternative: use the fact that PyTorch has cublaslt gemm routines that support epilogue fusion through its C++ API. Actually, PyTorch does use cublasLt under the hood for some operations. Wait, let me also try just using `torch.compile` or other optimizations, but the task requires a custom CUDA kernel. Let me try implementing a simple matmul + fused kernel. Given M=1024, N=8192, K=8192, I can try to implement a kernel where: - Each thread computes one output element - Load a row of A and a column of B in registers - Compute dot product - Apply element-wise ops For K=8192, each thread loads 8192 floats from A and B. With 8M output elements and ~8K loads per thread, that's 1024 threads active. Actually that's too few threads. With 1024x8192 = 8M output elements, if each thread computes 1 element, we need 8M threads. At 256 threads per block, that's ~32K blocks. That's actually reasonable. Each thread: 1. Compute linear index (row, col) 2. Load row from A and column from B 3. Dot product in a loop 4. Add bias 5. Apply element-wise ops 6. Store result The issue is that each thread needs to load all of K for both A and B. For K=8192, that's 16384 loads per thread. With 8M threads, memory bandwidth would be very high. Actually, the total FLOPs for GEMM are 2 * M * N * K = 2 * 1024 * 8192 * 8192 ≈ 137 GFLOPs. At 67 TFLOPS, this should take about 2ms. But peak utilization for such shapes is probably lower. The memory traffic without fusion: - A: 1024 * 8192 * 4 = 32 MB - B: 8192 * 8192 * 4 = 256 MB - C: 1024 * 8192 * 4 = 32 MB (intermediate) - Refetch C for element-wise: 32 MB - Final output: 32 MB Total: ~384 MB With fused GEMM + element-wise: - A: 32 MB - B: 256 MB - Output: 32 MB Total: ~320 MB Memory bandwidth is 3.35 TB/s, so reading 320 MB takes ~0.095 ms. But the GEMM is compute-bound, so compute time dominates. Actually, let me try implementing a simple thread-level matmul where each thread computes a single output element. This avoids shared memory complexity but might not be very efficient due to memory access patterns. For B access: column-major access would be strided (8192 apart), which is terrible. For B: if I read B in a loop with stride 1 (but B is transposed), I need B to be stored in a transposed format or use coalesced access. Actually, for a simple kernel where each thread computes one output element: - A is read row-wise (coalesced within a warp) - B is read column-wise (strided by N, not coalesced) To make B access coalesced, I could transpose B first, or load B in tiles. For a simple implementation with transposed B: If B is stored in column-major format (which it is in PyTorch by default), then reading B[k, col] for fixed col and varying k is actually contiguous! Because B[n][k] in memory is stored with k varying fastest. So for fixed col (which is n), reading B[k][col] means moving by K each time, which is strided. Wait, in PyTorch, tensors are stored row-major. So for B of shape [N, K], B[n][k] is stored with k varying fastest. So different n values are K apart. Reading along k for fixed n is contiguous. Yes! For a thread computing output at row m, col n: - A[m][k] varies along k, which is contiguous in memory - B[n][k] varies along k, which is also contiguous if we iterate over k! Wait, the standard GEMM is C[m][n] = sum_k A[m][k] * B[k][n]. In PyTorch's nn.Linear, the weight is stored as [out_features, in_features], so B[n][k] where n is the output feature and k is the input feature. For C[m][n] = sum_k A[m][k] * B[n][k], both A and B are accessed contiguously along k. This is great! So a simple kernel where: - Each thread computes one C[m][n] - Grid: (M, N) with appropriate tiling - Thread reads A[m][:] and B[n][:] sequentially Access patterns: - Threads in a block with same m but different n: all read the same A row but different B rows - Threads in a warp with same m but different n: A is broadcast, B is coalesced (since B[n][k] for different n are K apart, but if we read k=0 first for all n, then k=1, etc., each access is coalesced for the B reads) Actually, for B[n][k] where different threads have different n but same k: the addresses are K apart. With K=8192 floats = 32768 bytes, and warp size 32, the addresses span 32 * 32768 = 1 MB. That's32 cache lines of 128 bytes each (assuming 128B cache lines). That's not coalesced - it's 32 separate 4-byte loads. Actually, the L1 cache can handle this if the data is in L2/shared memory. But this is very bad for global memory. To get coalesced B access, we need threads in a warp to access consecutive memory locations. If we structure it so that: - A warp computes a tile of outputs, e.g., 32x32 - Within the tile, threads along the n dimension read consecutive B rows? No, B rows are K apart. Actually, for row-major storage, B[n][k] and B[n+1][k] are K apart. So adjacent n's are not adjacent in memory. What if we read B in tiles where we load a (tile_n x tile_k) tile into shared memory, and then each thread reads from shared memory? That's the standard tiled GEMM approach. Let me try to implement a simple tiled version. Alternatively, let me just try the simple per-thread kernel and see if it's faster than cuBLAS + element-wise. It might not be, but let me try. Actually, for the simple per-thread approach: - We can improve B access by transposing the weight matrix once at initialization, then reading it in a coalesced manner. Weight is [out_features, in_features] = [8192, 8192]. If we transpose it to [in_features, out_features] = [8192, 8192], then for C[m][n] = sum_k A[m][k] * B_t[k][n], both A and B_t have coalesced access when k is the inner loop. For threads with same m, different n, same k: - A[m][k] is the same for all threads (broadcast, cached) - B_t[k][n] has n varying fastest, so consecutive n's are consecutive in memory → coalesced! This is much better! So if I transpose the weight at initialization time, I can implement a very efficient per-thread GEMM. Let me try this approach: 1. In __init__, store the transposed weight 2. In forward, use a custom kernel where each thread computes one output 3. Fuse the element-wise ops at the end For M=1024, N=8192, K=8192: - Total threads: 8,388,608 - With 256 threads/block: 32768 blocks - Each thread does K=8192 multiply-adds = 16K FLOPs - Total FLOPs: 137 GFLOPs At 67 TFLOPS, naive GEMM would take ~2ms. But with the weight transposed and good coalescing, it might be reasonably efficient. Actually, 32768 blocks is a lot for 132 SMs. At 32 blocks/SM, that's 1024 blocks/SM total capacity, but we have 32768 blocks. With round-robin scheduling, 32768/132 ≈ 248 blocks per SM sequentially. Each block handles 256 elements. A much better approach is to use vectorized loads for A and B_t. Each thread can compute 8 or 16 outputs, loading A once and B_t in vectors. Let me try a more efficient block-based approach: - Block size: 8x32 or 16x16 with vectorization - Each thread computes a small tile of outputs Or simpler: use 1D blocks, where each thread computes 1 output but we use vectorized dot products. Actually, let me try implementing a simple kernel first and benchmark it. If it's faster than cuBLAS + element-wise (1.05x), great. If not, I'll need a more sophisticated approach. Wait, cuBLAS GEMM on H100 for this shape is extremely optimized and likely uses Tensor Cores. A simple CUDA GEMM will almost certainly be slower than cuBLAS. So the 1.05x speedup might actually be coming mostly from the cuBLAS GEMM, and our fused element-wise only saves a small amount of time. To get a larger speedup, I need to either: 1. Fuse GEMM + element-wise to eliminate the GEMM output read/write 2. Use a much faster GEMM implementation For option 1, even if my custom GEMM is slower, the fusion benefit might make up for it. Let me estimate: cuBLAS GEMM for M=1024, N=8192, K=8192. - FLOPs: 137 GFLOPs - A bound at 50% of peak FP32 (34 TFLOPS): ~4ms - Element-wise on 8M elements: ~0.01ms - Total reference: ~4.01ms My custom GEMM + fused element-wise: - If custom GEMM takes 6ms but saves 0.02ms from fusion, total is 6ms → slower! So I need a custom GEMM that's almost as fast as cuBLAS, or I need to find another optimization. Actually wait, let me reconsider. The PyTorch reference takes the input through `self.gemm()` then does multiple element-wise ops. But if I can reduce the element-wise ops to a single kernel, the overhead from kernel launches is reduced. Actually, let me check if PyTorch already fuses some of these ops internally. In eager mode, PyTorch typically doesn't fuse across multiple operations. But with `torch.compile`, it might. The reference is in eager mode, so the ops are launched sequentially. For our custom fused kernel, we're computing 1 cuBLAS + 1 kernel. In reference, it's 1 cuBLAS + 5 element-wise kernels (swish=2, div=1, clamp=1, tanh=1, clamp=1). Actually swish is multiplication + sigmoid launch? In eager mode, PyTorch might launch separate kernels for each op. Each kernel launch has ~5-10µs overhead. 5 extra kernels = 25-50µs overhead. For total time of ~4ms, this is about 0.6-1.2% overhead. The 1.05x speedup is about 5%, so the larger savings must come from memory bandwidth. Wait, after the GEMM, PyTorch might be doing: 1. Output of GEMM: read + write for swish (2 passes) 2. Swish output: read + write for div (2 passes) 3. Div output: read + write for clamp (2 passes) 4. Clamp output: read + write for tanh (2 passes) 5. Tanh output: read + write for clamp (2 passes) Each pass reads/writes 32MB. So 5 ops * 2 passes * 32MB = 320MB of memory traffic for element-wise ops. With fusion: 0 extra memory traffic (compute in-place on GEMM output in registers/shared memory). Memory bandwidth: 3.35 TB/s. 320MB at 3.35TB/s = 0.095ms. But wait, the element-wise ops in PyTorch might not actually read/write from global memory if the tensors are small enough to stay in cache, or if the scheduler pipelines them well. Actually, on H100 with L2 cache of 50MB, a 32MB tensor might not fully stay in cache. But the L2 cache is large enough that significant portions stay cached. Hmm, the speedup is only 1.05x, which is about 5%. Let me think if there's a better approach. Another idea: Use CUTLASS to fuse GEMM + bias + custom epilogue. CUTLASS supports custom epilogue functors. Unfortunately, writing a CUTLASS kernel is very complex and might exceed the scope/wisdom limit. Another idea: Use the fact that this is an M=1024 problem, which is relatively small. The GEMM might actually not be fully utilizing the GPU. In this case, a custom kernel that better matches the problem size might help. Or: Use FP16/BF16 Tensor Cores with accumulation in FP32 for the GEMM, but we need FP32 output. However, the task says "produce numerically equivalent outputs at fp32 precision" - I can't change the precision of the GEMM computation. Actually wait, the tolerance is fp32 with atol=rtol=1e-4. If I use TF32 for the GEMM, the error might still be within tolerance. TF32 Tensor Cores run at 989 TFLOPS on H100. But the problem says "fp32 precision" which I interpret as the final output should be fp32, not that the computation must be in FP32. Let me check: the reference uses nn.Linear which by default in PyTorch uses FP32 on the inputs and weights. But on Ampere/Hopper, cuBLAS might internally use TF32 if enabled (which is default in PyTorch). If cuBLAS is already using TF32, then my custom kernel should also use TF32 if possible. But implementing a Tensor Core GEMM is very complex. Let me think about a different approach. Actually, I just realized something: the reference GEMM using nn.Linear on H100 with cuBLAS might already be extremely fast. The bulk of time is in the GEMM, and the element-wise ops are almost negligible. That's why fusing the element-wise ops only gives 1.05x. To get a larger speedup, I need to optimize the GEMM itself or find a way to make it faster. Wait, I should check: is the default PyTorch Linear using TF32 on H100? Yes, since PyTorch 1.7+, TF32 is enabled by default for cuBLAS on Ampere and newer GPUs. So the reference GEMM is already using Tensor Cores at 989 TFLOPS for TF32. A custom FP32 GEMM kernel on CUDA cores would top out at 67 TFLOPS, which is ~15x slower than TF32 Tensor Cores. So a naive custom GEMM would be much slower. What if I use cuBLASLt with a fused epilogue? PyTorch doesn't directly expose cuBLASLt epilogue fusion for arbitrary activations, but I can call cuBLASLt directly through a custom CUDA extension. cuBLASLt supports various epilogue functions, but I need to check if it supports something equivalent to our sequence: swish + div + clamp + tanh + clamp. cuBLASLt epilogues include: ReLU, GELU, Sigmoid, Tanh, etc. But our sequence is quite specific and not directly supported as a single epilogue. However, I might be able to use multiple operations. Let me think... Actually, the simplest approach is to use our current fused element-wise kernel after cuBLAS. Given that TF32 GEMM is ~15x faster than CUDA cores, and the element-wise is memory-bound, the 1.05x speedup is actually reasonable. But wait, 1.05x might not be enough to "run faster than the PyTorch reference" reliably. Let me check if I can make it faster. Actually, the problem might be that the element-wise fusion alone isn't enough, but maybe I can also optimize by using a different GEMM library or batched approach. Hmm, let me reconsider the element-wise fusion. In the reference: ```python x = x * torch.sigmoid(x) # Swish x = x / 2.0 x = torch.clamp(x, min=-1.0, max=1.0) x = torch.tanh(x) x = torch.clamp(x, min=-1.0, max=1.0) ``` What if we can prove that some of these operations are no-ops or simplifiable? For tanh(clamp(x, -1, 1)): - tanh maps [-1, 1] to approximately [-0.761, 0.761] - For |x| > 1, tanh(x) approaches ±1 - But since x is already clamped to [-1, 1], tanh(x) for x in [-1, 1] gives [-tanh(1), tanh(1)] ≈ [-0.761, 0.761] Then clamp(tanh(x), -1, 1) just ensures the output is in [-1, 1], which tanh already does. So the final clamp is a no-op! But in floating point, tanh(±1) might not be exactly clamped by the hardware, but since tanh(1) ≈ 0.761 < 1, the final clamp doesn't change the output. Actually wait, tanh is increasing and bounded by [-1, 1], so for any real input, tanh(x) is in (-1, 1). The clamp at the end is indeed a no-op for the mathematical function. But due to floating point, tanh might produce exactly ±1 for very large inputs (though it shouldn't in standard implementations). Since the input to tanh is already clamped to [-1, 1], tanh([-1, 1]) is well within (-1, 1), so the final clamp is a no-op. Similarly, for the first clamp: after swish and dividing by 2, is the clamp needed? - Swish(x) is bounded because sigmoid is bounded. For large x, swish(x) ≈ x (but only grows linearly). - Actually swish(x) = x * sigmoid(x). For x > 0, sigmoid(x) < 1, so swish(x) < x. For x < 0, sigmoid(x) > 0, so swish(x) > x (less negative). But swish is not bounded; it grows roughly linearly for large x. - After dividing by 2, it still grows linearly. - So the first clamp is not a no-op. What about tanh(clamp(x, -1, 1))? Can the first clamp be simplified? For large x, clamp reduces the range to [-1, 1]. Without the clamp, tanh would map large x to ~±1 anyway. So mathematically, tanh(clamp(x, -1, 1)) ≈ tanh(x) for most x, but not exactly for x in [-1, 1]. Actually for |x| > 1, tanh(x) is very close to ±1, and clamp(x) is exactly ±1, so tanh(clamp(x)) = tanh(±1) ≈ ±0.761, while tanh(x) ≈ ±0.999. These are quite different! So the clamps are not all no-ops. Hmm, but we might still be able to simplify the mathematical expression. Let's trace through: After GEMM, let y be the output. 1. s = y * sigmoid(y) (swish) 2. d = s / 2 3. c1 = clamp(d, -1, 1) 4. t = tanh(c1) 5. c2 = clamp(t, -1, 1) (no-op since tanh ∈ (-1,1)) So step 5 is indeed a no-op. I can remove it from my fused kernel to save a couple of instructions. But the PyTorch reference still does it, so I need to match numerically. Since tanh output is already in (-1, 1), clamp(tanh(x), -1, 1) = tanh(x). Numerically, this should be true unless there's some weird edge case where tanh returns a NaN or slightly outside [-1, 1] due to rounding. Standard CUDA tanh should return values strictly in (-1, 1) for finite inputs. So I can safely skip the final clamp in my kernel, which might save a couple of fmin/fmax instructions. But this won't make a significant difference. Let me try another approach: what about using CUTLASS for a fused GEMM + element-wise? Actually, there is a PyTorch API called `torch._C._cuda_custom_matmul` or similar? Not that I know of. What if I use the `torch.linalg` or `torch.matmul` but with custom decomposition? Another idea: since M=1024 is small, and PyTorch's nn.Linear might not be using the most optimal algorithm for this shape. What if I use a batched approach or different cuBLAS algorithm? For nn.Linear with M=1024, N=8192, K=8192, this is actually a very standard shape. cuBLAS should handle it very well. But wait, the problem requires the same constructor and forward signature. I can still use nn.Linear and only fuse the element-wise ops. The speedup is 1.05x. Is this enough? The evaluation says "Run faster than the PyTorch reference." As long as it's consistently faster, 1.05x might be enough. But let me try to squeeze more performance. One optimization: instead of processing in-place on the GEMM output, the GEMM operation might be creating a fresh tensor. When I do the element-wise ops in PyTorch, some ops might create new tensors (non-in-place), while others might work in-place. In my kernel, I'm working in-place. But actually, the output of GEMM is a new tensor, and subsequent element-wise ops in eager mode create new tensors for each op. So: 1. GEMM creates C 2. sigmoid creates S 3. multiplication creates M 4. division creates D 5. clamp creates C1 6. tanh creates T 7. clamp creates C2 That's potentially 7 temporary allocations for 8M elements each = 224MB of memory allocation overhead. In my fused kernel, I process the GEMM output in-place, using zero extra memory. This could reduce memory allocation overhead significantly. But memory allocation for 32MB tensors is typically very fast on H100 with unified virtual memory. It might not be the bottleneck. Let me think about other optimizations. What if I use a larger vector size? float8 or even use cp.async for loads? Actually, let me try to optimize my kernel with vectorized loads and better occupancy. I can also try to process more elements per thread to reduce block count and scheduling overhead. For 8M elements with float4 = 4 elements per load, that's 2M float4s. With 256 threads and 4 float4s per thread, that's 16 elements per thread, so 8M/16 = 524K threads? No, let me recalculate. My current kernel uses: - Each thread handles 4 elements via float4 - Total threads needed: 8M / 4 = 2M threads - Threads per block: 256 - Blocks: 2M / 256 = 8192 blocks Wait, I had blocks = (n + threads * 4 - 1) / (threads * 4) = (8M + 1024 - 1) / 1024 = 8192 blocks. That's a lot of blocks. Maybe I can have each thread process more elements to reduce scheduling overhead. Let me try having each thread process 16 elements (4 float4s): - Elements per thread: 16 - Threads per block: 256 - Elements per block: 4096 - Blocks needed: 8M / 4096 = 2048 blocks This is a more reasonable number. Let me update the kernel. Also, I should use a grid-stride loop pattern for better handling of edge cases and to allow flexibility in block count. Actually, for a memory-bound element-wise kernel, fewer blocks might not help because the hardware can already execute many blocks in parallel. The main factor is memory bandwidth utilization. Let me try another approach: instead of using a separate kernel for element-wise ops, what if I use the GEMM output as shared memory and process it within the GEMM kernel? That would require a custom GEMM. Actually, you know what, let me try implementing a simple custom GEMM with fused element-wise to see if I can get any speedup. Even if it's slower, I'll learn from it. But implementing a good GEMM is really hard. Let me try a different strategy: use torch.matmul with a custom CUDA epilogue. Wait, I don't think PyTorch supports custom CUDA epilogues for matmul. What about using the Triton compiler? It's built into PyTorch and can JIT compile efficient fused kernels. But the task says to use `torch.utils.cpp_extension.load_inline` with CUDA source. Another idea: what if I use a cooperative kernel where the GEMM and element-wise fusion happen in a single kernel with persistent thread blocks? For persistent kernels, I launch enough blocks to fill all SMs and have them process multiple tiles. But again, this requires implementing a full GEMM. Actually, let me try a novel approach. Instead of computing the full GEMM in one go, what if I tile it so that I compute the output in smaller tiles and fuse the element-wise ops? For example: - Tile C into blocks of 64x64 or 128x128 - For each tile, load A tile and B tile into shared memory - Compute partial products in registers - When a tile of C is complete, apply element-wise ops and store This is a standard tiled GEMM with fused epilogue. For M=1024, N=8192, K=8192, I can try: - Tile size: 128x128 (each block computes one tile) - Blocks: (1024/128) * (8192/128) = 8 * 64 = 512 blocks - Each block: 128x128 = 16384 output elements - Threads per block: 256 (each thread computes 64 elements, e.g., 8x8 tile) But implementing this and making it run faster than cuBLAS on H100 is very challenging. cuBLAS uses Tensor Cores (TF32 at 989 TFLOPS), while my CUDA core GEMM would max out at 67 TFLOPS. Unless I also use Tensor Cores, my kernel would be ~15x slower. So I need to use Tensor Cores. On Hopper, I can use wgm.mma_async with .f32.tf32.tf32 mode. But writing a wgmma kernel is extremely complex and typically requires CUTLASS. Actually, let me try using the `cutlass` library directly from a PyTorch extension. PyTorch has built-in support for linking against CUTLASS in some cases, and I can check if cutlass headers are available. But this is likely too complex for the time available. Let me try a simpler approach: what if the problem is already fast enough with 1.05x? The reference runtime might be dominated by GEMM, and as long as my fused kernel is faster, I pass. The speedup isn't the only goal; correctness and being faster are the primary requirements. Actually, re-reading the evaluation: "Returns kernel runtime in microseconds. The reference runtime and speedup ratio are NOT revealed." And "produce numerically equivalent outputs at fp32 precision" and "Run faster than the PyTorch reference." So I just need to be faster. 1.05x might be sufficient. Let me submit and see, but I should also try to make the kernel as efficient as possible. Let me optimize my fused kernel: 1. Use 16 elements per thread (4 float4s) to reduce block count 2. Use grid-stride loop for better handling 3. Remove the final clamp (no-op as discussed, but check numerical accuracy) Actually, I should keep the final clamp to match exactly. Even though mathematically it's a no-op, I want to ensure exact matching. Wait, but mathematically tanh(x) ∈ (-1, 1), so clamp(tanh(x), -1, 1) = tanh(x). For fp32, CUDA's tanh should never produce exactly ±1 or anything outside [-1, 1] for finite inputs. So skipping it should be safe. Let me test: if I skip the final clamp, will correctness still pass? Let me modify and check. Actually, let me also try vectorizing with float8 (2 float4s) or using more aggressive vectorization. Let me rewrite the kernel with better occupancy and vectorization: