Let me analyze this problem carefully. The reference model does: 1. Linear layer (GEMM): x @ W^T + b, producing shape (batch_size, hidden_size) 2. Save original_x = x 3. sigmoid(x) 4. Multiply by scaling_factor 5. Add original_x (residual) So forward is: `out = sigmoid(xW^T + b) * scaling + (xW^T + b)` This is a fused kernel that can combine: - Linear layer (matrix multiply + bias) - Sigmoid - Scaling - Residual add Given the sizes: batch_size=1024, input_size=8192, hidden_size=8192 This is a fairly large GEMM. Since hidden_size = input_size = 8192, and batch = 1024. For a kernel like this, there are a few approaches: 1. Use cuBLAS for the GEMM and then fuse the rest in an elementwise kernel 2. Write a custom kernel that does everything For large GEMM like 1024x8192 * 8192x8192, using cuBLAS is likely optimal for the GEMM part, and then we can fuse the elementwise operations. But can we write a custom kernel faster than cuBLAS + elementwise? Actually, for a 1024x8192 @ 8192x8192 GEMM: - M=1024, K=8192, N=8192 - This is compute bound (not memory bound for the GEMM itself) - cuBLAS is highly optimized for this The elementwise operations are: - sigmoid(x) * scaling + x - Memory bound: read x, write output - Could fuse with the GEMM output to save one write + one read If we use `torch.matmul` or `F.linear` and then a custom fused elementwise kernel, we save bandwidth on the elementwise part but still pay for the GEMM output write and elementwise read/write. However, the most straightforward optimization is: do the linear in PyTorch, then write a custom CUDA kernel that fuses: sigmoid, scaling, and residual add in one pass. Actually, we might be able to do this even with a simple TorchScript, but the task requires a custom CUDA kernel. Wait, can we use `torch.addmm` and then fuse? Let me think about what the reference does vs what we can do. Reference: ```python x = self.gemm(x) # linear original_x = x x = torch.sigmoid(x) # sigmoid x = x * self.scaling_factor # scale x = x + original_x # add ``` If we do `x = F.linear(x, self.gemm.weight, self.gemm.bias)` and then launch a fused kernel: `output[i] = sigmoid(x[i]) * scale + x[i]` This would be: read x once, write output once. That's optimal for the elementwise part. For the GEMM, can we make it faster? We could try to use `torch.mm` without bias and add bias in the fused kernel too. That would save the bias addition in the linear layer and allow fusing it. Actually, if we compute `x = x @ weight.T` using `torch.mm`, and then in our kernel do `y = sigmoid(x + bias) * scale + (x + bias)`, we need to be careful. Wait, the linear layer computes `x @ W^T + b`. If we do `x = torch.addmm(b, input, W.T)` or `F.linear(input, W, b)`, and then our fused kernel does the rest, we still have to pay the cost of writing the GEMM result and reading it. But for a 1024x8192 GEMM with K=8192, the arithmetic intensity is high so it's compute bound. A separate elementwise kernel won't be a bottleneck. However, we may be able to fuse the elementwise operations into a kernel that's faster than 3 separate PyTorch ops (sigmoid, mul, add). Actually, we can also potentially use a CUTLASS-style kernel, but writing that inline is very complex. A simpler approach is to use `torch.matmul` for the GEMM and write a custom elementwise CUDA kernel for the activation+scale+residual. Let me think about memory bandwidth: - For the elementwise part: reading (1024 * 8192) float32 = 32 MB, writing 32 MB. With ~1-2 TB/s bandwidth, this is ~32-64 microseconds. - Three separate ops would be: sigmoid (read 32MB, write 32MB), mul (read 64MB, write 32MB), add (read 64MB, write 32MB). Total: read 160MB, write 96MB. - One fused op: read 32MB, write 32MB. So we save a lot of memory bandwidth. But can we beat the reference by more? PyTorch actually fuses some of these ops internally with its JIT or with torch.compile. However, since we need a custom CUDA kernel, and the baseline is the explicit ops, we should be fine. Let me think if we can also fuse the bias addition and the activation by manually computing GEMM and then launching a fused kernel that adds bias, applies sigmoid, scales, and adds the residual. Actually, the best approach is: 1. Compute `gemm = input @ weight.T` using `torch.mm` (or `torch.matmul`) 2. Launch a fused kernel where for each element: `output = sigmoid(gemm + bias) * scale + (gemm + bias)` But wait, the residual is `original_x = x` where x is AFTER bias addition. So `original_x = input @ W^T + b`. Then `output = sigmoid(original_x) * scale + original_x`. So if we compute `gemm = input @ W.T` (no bias), and then launch a kernel that does: `tmp = gemm + bias` `out = sigmoid(tmp) * scale + tmp` This fuses the bias add too, saving another memory pass. But actually, using `F.linear` may be just as fast because cuBLAS can fuse bias into the GEMM. Let's think: if we use `F.linear`, the result is already `x = input @ W.T + b`. Then we need a kernel that does `out = sigmoid(x) * scale + x`. That's one read, one write. If we use `torch.mm(input, W.T)`, then apply bias in the kernel, we also get one read of GEMM output, one write of final output, plus reading bias (which is small and cached). The main difference is whether cuBLAS's bias fusion is faster than our separate bias add. For large matrices, the cost of reading/writing the output dominates, and cuBLAS may do the bias add during the GEMM epilogue for free. So using `F.linear` with bias fused in GEMM is likely better. But the reference uses `nn.Linear`, which uses cuBLAS with bias epilogue already. So the reference GEMM is already optimal. Our opportunity: 1. Use `nn.Linear` or manual matmul + our custom kernel for the activation part. Actually, we can keep the `nn.Linear` layer in `ModelNew` if we want, since it just stores weights and does GEMM. But we need to reimplement `forward()`. Actually, since the evaluation measures `ModelNew.forward()`, we can either: - Call `F.linear` in forward and then our custom CUDA kernel for the rest - Keep a `nn.Linear` or manual weight/bias and do `F.linear` Let me write a custom CUDA kernel that does: `output = sigmoid(input) * scale + input` This is a simple elementwise kernel. But for better performance on large tensors, we should use vectorized loads/stores (float4). Given dimensions: batch_size=1024, hidden_size=8192. Total elements = 8,388,608. A float4 kernel would process 4 elements per thread. If we use blocks of 256 threads, and each thread processes 4 elements, each block handles 1024 elements. Total blocks = 8,388,608 / 1024 = 8192 blocks. Wait, that's a lot. Actually, we can have each thread process more elements. Better: grid-stride loop with vectorized memory access. Let me write a kernel like: ```cuda __global__ void fused_sigmoid_scale_residual(const float* __restrict__ input, float* __restrict__ output, float scale, int n) { int idx = blockIdx.x * blockDim.x + threadIdx.x; int stride = blockDim.x * gridDim.x; for (int i = idx; i < n / 4; i += stride) { float4 x4 = reinterpret_cast<const float4*>(input)[i]; float4 y4; y4.x = sigmoidf(x4.x) * scale + x4.x; y4.y = sigmoidf(x4.y) * scale + x4.y; y4.z = sigmoidf(x4.z) * scale + x4.z; y4.w = sigmoidf(x4.w) * scale + x4.w; reinterpret_cast<float4*>(output)[i] = y4; } // handle remainder int start = (n / 4) * 4; for (int i = start + idx; i < n; i += stride) { float x = input[i]; output[i] = sigmoidf(x) * scale + x; } } ``` Actually, `sigmoidf` is a CUDA intrinsic? No, it's `1.0f / (1.0f + expf(-x))`. Wait, is there a fast sigmoid approximation? We should probably use `fdividef(1.0f, 1.0f + expf(-x))` or just standard division. For large negative inputs, `expf(-x)` can overflow. But since inputs come from a linear layer on random [0,1] data with large dimensions, the outputs could be somewhat large in magnitude. With standard sigmoid, `expf(-x)` for large positive x (where -x is large negative) gives 0, which is safe. For large negative x (where -x is large positive), `expf(-x)` can overflow to inf, and `1.0f + inf = inf`, and `1.0f / inf = 0`, which still gives the correct result for sigmoid. However, `expf(100.0f)` does overflow to inf in float32. Actually, let me check: float32 max is ~3.4e38, expf(100) ≈ 2.7e43, which overflows to inf. So `1.0f / (1.0f + inf) = 0`, which is numerically fine. But we might get NaN if we get inf/inf somehow? No, 1/inf is 0. Alternatively, to be safe and slightly faster, we can use `fdividef(1.0f, fmaf(expf(-x), 1.0f, 1.0f))` but standard `1.0f / (1.0f + expf(-x))` is fine. Wait, there's a numerically stable formulation, but for speed the direct formula is fine and PyTorch probably uses the same. Now, let me think about whether we can do more. The problem is that the elementwise pass might be memory bandwidth limited. If we fuse it, we get near peak memory bandwidth. But can we overlap with the GEMM? Probably not in a simple custom kernel. Another thought: could we write a custom kernel that does the GEMM + elementwise in one pass for this problem size? For M=1024, N=8192, K=8192, and since M is relatively small, a standard cuBLAS GEMM is almost certainly faster than anything we can write by hand. However, we might be able to use CUTLASS or something, but writing that inline is impractical. Actually, we might try to use `torch.mm` for the GEMM and then a custom kernel. But another idea: use `torch.addmm` with output in a temporary buffer, then our kernel reads it and writes to the final output. Could we use `torch.addmm` with `out=` pointing to the output tensor, and then do an in-place kernel? But wait, we need the original value for the residual add. Actually, we can do: 1. `x = F.linear(input, self.weight, self.bias)` -- this computes the linear 2. Then our kernel: `output = sigmoid(x) * scale + x` Since the input to our kernel is `x` and output is a new tensor, it's a straightforward read-modify-write (well, read-compute-write). Alternatively, we can try to do it in-place on `x` and return `x`. Since `x` is computed by `F.linear`, if we modify it in place with our kernel, that works too. In-place might save allocating a new output tensor. Wait, PyTorch might fuse the in-place op? No, our custom kernel handles it. If we launch an in-place kernel reading and writing the same tensor, that's fine. But actually, `F.linear` might return a tensor that's needed for backward, but for forward only, in-place modification after is okay. However, for correctness and not interfering with autograd, we should be careful. The evaluation likely runs in `no_grad()` or `eval()` mode, but even if not, since `x` is produced by `F.linear`, modifying it in-place shouldn't cause issues with the preceding op, but the autograd graph might have references. To be safe, we can allocate a new output tensor, but that costs memory bandwidth for writing output. Actually, the output has to be written anyway. If we do in-place on `x`, it's one READ + one WRITE on the same memory. If we allocate separate output, it's READ from x, WRITE to output. Either way, one read and one write of the tensor data. But in-place avoids allocating a new tensor (which is a small overhead). Actually, for large tensors, the allocation overhead is negligible. Let's do out-of-place to be safe and simple. Wait, can we allocate `output` and then do the computation writing directly to it? Yes. Let me think about the kernel launch configuration. - n = 1024 * 8192 = 8,388,608 elements. - With float4, nvec = 2,097,152. - If we use 256 threads per block and 256 blocks, total threads = 65,536. Each thread would do 2,097,152 / 65,536 = 32 float4 loads. That gives good instruction-level parallelism. - Actually, we can use fewer blocks to reduce overhead, like 128 or 256 blocks on SMs. Let me check GPU specs first.

Let me analyze the problem: 1. We have a model that does: `y = gemm(x)`, then `output = sigmoid(y) * scaling_factor + y` 2. The dimensions are: batch_size=1024, input_size=8192, hidden_size=8192 3. This is essentially a fused GEMM + sigmoid + scale + residual add The operation is: `output = sigmoid(Wx + b) * s + (Wx + b)` To optimize this, I should fuse the element-wise operations (sigmoid, scale, add) with the GEMM output. The most efficient approach would be to: 1. Compute the GEMM using cuBLAS (via PyTorch matmul, or custom) 2. Fuse the remaining ops into a single kernel Or even better, use CUTLASS-style fusion but since we're writing a custom CUDA kernel with load_inline, we can: 1. Do the matmul with `torch.matmul` or `F.linear` 2. Apply a custom fused kernel for `sigmoid(x) * scale + x` Since the dimensions are large (8192 x 8192), the GEMM will dominate the runtime. But we can still save memory bandwidth by fusing the epilogue. Let me think about whether we can optimize the GEMM itself. For 8192x8192 matrix multiplied by 1024x8192: - M=1024, K=8192, N=8192 - This is a large GEMM that will be compute-bound or near bandwidth-bound Actually, let me calculate: - FLOPs = 2 * M * N * K = 2 * 1024 * 8192 * 8192 ≈ 137 GFLOPs - Memory for weights = 8192 * 8192 * 4 = 256 MB - Memory for input = 1024 * 8192 * 4 = 32 MB - Memory for output = 1024 * 8192 * 4 = 32 MB At 3.35 TB/s bandwidth, if we were just reading weights: 256MB / 3350GB/s ≈ 0.076 ms At 67 TFLOPS: 137 GFLOPs / 67 TFLOPs ≈ 2.04 ms Wait, that doesn't seem right. Let me recalculate: 137 GFLOPs / 67,000 GFLOPS/s = 0.00204 s = 2.04 ms Hmm, that still seems high. Actually, H100 has 67 TFLOPS for FP32 non-tensor, but 989 TFLOPS for TF32 tensor core. If cuBLAS uses tensor cores, the effective compute time would be much lower. Actually, M=1024, N=8192, K=8192: - In TF32 tensor core: 137 GFLOPs / 989 TFLOPS ≈ 0.138 ms But we also have to read weights (256MB) and input (32MB). At 3.35TB/s: - (256 + 32) MB / 3350 GB/s = 288 / 3350 ms ≈ 0.086 ms So the kernel is likely compute-bound if using tensor cores, or memory-bound if not. To get maximum speedup, I should: 1. Use cublas for the GEMM 2. Fuse the element-wise ops Can I do a custom CUTLASS kernel? That's very complex. Instead, let me try using torch.matmul and then a fused elementwise kernel. But wait - can I use `F.linear` and then a custom kernel? That's the simplest approach. Or, I could try to use `torch._cudnn_rnn_eq...` - no that's for RNNs. Let me think about writing a custom kernel that handles everything. For the GEMM, I could write a custom kernel using CUTLASS but `load_inline` makes this complicated. The simplest high-performance approach: 1. Use `torch.matmul` or `F.linear` for the gemm 2. Write a fused CUDA kernel for `y = sigmoid(x) * scale + x` Actually, wait - if I use torch's linear, PyTorch might already fuse some ops. But the reference doesn't fuse them. So even just fusing the elementwise ops after GEMM should help. For even more speedup, I could consider: - Using TF32 tensor cores by default (PyTorch does this) - Writing a custom kernel that loads the GEMM result and applies the fused ops Let me write a kernel where: 1. First do `x @ W.T + b` via torch.matmul or F.linear 2. Then launch a custom fused kernel: `output = sigmoid(x) * scale + x` The custom kernel would be simple - just load each element, compute sigmoid, scale, and add. Wait, can I also handle the bias addition in the fused kernel? Actually `F.linear` already handles bias. So after F.linear, I get `y`, then I apply `sigmoid(y) * s + y`. Kernel signature: ```cpp void fused_sigmoid_scale_add(const float* input, float* output, float scale, int n, int m) ``` Where n=batch_size, m=hidden_size. Let me write this. But I should consider if I can use vectorized loads/stores. For H100, float4 (16 bytes) loads would be good. Since hidden_size=8192 is divisible by 4, we can use float4. Let me think about grid/block configuration: - Each thread processes 4 elements (float4) - Total elements = 1024 * 8192 = 8,388,608 - Elements with float4 = 2,097,152 - Threads = 256 per block - Blocks = (2,097,152 + 255) / 256 = 8192 Wait, 8192 blocks might be a lot. Let me use more threads per block or process more elements per thread. Actually, for 8M elements: - Block size: 256 threads - Each thread: 8 floats (2 float4s) - Elements per block: 2048 - Blocks: 4096 Or maybe simpler: just 1024 threads per block, each handling 8 floats: - Elements per block: 8192 - Blocks: 1024 Let me think... Actually, to maximize occupancy, we want enough warps. With 1024 threads per block, that's 32 warps. On H100 with 132 SMs, we can run up to 32 blocks per SM (but limited by warps, each SM can run 64 warps for compute capability 9.0? Actually Hopper supports up to 64 warps per SM). So with 32 warps per block, we can run 2 blocks per SM. With 132 SMs, we need at least 264 blocks for full occupancy. 1024 * 8192 = 8,388,608 elements total If each thread handles 4 floats: 2,097,152 threads needed If 256 threads per block: 8192 blocks 8192 blocks is reasonable and provides good occupancy. Let me also consider writing a kernel that handles the GEMM + fusion using wmma or mma instructions... but that's very complex. Actually, wait. PyTorch's linear already uses highly optimized cuBLAS. So the main speedup from a custom kernel would be: 1. Fusing the elementwise ops (saves memory bandwidth) 2. Maybe doing something with the epilogue PyTorch's linear + separate sigmoid + mul + add reads/writes memory multiple times: - Linear writes output (32MB) - Sigmoid reads/writes (32MB) - Mul reads/writes (32MB) - Add reads/writes (32MB + 32MB for residual) Actually wait, PyTorch might fuse some of these automatically. Let me think about what actually happens: `x = self.gemm(x)` -> writes output `original_x = x` -> no copy, just reference `x = torch.sigmoid(x)` -> in-place? No, creates new tensor `x = x * self.scaling_factor` -> new tensor `x = x + original_x` -> new tensor Actually I don't think PyTorch fuses any of this because `sigmoid` might not be fused with pointwise ops. But actually PyTorch does have some fusion via the JIT/graph executor. In eager mode though, these might not fuse. So a custom kernel that uses F.linear and then fuses all elementwise ops in one kernel could save: - Read linear output once - Write final output once Instead of multiple read/writes. But if PyTorch is already using nvFuser or something, the speedup might be modest. Still, a direct custom kernel should be reliable. Let me write the kernel. Actually, one more optimization idea: can I keep the weight in a special format? Or use a persistent kernel? For a problem this size, let me first write a simple fused kernel approach and see how fast it is. ```python import torch import torch.nn as nn import torch.nn.functional as F from torch.utils.cpp_extension import load_inline cuda_source = ''' #include <torch/extension.h> #include <cuda_runtime.h> #include <math.h> __global__ void fused_sigmoid_scale_add_kernel(const float* __restrict__ input, float* __restrict__ output, float scale, int n_elements) { int idx = blockIdx.x * blockDim.x + threadIdx.x; if (idx < n_elements) { float val = input[idx]; float sig = 1.0f / (1.0f + expf(-val)); output[idx] = sig * scale + val; } } torch::Tensor fused_sigmoid_scale_add(torch::Tensor input, float scale) { auto output = torch::empty_like(input); int n_elements = input.numel(); const int threads = 256; const int blocks = (n_elements + threads - 1) / threads; fused_sigmoid_scale_add_kernel<<<blocks, threads>>>( input.data_ptr<float>(), output.data_ptr<float>(), scale, n_elements ); return output; } ''' ... class ModelNew(nn.Module): def __init__(self, input_size, hidden_size, scaling_factor): super().__init__() self.weight = nn.Parameter(torch.randn(hidden_size, input_size)) self.bias = nn.Parameter(torch.randn(hidden_size)) self.scaling_factor = scaling_factor def forward(self, x): y = F.linear(x, self.weight, self.bias) return fused_sigmoid_scale_add(y, self.scaling_factor) ``` Wait, the reference uses `nn.Linear`, which stores weight as (hidden_size, input_size). I should match this exactly. Actually, nn.Linear stores weight as (out_features, in_features), so yes, (hidden_size, input_size). But the reference generates random weights via `nn.Linear` initialization. Since `ModelNew` is instantiated with the same constructor and test framework calls `get_init_inputs()`, my module needs to be numerically equivalent. That means I need the same weight initialization or I need to load the weights. Actually, the test framework will do something like: ```python args = get_init_inputs() model = Model(*args) model_new = ModelNew(*args) # copy weights or compare outputs for same inputs ``` Wait, how are weights copied? Let me think... The test framework might just instantiate both and call forward. If `ModelNew` uses `nn.Parameter` initialized differently, outputs won't match. Looking at the problem, I think the test framework probably copies parameters from Model to ModelNew, or they might initialize Model, then instantiate ModelNew and copy state_dict. Let me assume they do something reasonable. Actually, they might just compare against the reference Model with same constructor args, but how would weights match? In these types of challenges, usually the framework does something like: 1. Creates `model = Model(*get_init_inputs())` 2. Creates `model_new = ModelNew(*get_init_inputs())` 3. Copies weights: `model_new.load_state_dict(model.state_dict())` Or they call both and compare outputs on the same inputs. Since `ModelNew` must accept same constructor args, I should use `nn.Linear` to ensure compatibility, OR I should create parameters that match `nn.Linear`'s state dict keys. Actually, let me just use `nn.Linear` inside `ModelNew.__init__` but keep it as a module. That way state_dict keys match exactly. ```python class ModelNew(nn.Module): def __init__(self, input_size, hidden_size, scaling_factor): super().__init__() self.gemm = nn.Linear(input_size, hidden_size) self.scaling_factor = scaling_factor # ... load cuda module ... ``` This ensures weight compatibility. For forward: ```python def forward(self, x): x = self.gemm(x) return fused_sigmoid_scale_add(x, self.scaling_factor) ``` This is clean and definitely correct. Now for optimization. Let me think about vectorization and whether I should use float4. Also, I might want to use `torch.cuda.get_device_capability()` or just trust that float4 works on H100. Let me write a better kernel with float4 vectorization: - Process 4 elements per thread with float4 - Use block size 256 or 512 - Grid size based on total elements / 4 For sigmoid, I need to compute element-wise. With float4: - Load float4 - Compute sigmoid for each component - Scale and add - Store float4 Actually, let me vectorize more aggressively. Can I use __ldg for read-only cache? The input is only read once, so __ldg might help. Also, I should consider whether to write a more optimized kernel that threads elements properly to avoid bank conflicts, etc. But for a simple element-wise op, a basic vectorized kernel should be near peak bandwidth. Wait, is there any chance to fuse the GEMM with the epilogue? In some frameworks you can do this with CUTLASS custom epilogues. But with `load_inline`, it's quite hard to include CUTLASS. However, we could try to use the CUTLASS header-only library if it's available in PyTorch's CUDA include path... But that's risky and complex. A safer approach is just fusing the element-wise ops. Let me think about whether PyTorch's eager mode already fuses some ops automatically. Actually, in PyTorch 2.x, `torch.compile` does fusion, but the reference model is in eager mode. The test likely runs the reference as-is in eager mode. One more thought: can we use TensorFloat-32? The default PyTorch linear might use TF32 if enabled. Let me check - by default `torch.backends.cuda.matmul.allow_tf32 = True` on Ampere/Hopper. So linear uses TF32. For the element-wise kernel, we're doing exact FP32 math. Let me write the kernel and try to compile it. Actually, I just realized: can I do better by doing the matmul manually in my kernel? For M=1024, N=8192, K=8192, a custom kernel could potentially go faster if I can overlap memory loads and compute, or if I use warp-specialization. But realistically, cuBLAS is heavily optimized and will be faster than anything I write quickly. So the strategy is: 1. Use `F.linear` for GEMM 2. Custom fused kernel for element-wise ops Let me write this now. I'll start with a simple non-vectorized version, compile, test, then optimize. Wait, actually there's one more thing. The PyTorch reference computes `sigmoid(x)` where x is the result of GEMM. The element-wise kernel will read x and write output. If we vectorize, we should get close to peak bandwidth. Peak bandwidth: 3.35 TB/s Kernel needs to read 32MB and write 32MB = 64MB total Theoretical minimum time: 64MB / 3.35TB/s = 19 microseconds cuBLAS GEMM for this size might take ~100-200 microseconds? Actually let me estimate: for M=1024, N=8192, K=8192 using tensor cores in TF32... probably around 50-150 microseconds. So the elementwise kernel at ~20-30 microseconds is a small portion. If PyTorch's eager mode does 3 separate kernels taking maybe 3 * 20 = 60 microseconds, we save ~40 microseconds. That's a modest speedup. Is there a way to get more speedup? Actually, let me reconsider the GEMM. What if I write a custom kernel that uses K-loop tiling and computes GEMM + fused epilogue? For a specific case like M=1024, N=8192, K=8192, I could write a reasonably efficient WMMA-based kernel, though it would be complex. Or maybe I can use `torch.mm` with specific settings... Another approach: since the problem size is fixed (batch_size=1024, input=8192, hidden=8192), I could potentially optimize specifically for these sizes. But the kernel should also work for the general case since the test might vary sizes. Actually, looking at the task description again, it says "Replace the forward computation of the PyTorch model below with a custom CUDA kernel." And `ModelNew` must "Accept the same constructor arguments as Model." The batch size might be fixed to 1024 though. Let me write the fused element-wise kernel first and see. Wait, I should check if there are other opportunities. For example, if I can avoid materializing the GEMM output and directly compute the fused output, that would save the 32MB write + 32MB read of GEMM output. But this requires fusing into the GEMM, which means CUTLASS or custom MMA. Let me try writing a simpler tiled GEMM kernel in CUDA for practice. For M=1024, N=8192, K=8192: If I use a block size of 128x128 or something, and shared memory tiling... That's actually quite doable. Let me think about the performance of a custom tiled GEMM vs cuBLAS. For a custom kernel without tensor cores: - 137 GFLOPs / 67 TFLOPS = 2ms... that's way slower than cuBLAS with tensor cores. With tensor cores using WMMA/mma.sync: - Hopper has specific instructions. Writing a Hopper-specific kernel with warpgroup MMA might get decent performance but it's very complex. So I think relying on cuBLAS for GEMM and optimizing the epilogue is the best approach. Actually wait - I should check if PyTorch has `torch.nn.functional.linear` and whether we can use `torch.addmm` to fuse bias. Then the post-GEMM ops are what we optimize. Let me write the code now. I'll use a vectorized float4 kernel. ```python import torch import torch.nn as nn import torch.nn.functional as F from torch.utils.cpp_extension import load_inline cuda_source = ''' #include <torch/extension.h> #include <cuda_runtime.h> #include <cstdint> __device__ __forceinline__ float sigmoid(float x) { return 1.0f / (1.0f + expf(-x)); } __global__ void fused_sigmoid_scale_add_kernel(const float* __restrict__ in_ptr, float* __restrict__ out_ptr, float scale, int64_t n) { int64_t idx = blockIdx.x * blockDim.x + threadIdx.x; // Process 4 elements per thread using float4 int64_t idx4 = idx * 4; if (idx4 + 3 < n) { float4 val = reinterpret_cast<const float4*>(in_ptr)[idx]; val.x = sigmoid(val.x) * scale + val.x; val.y = sigmoid(val.y) * scale + val.y; val.z = sigmoid(val.z) * scale + val.z; val.w = sigmoid(val.w) * scale + val.w; reinterpret_cast<float4*>(out_ptr)[idx] = val; } else { // Scalar fallback for tail for (int64_t i = idx4; i < n; i += blockDim.x * gridDim.x * 4) { if (i < n) { float v = in_ptr[i]; out_ptr[i] = sigmoid(v) * scale + v; } } } } // Scalar version for non-contiguous or other cases __global__ void fused_sigmoid_scale_add_kernel_scalar(const float* __restrict__ in_ptr, float* __restrict__ out_ptr, float scale, int64_t n) { int64_t idx = blockIdx.x * blockDim.x + threadIdx.x; if (idx < n) { float v = in_ptr[idx]; out_ptr[idx] = sigmoid(v) * scale + v; } } torch::Tensor fused_sigmoid_scale_add(torch::Tensor input, float scale) { TORCH_CHECK(input.is_cuda(), "input must be a CUDA tensor"); TORCH_CHECK(input.scalar_type() == torch::kFloat32, "input must be float32"); auto output = torch::empty_like(input); int64_t n = input.numel(); // Only use vectorized kernel if aligned and contiguous enough if (n % 4 == 0 && ((uint64_t)input.data_ptr()) % 16 == 0 && ((uint64_t)output.data_ptr()) % 16 == 0) { const int threads = 256; const int blocks = (n / 4 + threads - 1) / threads; fused_sigmoid_scale_add_kernel<<<blocks, threads>>>( input.data_ptr<float>(), output.data_ptr<float>(), scale, n ); } else { const int threads = 256; const int blocks = (n + threads - 1) / threads; fused_sigmoid_scale_add_kernel_scalar<<<blocks, threads>>>( input.data_ptr<float>(), output.data_ptr<float>(), scale, n ); } return output; } ''' ``` Wait, there's an issue with my tail handling. The scalar fallback inside the vectorized kernel doesn't make sense because I'm launching blocks based on n/4, so threads might not cover the tail properly. Let me fix: ```cpp __global__ void fused_sigmoid_scale_add_kernel(const float* __restrict__ in_ptr, float* __restrict__ out_ptr, float scale, int64_t n) { int64_t idx = blockIdx.x * blockDim.x + threadIdx.x; int64_t idx4 = idx * 4; if (idx4 + 3 < n) { float4 val = reinterpret_cast<const float4*>(in_ptr)[idx]; val.x = sigmoid(val.x) * scale + val.x; val.y = sigmoid(val.y) * scale + val.y; val.z = sigmoid(val.z) * scale + val.z; val.w = sigmoid(val.w) * scale + val.w; reinterpret_cast<float4*>(out_ptr)[idx] = val; } } ``` And launch blocks = (n / 4 + threads - 1) / threads. If n is divisible by 4 (which 1024 * 8192 = 8,388,608 is), then this is fine. 8,388,608 / 4 = 2,097,152. Blocks = (2,097,152 + 255) / 256 = 8192. That's 8192 blocks. With 132 SMs on H100, that's about 62 blocks per SM. That should be plenty for good occupancy. Wait, but actually I should think about whether I need tail handling. 1024 * 8192 is divisible by 4. But what if hidden_size isn't divisible by 4? The test might use different sizes. Let me just handle the tail explicitly. Actually, I can launch the vectorized kernel with `(n/4 + threads - 1) / threads` blocks and then conditionally handle the tail. Or I can launch `((n + 3) / 4 + threads - 1) / threads` blocks and check bounds inside. Let me do the latter for safety. Actually, let me keep it simpler. Just use a scalar kernel but with unrolled loads. Or use a grid-stride loop. Grid-stride loop is often best for element-wise ops: ```cpp __global__ void fused_sigmoid_scale_add_kernel(const float* __restrict__ in_ptr, float* __restrict__ out_ptr, float scale, int64_t n) { int64_t idx = blockIdx.x * blockDim.x + threadIdx.x; int64_t stride = blockDim.x * gridDim.x; for (int64_t i = idx; i < n; i += stride) { float v = in_ptr[i]; out_ptr[i] = sigmoid(v) * scale + v; } } ``` This is robust and lets us tune block/grid sizes independently. For peak bandwidth though, we want the vectorized version. Let me write a combined approach: process 4 elements at a time in a grid-stride loop. ```cpp __global__ void fused_sigmoid_scale_add_kernel(const float* __restrict__ in_ptr, float* __restrict__ out_ptr, float scale, int64_t n) { int64_t idx = (blockIdx.x * blockDim.x + threadIdx.x) * 4; int64_t stride = blockDim.x * gridDim.x * 4; for (int64_t i = idx; i + 3 < n; i += stride) { float4 val = reinterpret_cast<const float4*>(in_ptr)[i / 4]; val.x = sigmoid(val.x) * scale + val.x; val.y = sigmoid(val.y) * scale + val.y; val.z = sigmoid(val.z) * scale + val.z; val.w = sigmoid(val.w) * scale + val.w; reinterpret_cast<float4*>(out_ptr)[i / 4] = val; } // Scalar tail int64_t tail_start = (n / 4) * 4; for (int64_t i = tail_start + (blockIdx.x * blockDim.x + threadIdx.x); i < n; i += blockDim.x * gridDim.x) { float v = in_ptr[i]; out_ptr[i] = sigmoid(v) * scale + v; } } ``` Wait, the tail handling here is wrong because not all blocks reach the tail. Let me fix: Actually, a simpler and probably fast enough approach: just use float4 with a separate scalar tail kernel, or just process the tail in the same kernel but use proper indexing. Let me try: ```cpp __global__ void fused_sigmoid_scale_add_kernel(const float* __restrict__ in_ptr, float* __restrict__ out_ptr, float scale, int64_t n) { int64_t idx = blockIdx.x * blockDim.x + threadIdx.x; int64_t n_vec = n / 4; int64_t stride = blockDim.x * gridDim.x; // Vectorized part for (int64_t i = idx; i < n_vec; i += stride) { float4 val = reinterpret_cast<const float4*>(in_ptr)[i]; val.x = sigmoid(val.x) * scale + val.x; val.y = sigmoid(val.y) * scale + val.y; val.z = sigmoid(val.z) * scale + val.z; val.w = sigmoid(val.w) * scale + val.w; reinterpret_cast<float4*>(out_ptr)[i] = val; } // Scalar tail - only needed for last few elements int64_t tail_start = n_vec * 4; int64_t tail_idx = idx + tail_start; if (tail_idx < n) { float v = in_ptr[tail_idx]; out_ptr[tail_idx] = sigmoid(v) * scale + v; } } ``` Launch with grid size based on max(n_vec, some minimum). Actually, for simplicity and robustness, let me just use a scalar grid-stride loop. The compiler will vectorize if it can, and for this problem the element-wise portion is small compared to GEMM. Let me benchmark both. Actually, on H100, memory bandwidth is very high, and a scalar kernel might still be close enough. But let me go with float4 since it's easy. Wait, I just had another idea. Can I use `torch.compile`... no, the problem says write a custom CUDA kernel. Let me write the code and compile. One more consideration: the `load_inline` needs to be called during module loading. I need to make sure the compilation happens once. Also, I should use `cudaDeviceSynchronize()` somewhere? Actually `load_inline` generates pybind wrappers, and PyTorch CUDA ops are async. I don't need explicit sync in the kernel. Let me draft the code: ```python import torch import torch.nn as nn import torch.nn.functional as F from torch.utils.cpp_extension import load_inline CUDA_SOURCE = ''' #include <torch/extension.h> #include <cuda_runtime.h> #include <cstdint> __device__ __forceinline__ float sigmoid(float x) { return 1.0f / (1.0f + expf(-fabsf(x))); } ''' # Actually expf(-fabsf(x)) is wrong for sigmoid. Let me use standard formulation. # Standard sigmoid: 1/(1+exp(-x)) # For numerical stability: if x >= 0: z = exp(-x), return 1/(1+z) # else: z = exp(x), return z/(1+z) ``` Let me use a numerically stable sigmoid: ```cpp __device__ __forceinline__ float sigmoid(float x) { return 1.0f / (1.0f + expf(-x)); } ``` For FP32, `expf(-x)` can overflow for very negative x, but it just returns inf, and 1/(1+inf) = 0, which is correct. For very positive x, expf(-x) underflows to 0, and 1/(1+0) = 1, which is correct. So it's actually stable enough for FP32. Now the full code: ```python import torch import torch.nn as nn import torch.nn.functional as F from torch.utils.cpp_extension import load_inline CUDA_SOURCE = r''' #include <torch/extension.h> #include <cuda_runtime.h> #include <cstdint> __device__ __forceinline__ float sigmoid(float x) { return 1.0f / (1.0f + expf(-x)); } __global__ void fused_sigmoid_scale_add_kernel(float* __restrict__ out, const float* __restrict__ in, float scale, int64_t n) { int64_t idx = (int64_t)blockIdx.x * blockDim.x + threadIdx.x; int64_t n4 = n >> 2; int64_t stride = (int64_t)blockDim.x * gridDim.x; // Vectorized path for (int64_t i = idx; i < n4; i += stride) { float4 val = reinterpret_cast<const float4*>(in)[i]; val.x = sigmoid(val.x) * scale + val.x; val.y = sigmoid(val.y) * scale + val.y; val.z = sigmoid(val.z) * scale + val.z; val.w = sigmoid(val.w) * scale + val.w; reinterpret_cast<float4*>(out)[i] = val; } // Scalar tail int64_t tail_start = n4 << 2; int64_t tail_idx = idx + tail_start; if (tail_idx < n) { float v = in[tail_idx]; out[tail_idx] = sigmoid(v) * scale + v; } } torch::Tensor fused_sigmoid_scale_add(torch::Tensor input, float scale) { TORCH_CHECK(input.is_cuda(), "input must be CUDA tensor"); TORCH_CHECK(input.scalar_type() == at::kFloat, "input must be float32"); auto output = torch::empty_like(input); int64_t n = input.numel(); if (n == 0) return output; const int threads = 256; const int blocks = std::min((int)((n + threads - 1) / threads), 65535); fused_sigmoid_scale_add_kernel<<<blocks, threads>>>( output.data_ptr<float>(), input.data_ptr<float>(), scale, n ); return output; } ''' # Load inline - this needs to be called once fused_sigmoid_scale_add = load_inline( name="fused_sigmoid_scale_add", cpp_sources="", # or maybe cpp_sources="", cuda_sources=CUDA_SOURCE, functions=["fused_sigmoid_scale_add"], extra_cuda_cflags=["-O3", "--use_fast_math"], extra_cflags=["-O3"], ) class ModelNew(nn.Module): def __init__(self, input_size, hidden_size, scaling_factor): super(ModelNew, self).__init__() self.gemm = nn.Linear(input_size, hidden_size) self.scaling_factor = scaling_factor def forward(self, x): x = self.gemm(x) return fused_sigmoid_scale_add.fused_sigmoid_scale_add(x, self.scaling_factor) ``` Wait, `load_inline` returns a module-like object? Actually, looking at the torch docs, `load_inline` returns a module where the functions are exposed. Let me check. Actually, `torch.utils.cpp_extension.load_inline` returns a "module" object with the specified functions as attributes. So I should assign it to a variable and call `module.fused_sigmoid_scale_add(...)`. But wait - there's an issue. `load_inline` compiles at module load time. If I call it at module level, it should be fine. But the function object needs to be accessible from `ModelNew.forward`. Let me restructure slightly: ```python import torch import torch.nn as nn import torch.nn.functional as F from torch.utils.cpp_extension import load_inline _CUDA_SOURCE = r''' #include <torch/extension.h> #include <cuda_runtime.h> #include <cstdint> __device__ __forceinline__ float sigmoid(float x) { return 1.0f / (1.0f + expf(-x)); } __global__ void fused_sigmoid_scale_add_kernel(float* __restrict__ out, const float* __restrict__ in, float scale, int64_t n) { int64_t idx = (int64_t)blockIdx.x * blockDim.x + threadIdx.x; int64_t n4 = n >> 2; int64_t stride = (int64_t)blockDim.x * gridDim.x; for (int64_t i = idx; i < n4; i += stride) { float4 val = reinterpret_cast<const float4*>(in)[i]; val.x = sigmoid(val.x) * scale + val.x; val.y = sigmoid(val.y) * scale + val.y; val.z = sigmoid(val.z) * scale + val.z; val.w = sigmoid(val.w) * scale + val.w; reinterpret_cast<float4*>(out)[i] = val; } int64_t tail_start = n4 << 2; int64_t tail_idx = idx + tail_start; if (tail_idx < n) { float v = in[tail_idx]; out[tail_idx] = sigmoid(v) * scale + v; } } torch::Tensor fused_sigmoid_scale_add(torch::Tensor input, float scale) { TORCH_CHECK(input.device().is_cuda(), "input must be CUDA tensor"); TORCH_CHECK(input.scalar_type() == at::kFloat, "input must be float32"); auto output = torch::empty_like(input); int64_t n = input.numel(); if (n == 0) return output; const int threads = 256; const int blocks = std::min((int)((n + threads - 1) / threads), 65535); fused_sigmoid_scale_add_kernel<<<blocks, threads>>>( output.data_ptr<float>(), input.data_ptr<float>(), scale, n ); return output; } ''' _fused_module = load_inline( name="fused_sigmoid_scale_add", cpp_sources="", cuda_sources=_CUDA_SOURCE, functions=["fused_sigmoid_scale_add"], extra_cuda_cflags=["-O3", "--use_fast_math"], extra_cflags=["-O3"], ) class ModelNew(nn.Module): def __init__(self, input_size, hidden_size, scaling_factor): super(ModelNew, self).__init__() self.gemm = nn.Linear(input_size, hidden_size) self.scaling_factor = scaling_factor def forward(self, x): x = self.gemm(x) return _fused_module.fused_sigmoid_scale_add(x, self.scaling_factor) ``` I need to be careful about a few things: 1. `cpp_sources=""` - I think this is fine, or should I provide an empty string? The docs say it's required. 2. `extra_cuda_cflags` - should include `-O3` and `--use_fast_math` for speed. 3. The `functions` argument lists the functions to expose. 4. Make sure NOT to include `PYBIND11_MODULE`. Let me also consider whether to use the restrict keyword. I used `__restrict__` which is correct for CUDA. And I used `float* __restrict__ out` and `const float* __restrict__ in` in the kernel signature - that's good. But wait, `torch::empty_like(input)` creates a tensor with the same properties. But is it always contiguous? If input is contiguous (which it should be after linear), then output is too. If not, my float4 loads might fail. But `gemm(x)` returns a contiguous tensor for standard linear. Let me think about whether I should use `at::Tensor` or `torch::Tensor` - they're the same in modern PyTorch. `torch::Tensor` is fine. Actually let me reconsider the number of blocks. For n=8,388,608, with threads=256 and each thread handling 4 elements in the vectorized path, but with grid-stride loop, I launch `(n + 255) / 256 = 32768` blocks. But I cap at 65535, so that's fine. Actually, 32768 blocks with 256 threads each = 8,388,608 threads total. Each handles 4 elements via vectorized loop with stride = gridDim * blockDim. The stride would be 32768 * 256 = 8,388,608. Wait, so each thread would execute the vectorized loop once (for i = idx), and then check tail. That's perfect for n=8,388,608 because 8,388,608 / 4 = 2,097,152 exactly. So each of the 8,388,608 threads would try to access i = idx in the vectorized loop, but n4 = 2,097,152, so only the first 2,097,152 threads (blocks 0..8191) execute the vectorized path, and all others skip it. The remaining threads check tail. Hmm, that's a lot of threads (8M) but only 2M do actual work. The tail for n divisible by 4 is empty. This is somewhat inefficient because I'm launching 4x as many threads as needed for the vectorized path. Let me optimize: I should launch blocks based on n4 (the number of float4 elements), not n. ```cpp const int threads = 256; const int blocks = std::min((int)((n4 + threads - 1) / threads), 65535); if (blocks > 0) { fused_sigmoid_scale_add_kernel<<<blocks, threads>>>(...); } // Then handle tail with a separate small scalar kernel if needed. ``` But with the current grid-stride approach inside the kernel, if I launch based on n4, then: - Total threads = blocks * 256 ≈ n4 - Each thread executes the loop once for i = idx - Some threads might loop again if blocks is capped at 65535 Let me launch exactly based on n4: Actually let me redesign: launch `(n4 + threads - 1) / threads` blocks, and inside the kernel, threads only do vectorized work. Handle tail with a second tiny kernel launch. Or keep it simple: launch enough blocks for n4 but not more. ```cpp const int threads = 256; int blocks = (n4 + threads - 1) / threads; blocks = std::max(blocks, 1); // ensure at least 1 fused_sigmoid_scale_add_kernel<<<blocks, threads>>>(...); // Tail kernel int tail_count = n - (n4 * 4); if (tail_count > 0) { tail_kernel<<<1, threads>>>(...); } ``` This is cleaner. Let me rewrite: ```cpp __global__ void fused_kernel_vec(float* __restrict__ out, const float* __restrict__ in, float scale, int64_t n4) { int64_t idx = (int64_t)blockIdx.x * blockDim.x + threadIdx.x; int64_t stride = (int64_t)blockDim.x * gridDim.x; for (int64_t i = idx; i < n4; i += stride) { float4 val = reinterpret_cast<const float4*>(in)[i]; val.x = sigmoid(val.x) * scale + val.x; val.y = sigmoid(val.y) * scale + val.y; val.z = sigmoid(val.z) * scale + val.z; val.w = sigmoid(val.w) * scale + val.w; reinterpret_cast<float4*>(out)[i] = val; } } __global__ void fused_kernel_tail(float* __restrict__ out, const float* __restrict__ in, float scale, int64_t start, int64_t n) { int64_t idx = (int64_t)blockIdx.x * blockDim.x + threadIdx.x; int64_t stride = (int64_t)blockDim.x * gridDim.x; for (int64_t i = start + idx; i < n; i += stride) { float v = in[i]; out[i] = sigmoid(v) * scale + v; } } torch::Tensor fused_sigmoid_scale_add(torch::Tensor input, float scale) { ... int64_t n4 = n >> 2; const int threads = 256; int blocks = (n4 + threads - 1) / threads; if (blocks > 0) { fused_kernel_vec<<<blocks, threads>>>(out, in, scale, n4); } int64_t tail_start = n4 << 2; if (tail_start < n) { int tail_blocks = ((n - tail_start) + threads - 1) / threads; fused_kernel_tail<<<tail_blocks, threads>>>(out, in, scale, tail_start, n); } return output; } ``` This is much cleaner! For n = 8,388,608: n4 = 2,097,152. blocks = 8192. tail_start = n, so no tail kernel needed. But wait, can I fit 8192 blocks? The max is 65535 for 1D grid, so yes. Actually, I realize there might be an even better approach. Since we're doing GEMM + elementwise anyway, can I write a custom kernel that does the GEMM using `torch.mm` and then the elementwise in a single "in-place" operation on the output? I already thought about this - yes, with `F.linear` then my fused kernel. But the fused kernel allocates a new output tensor and writes to it. I could potentially reuse the GEMM output and overwrite it in-place: ```cpp __global__ void fused_kernel_inplace(float* __restrict__ ptr, float scale, int64_t n4) { ... } ``` Then in PyTorch: `x = self.gemm(x); return _fused_module.fused_sigmoid_scale_add_inplace(x, self.scaling_factor)` This saves the `empty_like` allocation and one memory write! Actually: - `self.gemm(x)` writes to its output tensor (32MB write, happens in cuBLAS) - My inplace kernel reads and writes the same tensor (32MB read + 32MB write) - Total: 64MB of memory traffic from my kernel perspective vs out-of-place: - `self.gemm(x)` writes output (32MB) - `empty_like` allocates new tensor (metadata only, no compute) - My kernel reads input and writes output (32MB read + 32MB write) - Total: 96MB memory traffic But actually, both approaches have the same total DRAM traffic because the GEMM output is in DRAM. If I do inplace, I read from DRAM and write back to DRAM. If I do out-of-place, I read from DRAM and write to another DRAM location. In-place might be slightly better because: 1. No allocation overhead 2. Better cache behavior (some cache lines might stay in L2) 3. Less memory pressure Let me do in-place! The output of `self.gemm(x)` is the input to my elementwise kernel, and I can overwrite it. ```python def forward(self, x): x = self.gemm(x) return _fused_module.fused_sigmoid_scale_add_inplace(x, self.scaling_factor) ``` This is equivalent to the reference since we don't need `original_x` anymore after using it. Wait, in the reference: ```python x = self.gemm(x) original_x = x x = torch.sigmoid(x) x = x * self.scaling_factor x = x + original_x ``` So `original_x` is used in the add. If I do inplace on `x`, I lose `original_x`. So I cannot do the entire operation in-place on a single buffer. I need both the original and the final output simultaneously during computation. Unless... I read the original value, compute the sigmoid and scaled version, add the original, and write back, all in one instruction per element. YES! I can do this in-place if I load the original value and immediately write the result. This is exactly what a single element-wise kernel does. For a single element: 1. Load `v` from memory 2. Compute `sigmoid(v) * scale + v` 3. Store result back to the same address This is perfectly valid and in-place! We don't need `original_x` in memory - it's just the value we loaded. So I can do: ```python def forward(self, x): x = self.gemm(x) return _fused_module.fused_sigmoid_scale_add_inplace(x, self.scaling_factor) ``` And the custom kernel overwrites `x` in-place. This is perfectly numerically equivalent and saves memory bandwidth. Wait - is there any reason to keep the original tensor? No, the reference doesn't return it. So the in-place version is better: - No extra allocation - Reads 32MB, writes 32MB - The GEMM output is read once and overwritten Actually, the GEMM writes 32MB to DRAM first, then my kernel reads 32MB and writes 32MB. Total GEMM + elementwise traffic = 32MB (GEMM write) + 32MB (read) + 32MB (write) = 96MB. For the naive reference: - GEMM writes 32MB - sigmoid reads 32MB, writes 32MB - mul reads 32MB, writes 32MB - add reads 64MB, writes 32MB Total: 32 + 64 + 64 + 96 = 256MB of memory traffic If PyTorch fuses some of these, it could be less. But with separate eager-mode ops, each creates a new tensor. So my approach saves a lot of memory bandwidth! But wait, what if the test framework checks for in-place modifications? Should be fine as long as output is correct. Let me write the in-place kernel: ```cpp __global__ void fused_sigmoid_scale_add_inplace_kernel(float* __restrict__ ptr, float scale, int64_t n4) { int64_t idx = (int64_t)blockIdx.x * blockDim.x + threadIdx.x; int64_t stride = (int64_t)blockDim.x * gridDim.x; for (int64_t i = idx; i < n4; i += stride) { float4 val = reinterpret_cast<float4*>(ptr)[i]; val.x = sigmoid(val.x) * scale + val.x; val.y = sigmoid(val.y) * scale + val.y; val.z = sigmoid(val.z) * scale + val.z; val.w = sigmoid(val.w) * scale + val.w; reinterpret_cast<float4*>(ptr)[i] = val; } // We need a separate tail kernel because this kernel only handles multiples of 4 // But actually, we could handle tail here if we launch based on n, not n4 } ``` For the tail, I should launch a separate kernel or handle it in the same kernel. Let me handle it properly. Actually, I can have a single kernel that uses grid-stride for the vectorized part and then a single-thread or small number of threads for the tail: ```cpp __global__ void fused_sigmoid_scale_add_inplace_kernel(float* __restrict__ ptr, float scale, int64_t n4, int64_t n) { int64_t idx = (int64_t)blockIdx.x * blockDim.x + threadIdx.x; int64_t stride = (int64_t)blockDim.x * gridDim.x; // Vectorized part for (int64_t i = idx; i < n4; i += stride) { float4 val = reinterpret_cast<float4*>(ptr)[i]; val.x = sigmoid(val.x) * scale + val.x; val.y = sigmoid(val.y) * scale + val.y; val.z = sigmoid(val.z) * scale + val.z; val.w = sigmoid(val.w) * scale + val.w; reinterpret_cast<float4*>(ptr)[i] = val; } } ``` And launch `blocks = (n4 + 255) / 256`. If there are tail elements, launch another small kernel. Since the test likely uses n divisible by 4 (1024*8192=8,388,608, divisible by 4), I could skip the tail. But to be safe, let me include it. Actually, I'll just use the single out-of-place kernel approach I had earlier but make it fuse in-place by using the same pointer for input and output. Or even simpler, just pass the same tensor twice. Wait, in PyTorch, if I modify a tensor in-place, I could potentially break gradient computation if autograd is enabled. But during inference (forward pass), the test likely checks numerical equivalence with `torch.no_grad()` or on the forward outputs directly. Actually, for nn.Module forward, autograd tracks operations. If I do an in-place CUDA kernel, autograd won't know about it. Hmm, does this matter? The test probably does `model.eval()` or just calls forward and compares outputs without backprop. Let me check the reference - it's just a forward pass. So autograd shouldn't matter for correctness testing. But to be safe, maybe I should do out-of-place with `torch.empty_like`. It's a bit slower but safer and still much faster than the reference. Actually, let me keep it out-of-place for safety. The memory allocation overhead is minimal and the bandwidth savings are already great. Wait, I just realized there could be another optimization. What about using `torch.addmm` with bias, but then fusing? In PyTorch, `F.linear(input, weight, bias)` is equivalent to `input @ weight.t() + bias`, which uses `addmm`. This is already highly optimized. Let me now think about whether I can include the bias in my fused kernel. Actually `F.linear` already handles bias addition internally. So the tensor I get is `Wx + b`, and my kernel operates on it. One more thought: what if I handle the GEMM myself with a custom kernel? For M=1024, K=8192, N=8192, I could write a custom kernel that does: 1. Load a tile of input (1024 x tile_k) 2. Load a tile of weight (8192 x tile_k) 3. Compute partial products 4. After full K reduction, add bias and apply sigmoid*scale + original But wait, the "original" here is `Wx + b`. So I need the result of the full GEMM before I can apply the epilogue. I can't easily fuse the epilogue into the GEMM reduction without first completing the K-dimension reduction. However, in a tiled kernel, after computing the full result for a tile of output, I can immediately apply the epilogue. So a custom GEMM kernel with fused epilogue would: 1. For each output tile `[m0:m1, n0:n1]`: - Accumulate `C = A[m0:m1, :] @ B[n0:n1, :].t()` over K tiles - Add bias - Apply `sigmoid(C) * scale + C` - Write to output This would save writing `C` to DRAM and reading it back! The GEMM result stays in registers/shared memory and we immediately apply the epilogue. For this specific problem size, could a custom tile GEMM kernel outperform cuBLAS? For M=1024, N=8192, K=8192, if we use a tile of 128x128: - Number of tiles = (1024/128) * (8192/128) = 8 * 64 = 512 tiles - Each tile requires reading A: 128 * 8192 = 1,048,576 floats = 4MB - Each tile requires reading B: 128 * 8192 = 1,048,576 floats = 4MB - Each tile writes C: 128 * 128 = 16,384 floats = 64KB - Total A reads = 8 * 4MB = 32MB (each row of A read once) - Total B reads = 64 * 4MB = 256MB (each col of B read 8 times for the 8 row tiles) Wait, that's not right. Let me think again. In a standard tiled GEMM: - For each tile of C (e.g., 128x128): - We stream through K in chunks - For each K chunk of size 32 (say): - Load A tile: 128x32 floats - Load B tile: 128x32 floats - Compute outer product and accumulate For the full K=8192, with K-tile=32, we have 8192/32 = 256 iterations per C-tile. The total A reads = M * K = 1024 * 8192 = 8,388,608 floats = 32MB The total B reads = N * K = 8192 * 8192 = 67,108,864 floats = 256MB Total memory reads for GEMM = 288MB Write output = 32MB A custom kernel would do exactly the same memory reads for the multiplication. But then it avoids the extra read of 32MB for the epilogue. However, cuBLAS is extremely well-optimized with tensor cores. For M=1024, N=8192, K=8192, it probably achieves very high utilization. A custom kernel without tensor cores would be much slower. With tensor cores, it gets complicated. I think for this problem, the simpler approach (cuBLAS + fused elementwise) is likely enough to beat the reference and is much more likely to be correct. Let me also consider whether I can use `torch.compile` to automatically fuse. But the problem explicitly asks for a custom CUDA kernel via `load_inline`, so that's what I need to do. Let me now write the code and compile. Actually, one more idea: can I use `torch._foreach` or other optimized APIs? No, custom CUDA kernel is required. Let me finalize my code. I'll use an out-of-place vectorized kernel for maximum safety. Wait, there's one more important consideration. Do I need to handle non-contiguous tensors? The output of `F.linear` is contiguous. And test inputs will be standard. So I don't need complex striding logic. But for robustness, I could use `input.contiguous()` or just rely on it being contiguous. Let me write the code now: ```python import torch import torch.nn as nn import torch.nn.functional as F from torch.utils.cpp_extension import load_inline _CUDA_SOURCE = r""" #include <torch/extension.h> #include <cuda_runtime.h> #include <cstdint> __device__ __forceinline__ float sigmoid(float x) { return 1.0f / (1.0f + expf(-x)); } __global__ void fused_sigmoid_scale_add_kernel(const float* __restrict__ in, float* __restrict__ out, float scale, int64_t n4) { int64_t idx = (int64_t)blockIdx.x * blockDim.x + threadIdx.x; int64_t stride = (int64_t)blockDim.x * gridDim.x; for (int64_t i = idx; i < n4; i += stride) { float4 val = reinterpret_cast<const float4*>(in)[i]; val.x = sigmoid(val.x) * scale + val.x; val.y = sigmoid(val.y) * scale + val.y; val.z = sigmoid(val.z) * scale + val.z; val.w = sigmoid(val.w) * scale + val.w; reinterpret_cast<float4*>(out)[i] = val; } } __global__ void fused_sigmoid_scale_add_tail_kernel(const float* __restrict__ in, float* __restrict__ out, float scale, int64_t start, int64_t n) { int64_t idx = (int64_t)blockIdx.x * blockDim.x + threadIdx.x; int64_t stride = (int64_t)blockDim.x * gridDim.x; for (int64_t i = start + idx; i < n; i += stride) { float v = in[i]; out[i] = sigmoid(v) * scale + v; } } torch::Tensor fused_sigmoid_scale_add(torch::Tensor input, float scale) { TORCH_CHECK(input.is_cuda(), "input must be CUDA tensor"); TORCH_CHECK(input.scalar_type() == at::kFloat, "input must be float32"); auto output = torch::empty_like(input); int64_t n = input.numel(); if (n == 0) { return output; } int64_t n4 = n >> 2; const int threads = 256; if (n4 > 0) { int blocks = (int)((n4 + threads - 1) / threads); blocks = std::min(blocks, 65535); fused_sigmoid_scale_add_kernel<<<blocks, threads>>>( input.data_ptr<float>(), output.data_ptr<float>(), scale, n4 ); } int64_t tail_start = n4 << 2; if (tail_start < n) { int tail_blocks = (int)(((n - tail_start) + threads - 1) / threads); tail_blocks = std::min(tail_blocks, 65535); fused_sigmoid_scale_add_tail_kernel<<<tail_blocks, threads>>>( input.data_ptr<float>(), output.data_ptr<float>(), scale, tail_start, n ); } return output; } """ _fused_module = load_inline( name="fused_sigmoid_scale_add", cpp_sources="", cuda_sources=_CUDA_SOURCE, functions=["fused_sigmoid_scale_add"], extra_cuda_cflags=["-O3", "--use_fast_math"], extra_cflags=["-O3"], verbose=False, ) class ModelNew(nn.Module): def __init__(self, input_size, hidden_size, scaling_factor): super(ModelNew, self).__init__() self.gemm = nn.Linear(input_size, hidden_size) self.scaling_factor = scaling_factor def forward(self, x): x = self.gemm(x) return _fused_module.fused_sigmoid_scale_add(x, self.scaling_factor) ``` Wait, I need to double check: `load_inline` with `cpp_sources=""` and `cuda_sources="..."` - is this the right signature? Let me check the PyTorch docs. Looking at my knowledge: ```python torch.utils.cpp_extension.load_inline(name, cpp_sources, cuda_sources=None, functions=None, extra_cflags=None, extra_cuda_cflags=None, extra_ldflags=None, extra_include_paths=None, build_directory=None, verbose=False, with_cuda=None, is_python_module=True, keep_intermediates=True) ``` So yes, `cpp_sources=""` is correct, `cuda_sources=_CUDA_SOURCE` is correct. Actually wait, `cpp_sources` is required and should be C++ host code, while `cuda_sources` is for `.cu` files. Since my kernel is entirely in CUDA source, I can set `cpp_sources=""`. But sometimes there needs to be at least something... Actually, if all code is in `cuda_sources`, the C++ wrapper might need some declarations. But `load_inline` auto-generates the pybind11 bindings based on `functions`. Hmm, actually I think `functions` tells `load_inline` which functions to expose from the compiled code. The pybind11 wrapper is auto-generated. So as long as the functions are defined in `cuda_sources` (which gets compiled as a `.cu` file), it should work. But wait, `torch::Tensor` types are used in the CUDA code. That's fine because torch/extension.h is included. Let me also check: do I need to use `at::Tensor` or `torch::Tensor`? Both work in the torch namespace. One thing I should be careful about: `input.is_cuda()` checks if it's on CUDA. But the input might be on a specific device. The test likely uses the default CUDA device. Also, I use `at::kFloat` to check dtype. `input.scalar_type() == at::kFloat` checks for float32. Actually, I should also consider whether there could be issues with `load_inline` name collisions. If I run this multiple times in the same process, `name="fused_sigmoid_scale_add"` might conflict. But in the test environment, it should be loaded once. Let me compile this and check correctness! Wait, I also want to make sure about `extra_cuda_cflags`. For H100 (sm_90), PyTorch should automatically add the right architecture flags. I shouldn't need to specify `-arch=sm_90`. Actually, let me try compiling this first. I'll write the code and pass it to `compile_kernel`. Wait, the `compile_kernel` tool takes the full Python source of the ModelNew kernel file. So I need to provide the entire Python module that defines `ModelNew`. Let me create the full file content and try to compile. Actually, looking at this more carefully, I wonder if we can do even better by writing a custom kernel that handles the matmul via `torch.addmm` with beta=0 and alpha=1, then our epilogue... no that's the same. What if I try to fuse bias addition too? But `F.linear` already fuses it. Is there a way to use CUTLASS through PyTorch? PyTorch has some CUTLASS integration but it's not directly accessible via `load_inline` without the headers. Actually, PyTorch's CUDA extensions can include headers from PyTorch's include path. PyTorch includes CUTLASS headers if installed. But that's complex and risky. Let me also consider writing the entire forward pass in a single custom kernel using `cublasSgemm` from within the CUDA extension. This would let me call cuBLAS and then immediately launch my kernel. Actually, I can do this in the host wrapper: ```cpp torch::Tensor custom_forward(torch::Tensor input, torch::Tensor weight, torch::Tensor bias, float scale) { auto gemm_output = torch::addmm(bias, input, weight.t()); return fused_sigmoid_scale_add(gemm_output, scale); } ``` And expose this as a single function. But there's no real advantage over calling `F.linear` and then my kernel in Python. Wait, there might be an advantage: if I do it all in one C++ function, I could potentially launch the GEMM and the elementwise kernel with different streams or... no, they need to be sequential anyway. What about using `torch.mm_out` to write directly into a pre-allocated buffer, then launch my kernel? In Python, `torch.mm` allocates output. But `torch.addmm(bias, input, weight.t(), out=output)` could write to a pre-allocated tensor. This avoids one allocation for the GEMM output! Then I can re-use or overwrite that tensor. Let me think: if I pre-allocate the output and use `torch.addmm`, then launch my kernel in-place on that output, I avoid: 1. The allocation overhead of `self.gemm(x)` output 2. The allocation overhead of my `empty_like` Actually, `self.gemm(x)` might already be optimized. But if I do: ```python output = torch.empty(batch_size, hidden_size, device=x.device, dtype=x.dtype) torch.addmm(self.gemm.bias, x, self.gemm.weight.t(), out=output) return _fused_module.fused_sigmoid_scale_add_inplace(output, self.scaling_factor) ``` Wait, I can do `addmm` with `beta=1` to add bias... actually `F.linear` computes `input @ weight.T + bias`. `torch.addmm(bias, input, weight.t())` does exactly this when bias is 1D and broadcasting works. Hmm, but PyTorch's `nn.Linear` might add some optimizations. The simplest approach is to just use `self.gemm(x)` and my out-of-place kernel. Allocation overhead for 32MB is very small. Actually, I just realized - I can do this all in-place by using the same tensor if I read before write: ```cpp __global__ void fused_sigmoid_scale_add_inplace(float* ptr, float scale, int64_t n4) { // read-modify-write } ``` And in Python: ```python x = self.gemm(x) _fused_module.fused_sigmoid_scale_add_inplace(x, self.scaling_factor) return x ``` This is safe for inference because each element is loaded, computed, and stored independently. No other thread accesses the same element. For autograd, in-place ops can cause issues with gradient computation, but for the forward-only correctness test, it should be fine. And this saves one tensor allocation (the output of empty_like) and one memory write! But to be maximally safe with autograd, I could wrap the call in `with torch.no_grad()`... no, that would break gradients for the whole model if users need them. Actually, PyTorch's autograd doesn't track custom CUDA kernels by default. So if I do an in-place custom kernel, autograd won't track it. This means backpropagation through ModelNew would not work correctly. But the test is only for forward correctness. However, the problem says "Replace the forward computation". It might not care about backward. To be safe, let me just do out-of-place. But wait, if the test framework does `model(x)` and compares, and doesn't do backward, then in-place is fine. Let me do in-place to save memory bandwidth. Actually, the time difference between in-place and out-of-place is probably very small for this kernel since we're already memory-bound. But the allocation overhead might matter. Actually, let me reconsider. `torch.empty_like` allocates from the CUDA memory pool, which is very fast. The actual DRAM write happens in the kernel. So for bandwidth: - Out-of-place: reads 32MB, writes 32MB - In-place: reads 32MB, writes 32MB Wait, for in-place, the kernel reads 32MB and writes 32MB to the same locations. For out-of-place, it reads 32MB from one location and writes 32MB to another. From a DRAM bandwidth perspective, in-place might actually be slightly worse due to write-after-read dependencies, but for element-wise ops with no reuse, it might be similar. Actually, for modern GPUs, writing to a separate output buffer is often just as fast. The main savings of in-place is avoiding the allocation call and the memory footprint. I'll stick with out-of-place for maximum safety. The speedup over the reference should still be substantial because the reference does multiple separate kernels. One more thing: what about using `torch.cuda.nvtx` markers? No, unnecessary. Let me now write the final code and compile it. I want to make sure the `load_inline` call works correctly. Actually, I notice in my previous draft I had `input.is_cuda()` which is correct. But I should use `input.device().is_cuda()` to be extra safe, though both work. Let me also check if the alignment check is needed. `float4` loads require 16-byte alignment. For `1024 x 8192` float32 tensors, the first element is typically 512-byte aligned (due to PyTorch's allocator). So 16-byte alignment is guaranteed. But I should check to avoid crashes if something weird happens. Actually, let me just add alignment check: ```cpp bool aligned = (reinterpret_cast<uintptr_t>(input.data_ptr<float>()) % 16 == 0) && (reinterpret_cast<uintptr_t>(output.data_ptr<float>()) % 16 == 0); if (aligned && n4 > 0) { // vectorized } else { // scalar } ``` This is robust. Alright, let me assemble the full Python file content and try to compile it. Wait, I realize I should also think about using `__ldg` for loads. Since we're doing read-only loads from `in`, `__ldg` can use the read-only data cache: ```cpp float4 val = __ldg(reinterpret_cast<const float4*>(in) + i); ``` This could improve performance on some architectures. On H100, global loads are already cached well, but `__ldg` bypasses L1 and uses the read-only cache (texture cache). It might help for streaming data where no reuse is expected. Let me include it. Actually, `__ldg` only works with scalar types (float, int, etc.), not with `float4`. We'd need to use `__ldg` for each component or use inline PTX. For float4, regular loads are fine - they compile to vectorized memory instructions `LDG.E.128`. So let me skip `__ldg` for float4. One more optimization: use `#pragma unroll` if I had loops inside, but my inner loop is the stride loop and the trip count varies. Actually, for typical grid-stride loops where blocks > 1 and stride is large, each thread only does 1 or very few iterations. So unrolling doesn't help. For n4 = 2,097,152 with blocks=8192, threads=256: - Total threads launched = 2,097,152 - Each thread handles exactly 1 float4 - No loop iterations needed! Wait, no. blocks = (n4 + 255) / 256 = 8192. With 256 threads per block, total threads = 2,097,152 = n4. So each thread handles exactly 1 float4! No loop needed. For tail: tail_count = 0, so no tail kernel launched. This is optimal! Let me make sure the blocks calculation is correct. ```cpp int blocks = (int)((n4 + threads - 1) / threads); ``` For n4 = 2,097,152 and threads = 256: blocks = 8192. Yes, each thread handles exactly 1 float4. Perfect. For a different size where n4 < 256, we'd have 1 block with some threads idle. If n4 > 65535 * 256 = 16,777,216, we'd cap at 65535 blocks and the grid-stride loop would handle multiple elements. 65535 * 256 = 16,777,216. For n4 up to 16M, each thread handles 1 element. Beyond that, multiple iterations. Our n4 is 2M, so fine. Great! Let me finalize and compile. Wait, I also want to make sure about `extra_cuda_cflags=["-O3", "--use_fast_math"]`. `--use_fast_math` enables faster but less precise `expf`. For sigmoid, `expf` with `--use_fast_math` should be acceptable within fp32 tolerance (`atol=rtol=1e-4`). Actually, let me think... `__expf` (fast math) is accurate to about 2-3 ULPs. For sigmoid, the error might accumulate. But fp32 tolerance is 1e-4, which is quite loose compared to the ~1e-7 precision of fp32. So `--use_fast_math` should be fine. Actually, let me check: `sigmoid(2.0) = 0.880797...` With fast math, it might be off by a small amount, well under 1e-4. Let me compile without `--use_fast_math` first, then add it later if needed. Actually, for maximum correctness, I'll skip `--use_fast_math` to start. We can add it later. Wait, one more thing. I should make sure the kernel handles the case where input and output might overlap. For safety, with out-of-place, they won't overlap. Here's my final code. Let me compile:

The issue is that I need to provide a C++ wrapper function, not just the CUDA kernel. The `load_inline` creates a C++ binding but the function must be declared/defined in the C++ source. Let me fix this - I need to provide a C++ header in `cpp_sources` and the CUDA implementation in `cuda_sources`. Actually, looking at the error more carefully - the function `fused_sigmoid_scale_add` is not visible from the generated `main.cpp`. When using `load_inline`, I should put the actual function declaration in `cpp_sources` or the CUDA code without torch extensions. Wait, looking at load_inline docs: the `cuda_sources` is compiled as .cu file, and functions must be declared/defined in a way that the auto-generated bindings can see them. The binding file is a cpp file that needs the function declaration. Let me restructure: put the declaration in `cpp_sources`, and the kernel + function definition in `cuda_sources`. Or, put everything in `cpp_sources` and `cuda_sources` correctly. Actually, I think the right approach with `load_inline` is: - `cpp_sources` should contain the function declaration (C++ header) - `cuda_sources` should contain the CUDA kernel and the host function definition Or, I can declare the function in `cpp_sources` and define it in `cuda_sources`. Wait, actually, the simplest approach is to put all CUDA code (including the wrapper) in `cuda_sources` but also provide the function declaration in `cpp_sources`. Let me try that.

The kernel compiles and is correct. But I want to see if I can optimize it more. Let me think about the GEMM issue: the gemm is a large matmul (1024 x 8192) @ (8192 x 8192). The reference does the matmul, then 3 elementwise ops. My current approach does the matmul in PyTorch (cuBLAS) and then the elementwise ops fused in a custom kernel. The real optimization here might be to fuse the elementwise ops into the GEMM using CUTLASS or by writing a more efficient kernel. However, writing a custom GEMM is extremely complex for Hopper. But wait - the elementwise ops are simple. The main win is avoiding multiple memory roundtrips. My kernels achieves that. But for a 1024x8192 output (8M floats = 32MB), the elementwise kernel itself is memory bandwidth bound. Let me think if I can do better: 1. Could I fuse directly into the Linear layer output? Not without modifying PyTorch internals. 2. Could I use cublasLt epilogue fusion? That's possible through torch.matmul with a custom epilogue, but PyTorch's Linear doesn't expose it easily. For a purely elementwise kernel, I want to maximize memory throughput. On H100, we have 3.35 TB/s. For 32MB read + 32MB write = 64MB total, at full bandwidth that would take ~19 microseconds. But the kernel launch overhead and PCI overhead can add significant time. Let me optimize my current kernel: - Use more blocks (grid-stride with enough blocks to saturate the GPU) - Perhaps use vectorized loads - Ensure alignment I can also try compute-sanitizer or profiling. Actually, let me improve the kernel by: 1. Using a much larger number of blocks (like 132 SMs * some occupancy) 2. Using grid-stride loops more aggressively 3. Maybe using `__launch_bounds__` to help the compiler But there's another optimization: since this is very simple, maybe I should just use `torch.compile` or a Triton kernel. But the instructions say to use `torch.utils.cpp_extension.load_inline`. Wait, I can fuse into cublasLt via torch's `_cublas` or similar. Actually, in PyTorch 2.x, there might be a way to use `torch.mm` with a bias and epilogue. But for a general `nn.Linear`, the bias is applied. Our model doesn't use a bias by default (nn.Linear defaults to bias=True). Let me check... Actually, `nn.Linear(input_size, hidden_size)` defaults to `bias=True`. So there's a bias term. In the reference: `x = self.gemm(x)` includes the bias addition. Then `original_x = x` saves that value, then sigmoid, scale, and add back original. So x = gemm(x) + bias. Then original_x = x. This means we need to save the bias-add result. We can't easily fuse the epilogue into cuBLAS without control over the epilogue. Let me think about another approach: could we use torch's `addmm` or call `F.linear` then fuse? Actually, wait. The gemm itself is (1024, 8192) @ (8192, 8192). On H100, this should be very fast in cuBLAS. The main overhead is the 3 extra memory operations. My kernel fuses them. How fast is the reference? Probably dominated by the gemm. Let me see if I can measure potential speedup. If gemm takes X time and the elementwise ops take Y time, then the theoretical max speedup is (X+Y)/X. For a 1024x8192x8192 matmul, using cuBLAS on H100, it should take maybe 100-200 microseconds. The elementwise ops on 8M elements should take maybe 30-50 microseconds. So speedup might be small. To get more speedup, maybe we should try to use torch.compile for the whole module. But torch.compile is allowed... actually, I could use torch.compile on top. But the focus is on the CUDA kernel. Let me think about another optimization: using Tensor Cores or doing the epilogue with a custom kernel that processes larger tiles with shared memory. But for elementwise operations, shared memory doesn't help much except for caching, and we're streaming. Actually, let me check if I can improve the kernel even further. The elementwise kernel is very simple. Let me try using `__restrict__` and ensuring contiguous memory. Also, H100 has 132 SMs. I should launch many blocks. Currently I use `min(blocks, 65535)`, which is a lot. But with grid-stride loops, each thread processes many elements. Maybe I should use fewer threads per block (say 128) to get more blocks and better occupancy. Let me make a more optimized version: - 128 threads per block - Unroll by processing more elements per thread - Still use float4 loads Actually, let me try a different approach. For the full forward pass, maybe torch.compile could fuse the entire thing and be very fast. But we need to use a CUDA kernel with load_inline. Hmm, wait... what if I use cublas for the matmul inside the custom kernel? Then I can fuse the epilogue. But writing a custom cublas call + epilogue in CUDA would be complex but possible. I can call cublasSgemm or cublasGemmEx inside the kernel... no, that's not how it works. cublas is a library call from host, not device. Actually, I could have the custom module call cublas from CPU side, then my own kernel. But that's already what PyTorch does. Let me think about CUTLASS. PyTorch 2.x has some CUTLASS integration, but it's hard to use directly from load_inline. What if I just use ternary operators to fuse the sigmoid more carefully? Or use the fast approximate math functions? Actually, for sigmoid, expf is already fast. We could use `__expf` (intrinsic, less accurate) but we need fp32 accuracy within 1e-4, so we should stick with `expf`. Let me try another optimization: use a larger block count, use 128 threads, and more elements per thread. But also ensure good alignment. Actually, let me try something else. What if my kernel operates on a non-contiguous tensor? The output of nn.Linear is contiguous, so it's fine. I think the kernel is as good as it gets for the elementwise part. Let me check if maybe we can do the matmul more efficiently... Actually, batch_size is 1024, input_size is 8192, hidden_size is 8192. 1024 * 8192 * 8192 * 2 = ~137 billion operations. At 1979 TFLOPS (bf16 tensor core), that's ~69 microseconds theoretical. At 989 TFLOPS (tf32), it's ~138 microseconds. At 67 TFLOPS (fp32 non-tensor), it's ~2ms. But PyTorch's nn.Linear typically uses TF32 by default on Ampere/Hopper. So the matmul should take < 200 μs. The 3 elementwise ops on 8.4M elements (1024*8192): - Each reads/writes once: 3 reads + 3 writes = 6 * 32MB = 192MB - At 3.35 TB/s = 57 μs Fused: 1 read + 1 write = 2 * 32MB = 64MB = 19 μs Savings: ~38 μs. Total reference: ~200 + 57 = 257 μs. My code: ~200 + 19 = 219 μs. Speedup: 1.17x To get more speedup, maybe I should also fuse the bias into the elementwise kernel... wait, bias is already inside the Linear (nn.Linear defaults to bias=True), so the gemm output includes bias. That's fine. But what if there's no weight grad computation issue? The model is in eval mode for inference, so no gradients. Another thought: maybe I can use torch.compile on my ModelNew's forward to let PyTorch optimize the matmul + elementwise. But that uses Triton, not our custom CUDA kernel. The task says "Replace the forward computation with a custom CUDA kernel." OK, let me try to see if using torch.compile or some other optimization could help. Actually, I could write a custom kernel that uses CUTLASS-like approach but that's too complex. Wait, could I do the matmul in a custom way using Triton? No, I need to use load_inline. Hmm, let me re-examine the reference more carefully: ```python x = self.gemm(x) # Linear: y = xA^T + b original_x = x # Save reference x = torch.sigmoid(x) # sigmoid x = x * self.scaling_factor # scale x = x + original_x # residual add ``` Could we do: `x = x * self.scaling_factor + original_x` after the sigmoid? No, the scaling is on the sigmoid output. Wait actually: `sigmoid(x) * scale + x` is exactly what my kernel does. What about using the GeLU approximation trick? No, sigmoid is exact. Could I use `torch.sigmoid(x) * scale + x = x + scale * sigmoid(x)` and maybe rewrite sigmoid? Not really. Actually, I have an idea. Can I use cublasLt epilogue fusion? PyTorch 2.1+ has `torch._cublas` or `_cublaslt` that might allow this. But it's internal and undocumented. Let me think... Another idea: use half-precision or bf16 for the elementwise kernel. But nn.Linear output is fp32 by default. We need to match fp32 output. What if the weights are in fp32? The nn.Linear stores weights in fp32 and does fp32 matmul. So everything is fp32. Actually, wait. cuBLAS with TF32 is a settings thing. PyTorch enables TF32 on matmul by default. So the Linear uses TF32, which is fast but still outputs in fp32 containers. Then the elementwise ops are on fp32. My kernel is fine. But can I get more speedup? Maybe by launching more blocks and using 128 threads to increase occupancy. Or by using cooperative groups. Actually, I think the real question is: can I make this much faster by also optimizing the Linear layer? What if I use `torch.nn.functional.linear` with explicit `matmul` and then process? No, same thing. What about using an even more optimized elementwise kernel? For H100 (132 SMs), I should have enough blocks. With 256 threads and 65535 blocks, that's plenty. But maybe I should not use grid-stride loops and instead just assign one element per thread (or one float4 per thread). This avoids the loop overhead and might be better for simple elementwise ops. Actually, for streaming kernels, having each thread process exactly one item (or one vector) and having exactly enough blocks can be better because the GPU can distribute work without loop overhead. Let me try a simpler kernel: each thread processes one float4, with just enough blocks to cover the work, no grid-stride loop. Actually, modern GPUs benefit from having many blocks for latency hiding. But for memory bandwidth, occupancy is key. With 256 threads and 132 SMs, if each SM can run 2048 threads, we need at least 1024 blocks. 65535 is plenty. But the grid-stride loop might add some overhead. Let me try both approaches. Also, can I use `__launch_bounds__(256)` to help the compiler? And one more thing: I should use `cudaStream_t stream = at::cuda::getCurrentCUDAStream();` and launch in that stream. Let me refine the kernel and also check if compiling with `--use_fast_math` helps. Another thought: in PyTorch, the bias addition in nn.Linear might be fused with the matmul using cublasLt. Then the sequence is: 1. Fused matmul+bias (one operation) 2. Sigmoid 3. Scale 4. Add My kernel fuses 2-4. But can we fuse 1-4? Only if we can pass the epilogue to cublasLt. PyTorch does use cublasLt for Linear, and it supports bias via epilogue. But PyTorch doesn't expose a way to add a custom epilogue after bias. Wait, let me check reference more carefully. If PyTorch is already fusing matmul+bias, then the memory for step 1 is never written to global memory! It goes straight to the output tensor. Then step 2 reads from that. So actually, the reference does: 1. matmul+bias (fused, writes output Y) 2. read Y, compute sigmoid, write Z 3. read Z, multiply by scale, write W 4. read W and Y, add, write output Wait, but `original_x = x` means x is stored. If x is the result of gemm, and gemm is fused with bias, x is still a tensor. Then sigmoid reads x and writes to a new tensor (in-place or out-of-place). Actually, PyTorch might do in-place operations where possible. Let me trace: - `x = self.gemm(x)` -> x is new tensor (output of linear) - `original_x = x` -> reference to same tensor - `x = torch.sigmoid(x)` -> x is now a new tensor - `x = x * self.scaling_factor` -> x is now a new tensor - `x = x + original_x` -> x is now a new tensor So the reference does 4 kernel launches: 1. Linear (matmul + bias) 2. Sigmoid 3. Mul 4. Add PyTorch might fuse some of these if it sees the pattern, but typically without torch.compile, it won't. So reference truly does 4 separate passes (one compute-heavy, three memory-bound). My code does: 1. Linear 2. My fused kernel So my speedup is from fusing 3 memory-bound passes into 1. The theoretical speedup would be: if Linear takes time T_m and each memory pass takes time T_e, reference takes T_m + 3*T_e, mine takes T_m + T_e. Speedup = (T_m + 3*T_e) / (T_m + T_e). Given T_m ~ 150 μs and T_e ~ 30 μs, speedup = (150 + 90) / (150 + 30) = 240/180 = 1.33x. That's decent but not huge. Can I get more? Only if I can also optimize the Linear layer itself. But using PyTorch's Linear already uses the best available cuBLAS/cuBLASLt. Actually, wait - could I use the fact that `nn.Linear` stores weights as (hidden_size, input_size) and maybe the weight matrix isn't the optimal layout? No, PyTorch handles that. Hmm, let me think if there's another approach. What if I write the matmul in the CUDA kernel too? Since this is a specific problem with known sizes, I could write a simple block-tile matmul. But writing a high-performance matmul for 8192x8192 is very hard and won't beat cuBLAS. Actually, wait. There's a trick: if I compute the matmul and immediately apply the epilogue without writing intermediate results. But this requires either modifying how PyTorch calls cuBLAS or writing a custom kernel that calls cuBLAS then my kernel back-to-back with zero overhead. The latter is what I'm already doing. The overhead of a kernel launch is ~5-10 microseconds, negligible compared to 150+ μs. So 1.33x is probably what I'll get. But let me optimize the elementwise kernel to ensure it actually achieves full memory bandwidth. One optimization: use `__ldg` for read-only loads. The input to my kernel is only read. `__ldg` tells the compiler to use the load-through-global cache. Another: for the tail handling, rather than a separate kernel, do it in the same kernel by having a branch. Wait, for a tensor with 1024*8192 = 8388608 elements, this is perfectly divisible by 4 (8388608/4 = 2097152). So there's no tail! Let me verify: 1024 * 8192 = 8388608. 8388608 / 4 = 2097152 exactly. So no tail needed. This means I can simplify the kernel significantly. Let me write a cleaner, more optimized kernel: 1. No tail needed 2. Use __ldg (or const __restrict__) 3. Use exactly enough blocks with grid-stride, or use a single block per element approach 4. Maybe use 128 threads for better occupancy Occupancy calculation: H100 has 64K registers per SM, max 255 per thread. With 256 threads, if the kernel uses 32 registers, that's 8192 registers per block. 64K/8K = 8 blocks per SM. Max 32 blocks per SM, so register-limited to 8. Threads per SM = 256 * 8 = 2048, which is the max threads per SM. So 100% occupancy is possible with 256 threads if registers <= 32. But if the compiler uses more registers, occupancy drops. Using 128 threads might allow more blocks and hide latency better. Let me try 128 threads and see if it's faster. Actually, for a simple memory-bound kernel, the main thing is just saturating memory bandwidth. Occupancy beyond some threshold doesn't matter for bandwidth. What matters is having enough concurrent memory requests. On H100, each SM has L1 cache that can handle many concurrent threads. With 132 SMs and a simple kernel, even 132 blocks might be enough to saturate bandwidth if each block has enough threads and each thread makes enough requests. But with 128 threads and float4, each thread reads 16 bytes. To saturate 3.35 TB/s with global memory latency of ~300ns, we need ~1MB in flight. If each thread can have ~10 requests in flight, that's 1280 bytes per thread. With 128 threads, that's ~164KB per block. We'd need ~6-7 blocks per SM to saturate, which is fine with 8+ blocks per SM. OK, let me write a clean kernel and test it. Then let me also think about any other optimizations. Actually, another thought: PyTorch's `torch.sigmoid` might have special fast paths. But since I'm just doing elementwise sigmoid, it's probably similar. Let me also ensure the tensor is contiguous and aligned (which it should be for Linear output). Here's my refined kernel plan: - Check if numel % 4 == 0 (which it is for our test sizes) - Use float4 loads - Use 128 or 256 threads per block - Launch with enough blocks to cover all SMs - Grid-stride loop for simplicity Let me try both 128 and 256 threads. Actually, let me just try 256 threads first since it compiled and passed. Wait, I should also make sure I'm using the right CUDA stream. Without specifying the stream, kernel launches use the default stream. In PyTorch, operations should be on the current stream. I should probably use `at::cuda::getCurrentCUDAStream()` explicitly. Let me write a cleaner version and test performance. Actually, one more optimization: using `fast_math` flag with `--use_fast_math`. This makes expf faster but less accurate. Need to check if accuracy is still within 1e-4. For sigmoid(x) = 1/(1+exp(-x)): - With fast math, expf might have slightly larger error. - For typical values of our Linear output (which are random-ish), the error should be small. - But let's check. Actually, let me first try without fast math and see if the speedup is enough. If not, I can try fast math. Let me also think about another approach: what if we don't do `x = self.gemm(x)` in PyTorch but instead write a custom kernel that does the entire thing? For the matmul, we cuold use CUTLASS or a tiled custom implementation. But for 8192x8192, custom won't beat cuBLAS. Hmm, but what about chunking? What if we do the linear and then the elementwise in chunks to keep data in cache? That wouldn't help for a single forward pass on 32MB - it fits in L2 cache (50MB on H100). Actually, 32MB might fit in L2! If so, the elementwise operations in the reference might already be L2-bound rather than HBM-bound. Wait, H100 L2 cache is 50MB per SM partition or total? Actually H100 has 60MB of L2 cache total. 32MB input + 32MB output = 64MB for all intermediate tensors. But they're not all needed simultaneously. In the reference: 1. Linear writes 32MB output Y. This goes to HBM. 2. Sigmoid reads Y (32MB) from HBM, computes Z, writes Z (32MB) to HBM. 3. Mul reads Z, writes W. 4. Add reads W and Y, writes output. If the subsequent operations happen immediately, some of Y might still be in L2 when Sigmoid reads it. But PyTorch's eager mode has kernel launch overhead between operations, so L2 might not retain everything. With torch.compile, these would be fused. Without it, they aren't. My fused kernel does: read Y (32MB), write output (32MB). So it's 64MB of memory traffic. For reference steps 2-4: read Y, write Z, read Z, write W, read W and Y, write output. That's Y read twice, Z read once, W read once, plus Z, W, output writes = 5 reads + 3 writes = 8 * 32MB = 256MB of traffic. Wait, let me recount: - Sigmoid: read Y, write Z → 64MB - Mul: read Z, write W → 64MB - Add: read W, read Y, write out → 96MB Total: 224MB My fused: read Y, write out → 64MB That's a 3.5x reduction in memory traffic for the elementwise part! If elementwise takes 3*30 = 90 μs in reference, and my fused takes 25 μs, the speedup for the elementwise part is 3.6x. Overall speedup: (150 + 90) / (150 + 25) = 240/175 = 1.37x. Alright, that's reasonable. Actually, I realize I should also make sure the Linear weight matrix is in the optimal format. PyTorch typically transposes for cublas. But nn.Linear already stores weights as (out_features, in_features), which is the right layout for cublas (A^T). OK, let me refine my kernel with proper stream handling and test it. But first, let me submit what I have and see. Actually no, let me optimize more. One more thing: could I use C++17 or C++14 features? That's fine. Actually, let me reconsider the kernel. The elements after linear are going to be somewhat randomly distributed. But `sigmoid` for large values saturates to 0 or 1. That's fine. Could I use vectorized math? expf is scalar. Wait, I should also check if PyTorch is already doing some fusion. In PyTorch eager mode, without torch.compile, the operations aren't fused. But there might be some minor optimizations. Still, our kernel should be faster. Let me write the final optimized code. Actually, I wonder if I can make this even faster by doing the matmul and the elementwise in one go by tiling - but that requires writing a custom matmul or using CUTLASS. Wait, PyTorch 2.x integrates with Triton, and Triton can write fused matmul + epilogue. But the assignment specifically asks for `load_inline` with CUDA C/C++. Another idea: can I use `torch.addmm` with `alpha` and `beta` somehow? No, the epilogue is sigmoid. What about using `torch.nn.functional.linear` then immediately calling my kernel with `out` parameter to avoid allocating a new output tensor? In my current code, I'm allocating a new output tensor. I already do `torch::empty_like(input)` which allocates the output. Could I do the operation in-place? Like `x = sigmoid(x) * scale + x` writing back to x. But the reference returns a new tensor for each op, so as long as the values are correct, in-place is fine for inference. But for training (gradients), in-place might cause issues. However, the assignment is only about `forward()`. So in-place could be fine, but it doesn't save much - it's the reads that matter more than the writes. Actually, for memory bandwidth, going from input->output (2x traffic) vs input->same (1x traffic) is a big difference. If I do in-place, it's 1x read + 1x write = same as new output. The total HBM traffic is the same. Hmm, unless the cache is hot. But we already accounted for that. Let me make sure the kernel uses ILP (instruction-level parallelism) by processing multiple elements per thread in the grid-stride loop. Actually, with grid-stride loops, the compiler unrolls some iterations. But I can manually unroll. For example, each thread could process 4 float4s per iteration: ```cpp for (int64_t i = idx; i < n4; i += stride) { float4 v0 = load(i); float4 v1 = load(i + stride/2); // compute... store(i, v0); store(i + stride/2, v1); } ``` This gives more independent loads to overlap memory latency. On H100 with high bandwidth, this might help saturate the memory subsystem. Let me write an unrolled version. Actually, let me think about grid size. With 132 SMs and wanting 8 blocks per SM, I'd need ~1056 blocks. With n4 = 2,097,152 elements and 256 threads, that's 8192 blocks at 1 element per thread, which is plenty. With a grid-stride loop where each thread handles 1 element but there are many blocks per SM, we get good parallelism. The issue with grid-stride is that the stride is the entire grid size. If we have 8192 blocks * 256 threads = 2,097,152 threads, that's exactly n4. So each thread does 1 float4. No grid-stride loop needed! Well, unless we use fewer blocks. Wait, I was using `blocks = min((n4 + threads - 1) / threads, 65535)`. For n4 = 2,097,152 and threads = 256, blocks = 8192. That's exactly n4. So each thread handles exactly 1 float4. The grid-stride loop `i += stride` won't execute more than once because `idx` ranges up to n4 - 1. So actually, there IS no loop iteration! The grid is sized exactly for the problem. That's good. But could I use fewer blocks and a grid-stride loop? Like blocks = 132 * 4 = 528, then each thread loops ~4 times. This might have slightly better characteristics because the GPU can finish blocks and free resources. But for memory bandwidth, it shouldn't matter much. Actually, with exactly 1 element per thread and 8192 blocks (62 blocks per SM on average), each SM gets many blocks. Since max blocks per SM is 32, and 62 blocks spread over 132 SMs is less than 1 per SM on average... wait, 8192 / 132 = 62 blocks per SM. But max is 32 blocks per SM. So actually, the GPU would only run 32 * 132 = 4224 blocks concurrently, and the remaining blocks would queue up. That's fine. So with 8192 blocks, we're at the max concurrent blocks (assuming each can fit). With 256 threads that's 8192 threads per SM concurrent max, but max threads is 2048. So actually, each SM can only run 8 blocks of 256 threads. 132 * 8 = 1056 blocks concurrently. The remaining ~7100 blocks are queued. So if we reduce blocks to ~1024, we can still saturate all SMs, and each thread processes ~2048 float4s (8192 elements). With a loop of 2048 iterations, that might have some loop overhead but it also helps amortize kernel launch and scheduling overhead. For memory bandwidth bound kernels, larger grids with less work per thread can sometimes be better because the GPU scheduler can hide latency better with more blocks. But 1056 concurrent blocks of 256 threads each is already 270K threads, which should be plenty for H100. Actually, there's another consideration: PyTorch tensors might not be 16-byte aligned unless the first and last dimensions are multiples of 4. For linear output of shape (1024, 8192), with 8192 columns, the row stride is 8192 floats = 32768 bytes, which is divisible by 16. And there are 1024 rows. So the whole tensor is perfectly aligned. Float4 loads are safe. OK, I think my kernel is about as good as it gets for the elementwise part. Let me just make sure it compiles cleanly and handles edge cases. For edge cases (non-multiple of 4), I added a tail kernel. But for our actual test case, we don't need it. One more thing: I can remove the tail and just use a single kernel for everything. Or I can keep the tail for safety. Let me keep it for generality. Actually, looking at the specs again: H100. Let me make sure I use `--maxrregcount=64` or something if needed. But the compiler should do a good job. Let me compile a slightly refined version with explicit CUDA stream use and see if there are any issues. Actually, I can also try using `AT_CUDA_CHECK` or `C10_CUDA_KERNEL_LAUNCH_CHECK` to verify kernel launches. But for performance, we don't want these in the final code. Let me just keep the kernel simple. The key optimization is correct. Wait, there's another important optimization I haven't considered: the matmul itself. What if I ensure PyTorch uses TF32 for the matmul? It does by default on H100. But what if I can use FP16/BF16 for the matmul and then the elementwise in FP32? No, because nn.Linear is FP32, and the model weights are FP32. Converting to FP16/BF16 would change the numerical output beyond tolerance. But, what about using TF32 (which is already the default)? That uses Tensor Cores and is fast. The output is still in FP32 containers. Another idea: what if the reference implementation doesn't create `original_x = x` as a copy, but just a reference? In Python, `original_x = x` is just a reference to the same tensor object. Then `x = torch.sigmoid(x)` creates a new tensor. Then `x = x * scale` creates another. Then `x = x + original_x` creates the final result but reads from `original_x` which is the output of Linear. So memory allocations in reference: - Output of Linear: x - Output of sigmoid: new tensor - Output of mul: new tensor - Output of add: new tensor That's 4 allocations. My code: - Output of Linear: x - Output of fused kernel: new tensor That's 2 allocations. Fewer allocations = less overhead. Could I do in-place on x? Like `x = sigmoid_inplace(x); x.mul_(scale); x.add_(original_x)`? But sigmoid_inplace might not be available, and PyTorch in-place ops have limitations for gradients. Actually, I realize: `torch.sigmoid` does not have an in-place variant. There is `torch.sigmoid_`? No. Actually there is `torch.sigmoid` and tensor.sigmoid but no in-place version. And `x.mul_(scale)` would work in-place, but then `x.add_(original_x)` reads `original_x` and modifies `x` in place. But the fused kernel already avoids separate allocations. I think my kernel is good. Let me just refine it with some minor improvements and submit. Actually, let me see if there are any other optimization opportunities I missed. What about using `torch.empty` instead of `torch.empty_like` to avoid copying stride information? No, `empty_like` is efficient. What about directly using `F.linear` instead of `nn.Linear`? Same thing. What about ensuring contiguous input? The test inputs from `get_inputs()` are created via `torch.rand(batch_size, input_size)` which is contiguous. OK, I think the best I can do is optimize the elementwise kernel. Let me write a cleaner version that might be marginally faster. Let me also consider using `asm` or PTX, but that's overkill. Another thought: the sigmoid function. Can I make it faster? ```cpp __device__ __forceinline__ float fast_sigmoid(float x) { return 1.0f / (1.0f + __expf(-x)); } ``` `__expf` is the fast intrinsic. It's less accurate but much faster on some architectures. On H100, the regular `expf` is already hardware-accelerated and quite fast. But `__expf` might still be faster. Let me check if accuracy is sufficient. For x in a reasonable range, `__expf` error is typically < 1 ULP or a few ULPs. For sigmoid, the derivative is small when x is large, so errors there matter less. Near x=0, sigmoid(0)=0.5, and we'd want accurate values. Actually, let me check: `__expf` error can be up to ~2-3 ULPs. For sigmoid, this translates to maybe 1e-7 relative error, which is well within 1e-4. Wait, but `__expf` doesn't handle denormals or overflow correctly. For large negative x, `__expf(-x)` might overflow to inf, causing sigmoid to be 0. But mathematically, sigmoid(large negative) ≈ 0 anyway. For large positive x, `__expf(-x)` underflows to 0, and sigmoid is 1. That's fine. What about moderate values where exp is close to 1? Like x = 1, exp(-1) ≈ 0.368. Small error here might cause noticeable error in sigmoid. Let me test `__expf` accuracy for sigmoid. Actually, I should just try it and see if it passes correctness. But before that, let me also think about whether the elementwise kernel is the bottleneck or if I can do something about the Linear. Wait a second - I could try to use `torch.compile` on the Linear operation to see if PyTorch 2.x generates better code. But if I use torch.compile on my ModelNew, it might optimize things further. However, torch.compile with Inductor might not play well with custom CUDA extensions. Actually, a better idea: what if I call `torch._C._cuda_getCurrentRawStream` or use `torch.cuda.current_stream()` in Python? The kernel already launches on the default stream. But PyTorch might be using a different stream context. In PyTorch, operations are launched on the current CUDA stream. The at::cuda::getCurrentCUDAStream() gets this stream. So I should use it. But by default, kernels launched without a stream argument use stream 0 (the legacy default stream), which is synchronous with respect to host but asynchronous with compute operations on the same device. In PyTorch, if you're not using CUDA streams explicitly, everything is on the default stream. However, PyTorch might be using its own default stream which is not CUDA stream 0. So I should use `at::cuda::getCurrentCUDAStream()` explicitly. Actually, looking at PyTorch's source code, most custom ops use `at::cuda::getCurrentCUDAStream()` to get the right stream. Let me update my kernel to use the explicit stream. One more optimization for streaming kernels: use `__launch_bounds__(256, 4)` to tell the compiler there can be 4 blocks per SM. This might help it optimize register usage. OK, here's my refined plan: ```cpp __launch_bounds__(256, 4) __global__ void kernel(...) { ... } ``` Actually, max blocks per SM is 32, but with 256 threads and register limits, we might only fit 8. Specifying 4 might cause the compiler to use fewer registers, allowing better occupancy. But for memory bandwidth, we want the kernel to launch with as many blocks as possible. Actually, occupancy isn't the main limiter for bandwidth. What matters is having enough warps to cover latency. Let me just try a version with explicit stream and see if correctness still passes. I also want to try `__expf` to see if it speeds things up. Actually, let me just go ahead and write a more optimized kernel with __expf and explicit stream, then test correctness. Wait, __expf might cause numerical issues. Let me test without it first, and then with it as a variant. I'll write a kernel that uses a template parameter for the exp function, or just hardcode it. Let me write two versions or make it configurable. Let me write the version with `expf` (accurate) and submit that to get a baseline. Then I can try `__expf` if needed. Actually, compiling and running takes turns. Since I have limited turns (10), I should be efficient. Let me write what I think is the best version and submit it. But first, let me think about whether there are any other fusion opportunities. Gemm —> Sigmoid —> Scale —> Add with residual. Could we rewrite the math to simplify? y = sigmoid(x) * s + x This can also be written as: y = x + s * sigmoid(x) Or: y = x + s / (1 + exp(-x)) I don't think there's any algebraic simplification. What if we fuse x + s * sigmoid(x) directly with the matmul? Like custom epilogue. But again, that requires custom matmul. Another thought: what if we precompute the sigmoid of the bias and weight columns? No, because x varies per input. Actually wait, is there a way to reformulate this as a GELU or other activation with known fast approximations? No, sigmoid is fundamentally different. Let me try another approach: what if the test sizes are such that we can use shared memory to tile the computation? Actually, if the linear output were smaller, we could keep it in shared memory. But 32MB is too big for shared memory (only 227KB per block). What about using L2 cache persistence? H100 supports this but it's complex and might not help for a single pass. I think the best approach is a clean, well-optimized elementwise kernel. Let me write it with explicit stream. I'll also make sure to handle non-contiguous tensors properly by using TensorAccessor or just falling back to PyTorch for edge cases. For simplicity and robustness, let me also ensure the tensor is contiguous before passing to the kernel. In my case, Linear output is contiguous, so it's fine. Actually, one very important thing: I should use `c10::cuda::CUDAGuard device_guard(input.device());` to ensure we're on the right device. Let me write the final, polished version. I'll try the `__expf` variant since I think it will be faster and still pass. If not, I can switch back. Actually, wait. On Hopper (sm_90), the regular `expf` might compile to the same instruction as `__expf` because the hardware has dedicated FP32 exponential units. Or there might still be a difference in range handling and accuracy. Let me check with a quick compile and correctness test using `__expf`. Wait, I should first check: does the H100 compiler actually differentiate between `expf` and `__expf`? In many cases on newer GPUs, `expf` is already heavily optimized. Let me just use `expf` for safety. The kernel is memory-bound anyway, so the difference in exp speed might not matter much. Actually, for a memory-bound kernel, the ALU operations (expf, division, etc.) can be completely hidden behind memory latency. So using `__expf` vs `expf` might not make a measurable difference at all. Let me verify: for float4, each thread reads 16 bytes and writes 16 bytes. At 3.35 TB/s, that's 3350 GB/s. For 64MB total traffic, 64/3350 = 19 microseconds. In that time, each thread processes 4 elements. With 256 threads in a block, that's 1024 elements per block. For 8192 blocks, total elements = 8,388,608. In 19 microseconds at 1.8 GHz clock, each thread gets ~34,000 cycles. For 1 float4 = 4 elements, we do 4 sigmoids = 4 expf + 4 adds + 4 divides + more ops. On H100, FP32 exp is ~10-20 cycles. Divide is ~10 cycles. So total ALU might be ~100 cycles per element = ~400 cycles per float4. That's negligible compared to 34,000 cycles per thread if the memory bandwidth is the bottleneck. Wait, but that's if we have enough threads. With 8192 blocks * 256 threads = 2M threads, and only 132 SMs, each thread runs for 19us * 1.8GHz / 2M * 132... actually this is confusing. The point is: for a simple elementwise kernel, memory bandwidth is almost certainly the bottleneck. The compute cost of sigmoid is tiny in comparison. So optimizing the ALU won't matter much. The only way to speed up the elementwise kernel further is to reduce memory traffic. Can we do that? If we do in-place: read-modify-write in one operation. But we still need to read and write the same amount of data. The only difference is that the old tensor memory can be reused instead of allocating new memory. But that affects allocation overhead, not memory bandwidth. Unless... can we avoid writing the Linear output and instead write directly into our output with the epilogue fused? That's what we'd need a custom matmul for. Wait, actually, there might be a way to do the GEMM in PyTorch but also specify an output tensor. `nn.Linear` doesn't support an `out=` argument. But `torch.addmm` does! The operation is `y = x @ A.T + b`. We could compute this with `torch.addmm(bias, x, weight.t())` and pass the output tensor... but we can't easily fuse the epilogue. Actually, wait! PyTorch has `torch.matmul` and other ops, but none with epilogue fusion. So I think the memory traffic for the elementwise part is at a minimum with our fused kernel: 1 read + 1 write per element. Could I do computation in-place to avoid one allocation? Yes, I could re-use `x` for the output: ```cpp torch::Tensor fused_sigmoid_scale_add(torch::Tensor input, float scale) { // Modify input in-place... wait, but we need original_x = input // If we modify input in-place, we lose original_x // So no, in-place doesn't work for this specific formula } ``` Actually, if we process element by element, we read input[i], compute output[i], and write to a new location. We can't write back to input because other threads might still need to read their elements. Well, if each element is independent, we CAN write back to the same tensor because we only read `input[i]` once and write once. But `original_x` is the same tensor. If we overwrite `input` with the result, that's fine as long as: 1. We're done reading the input element before writing the output. 2. We don't need the original tensor afterward. Since each thread processes independent elements, and reads `input[i]` before computing and writing, there's no data race for a single element. However, there might be cache coherency considerations. Writing to the same address we just read might have slightly different performance. But the main point: can we write in-place to save one allocation? Yes! Instead of allocating a new output tensor, we can allocate output as `input.clone()` and then... wait, `input.clone()` requires a copy, which is extra memory traffic. If we just write back to `input`, we avoid allocating `output`. But `input` is the result of Linear, and if we modify it in-place, that's fine for inference. Wait, but if we write back to `input`, the reads and writes are to the same memory location, but through different cache lines. The memory traffic is still 1 read + 1 write per element. The total bytes moved through HBM is the same. Actually, there could be a benefit: if the L1 cache keeps the read data, the write might hit L1 and not need to go all the way to HBM. But for sequential access patterns with float4, the cache lines are full and will be written back anyway. No, the difference is that writing to a separate tensor means the reads come from one set of memory pages and the writes go to another set. The memory controller can handle this well. Writing to the same tensor might also be fine. But there's no bandwidth savings from in-place. The only savings are the allocation overhead. For the purpose of optimization, let me use a separate output tensor to keep it clean. Allocation overhead is tiny compared to the kernel execution. Alright, I'm going to write one more refined version. Let me think if there's any architectural optimization for H100 specifically. H100 has: - 132 SMs - Special memory/cache architecture - Can benefit from persistent kernels or grid-level synchronization For a simple elementwise kernel, persistent kernels don't help. Another idea: use `__grid_constant__` or other Hopper-specific features? Probably overkill and not supported by the compiler version. OK, I'll go with a clean, simple kernel. But I'll also add one more optimization: rather than two kernels (main + tail), I'll use a single kernel that handles both float4 and scalar elements. The branch can be per-thread based on whether the index is in the float4 region or tail region. Actually, a simpler approach is to compute the number of float4 elements and pass that. If the tensor is not aligned, fall back to scalar mode. But for our test case, it's aligned and divisible by 4. Let me just write a robust single-kernel solution. Actually wait, I want to try something else. What if I don't even use float4? What if I just use scalar loads/stores but with a very large grid? On modern GPUs, the L1 cache handles scalar loads well and coalesces them automatically. Float4 might not be necessary. But float4 reduces instruction count, so it's probably still beneficial. Actually, on Hopper, vector loads of 128 bits (float4) or 256 bits might even have higher throughput. Let me stick with float4 and a tail handler. One final idea: could I structure the kernel with actual bounds checks in a single kernel to avoid launching two kernels? ```cpp __global__ void kernel(const float* in, float* out, float scale, int64_t n, int64_t n4) { int64_t idx = blockIdx.x * blockDim.x + threadIdx.x; int64_t stride = blockDim.x * gridDim.x; for (int64_t i = idx; i < n4; i += stride) { float4 val = reinterpret_cast<const float4*>(in)[i]; val.x = sigmoidf(val.x) * scale + val.x; ... reinterpret_cast<float4*>(out)[i] = val; } int64_t tail_start = n4 * 4; for (int64_t i = tail_start + idx; i < n; i += stride) { float v = in[i]; out[i] = sigmoidf(v) * scale + v; } } ``` The tail loop only executes for threads where `tail_start + idx < n`. For our test case, n4*4 == n, so no thread executes the tail. The compiler likely optimizes this away. But there's still the overhead of the second loop in the kernel code. Maybe it's negligible. Alternatively, I can have two completely separate kernel calls. That's what I had. Actually, with 0 tail elements, the second kernel launch is wasteful. It does nothing but still consumes launch overhead (~5 μs). Not significant, but unnecessary. For our exact test case, we know n4*4 == n. But to be general, I'll keep the tail. Let me write a single-kernel version that handles both, which avoids the second launch entirely. Here's the plan: 1. Main computation as a grid-stride loop over n4 2. After the main loop, check if there's a tail and compute tail elements in a grid-stride loop The tail loop condition naturally handles the case where n4*4 == n (no tail). This is cleaner. Let me write it. Actually, one issue: if `n4` is very large and the tail is 0-3 elements, the tail loop check `tail_start + idx < n` would rarely be true, but all threads would still compute `tail_start + idx` unnecessarily. However, this is negligible. Or I could put an `if (tail_start < n)` before the tail loop so that most warps skip it entirely when there's no tail. The compiler can optimize this. Let me write: ```cpp __global__ void kernel(...) { int64_t idx = ...; int64_t stride = ...; // Main loop for (int64_t i = idx; i < n4; i += stride) { ... } // Tail loop int64_t tail_start = n4 * 4; if (tail_start < n) { for (int64_t i = tail_start + idx; i < n; i += stride) { ... } } } ``` This is clean and avoids a second kernel launch. Alright, let me also add proper device guard. And one more thing: since this is essentially a simple elementwise operation, maybe I can just use `torch.jiterator` or a Triton kernel. But the assignment specifically says to use `load_inline`. So I'll stick with CUDA. Actually, wait - I just realized something important. The elementwise kernel can be written much more simply using `AT_DISPATCH_FLOATING_TYPES_AND2` and a generic elementwise kernel. But writing the raw CUDA gives better performance. Let me now write the final kernel. I'll also try using `__expf` to see if we can get any speedup. If the correctness check fails, I'll switch back. Actually no, let me not waste turns - I'll use `expf` which is safe. Wait, let me think about this: on Hopper, the `std::exp` or `expf` function from libdevice vs the intrinsic `__expf`. The intrinsic is faster but less accurate. On Ampere and Hopper, `expf` maps to `ex2.approx.f32` when using `--use_fast_math`. Without `--use_fast_math`, it might use a more accurate implementation. Actually, I should check if adding `--use_fast_math` affects compile time or performance. Let me try compiling with `--use_fast_math` and see. Hmm, but the compile tool just checks compilation, not speed. And if accuracy fails, I've wasted a turn. Let me skip fast_math for now. One more crazy idea: can I make the GEMM faster by using bfloat16 internally? Like, convert input to bf16, do matmul in bf16, convert back to fp32, then do the elementwise. This would use Tensor Cores at full speed. But bf16 matmul might have enough error to fail the 1e-4 fp32 tolerance. Let me think: bf16 has ~3 decimal digits of precision. For values in the range [0, 1], the error might be ~1e-3. But with accumulation in fp32 (which cuBLAS does for bf16), the error on the final result might still be around 1e-3, which exceeds 1e-4. So that's probably not safe. But wait, what if we use TF32? TF32 has ~4 decimal digits. The matmul accumulation error for a 8192-length dot product in TF32 is typically around 1e-3 to 1e-4 relative error. PyTorch uses TF32 by default for matmuls on H100. So the reference is already using TF32! And the accuracy tolerance is 1e-4, which is tight for TF32. Wait, but tolerance for fp32 is `atol=rtol=1e-4`. If the reference itself has TF32 error of ~1e-4, then my code must match the reference output exactly (or within 1e-4). Since I'm using the same `nn.Linear` as the reference, the matmul output should be the same. My elementwise kernel uses fp32 math, so it's exact. Actually, I should make sure the Linear layer in my ModelNew produces the same output as the reference. Since we're both using the exact same `nn.Linear` with the same weights, yes. Now, what if I changed the Linear implementation? Then I'd have to match the TF32 reference output, which is tricky. Better to keep PyTorch's Linear. Alright, I'm confident in my approach. Let me write the final polished kernel. I'll use a single kernel call with both main and tail in one kernel, and add a device guard and stream. Actually, I just realized: I need to be very careful about the number of blocks. If I use too many blocks, I might hit some limit. But with the single-kernel approach and grid-stride, let me compute blocks = (n4 + threads - 1) / threads for the main part, and that's it. Wait, for the tail, I need enough blocks to cover the tail elements too. But if I compute blocks based on n4, and the tail has up to 3 elements, those blocks also cover the tail because the tail loop has its own grid-stride with the same grid size. Actually no - the tail requires the same grid size. If blocks are based on n4, then for the tail, we still have `blocks * threads` threads, which can easily handle 3 elements. So it's fine! Let me write the code. Wait, the formula for blocks: if n4 = 2,097,152 and threads = 256, blocks = 8192. That's fine. If I use 128 threads: blocks = 16384. Also fine, still under 65535. I'll try 256 threads first. Let me also check: for very small tensors, blocks might be 0. So I need to handle n4 == 0. For n < 4, n4 = 0, and tail_start = 0 < n, so tail handles everything. But blocks = 0, which means no kernel launch! So I need to launch at least 1 block if there's any work. So: `int blocks = max((int)((n4 + threads - 1) / threads), 1)`. Wait, but if n4 == 0 and n > 0, the tail needs blocks. How many blocks for tail? `(n + threads - 1) / threads`. But we're using `n4` for blocks. If n4 == 0 and n = 3, and we launch 1 block (because max(..., 1)), then: - Main loop: `i < n4` -> `i < 0`, so no thread executes main loop - Tail: `tail_start = 0`, `i = 0 + idx`, if idx < 3, process elements 0, 1, 2. That works! So I can use: `blocks = max(1, (int)((n4 + threads - 1) / threads))` and handle both main and tail in the same kernel. Let me write this. Actually, there's an even simpler approach: just use `blocks = (n + threads - 1) / threads` and have each thread handle one element (no float4). Simpler code, but 4x more instructions for loads/stores. For bandwidth-bound, the instruction overhead is probably not the bottleneck. But float4 does reduce the number of threads and blocks needed. On H100 with 132 SMs, even 8 million threads (scalar) = 8M/256 = 31K blocks. That's under 65535 and plenty. Actually, with 256 threads per block and 8M elements, we need 32768 blocks, which is under 65535. Each block handles 256 elements. With 132 SMs, we have ~248 blocks per SM queued. Max concurrent blocks per SM is 32, so 248 won't all run concurrently, but they'll pipeline. Wait, 32768 blocks on 132 SMs means ~248 blocks per SM on average. With 32 max concurrent blocks per SM, the GPU queues the rest. Each block processes 256 elements. Once a block finishes, the next is scheduled. Actually, maybe using scalar elements with many blocks is better or worse than using float4 with fewer blocks. The difference is: - Scalar: 8M threads, each reads 4 bytes. Need to issue 8M loads. More instructions but same total memory traffic. - Float4: 2M threads, each reads 16 bytes. Fewer instructions, same memory traffic. Since the kernel is memory-bound, float4 should be slightly better due to fewer instructions, but the difference might be small. I'll stick with float4. Let me now write the final code. Actually, one more consideration: the blocks per SM limit. With 256 threads and 32 blocks per SM, that's 8192 threads per SM, which equals the max threads per SM (2048 on H100? Let me check: H100 has max 2048 threads per SM). So 32 blocks of 256 threads = 8192 threads, which EXCEEDS 2048! That means we can't have 32 blocks of 256 threads concurrently. Max blocks per SM with 256 threads = 2048 / 256 = 8 blocks. Max with 128 threads = 2048 / 128 = 16 blocks. Max with 64 threads = 32 blocks. So with 256 threads, we're limited to 8 blocks per SM concurrently. With 8192 blocks total and 132 SMs, we have 62 blocks per SM queued, but only 8 run concurrently. With 128 threads: 16 blocks per SM concurrently, 16384 total blocks = 124 per SM queued. The question is: does this difference matter for bandwidth? Probably not much, as long as we can hide latency. With 8 blocks * 256 threads = 2048 threads per SM, that's max threads. The GPU should be able to hide latency well with 2048 threads per SM. For float4 loads, each warp (32 threads) loads 128 bytes. With 64 warps per SM (2048 threads / 32), that's 64 concurrent warps. Each warp can have multiple memory requests in flight. This should easily saturate memory bandwidth. So 256 threads is fine. Let me stick with 256. Alright, final code:

Let me think about optimizations. The current kernel is correct. Let me try to optimize it further. The key insight is that for H100, we should be launching enough blocks to saturate the GPU. With the current approach of single blockDim of 256 and only up to 65535 blocks, we might not be fully saturating the GPU for very large inputs. The input size is (batch_size, hidden_size) = (1024, 8192), so there are 8,388,608 elements. n4 = 8,388,608 / 4 = 2,097,152 float4 elements. With 256 threads and 65535 blocks, that's 16,776,960 threads launched. Since we have 2,097,152 elements, each thread does only 0.125 iterations on average. Given the grid-stride loop, each thread would process very few elements. Actually for better performance, we should: 1. Use more threads per block (e.g., 512) 2. Or just use a simple 1D grid without grid-stride loops 3. Or use a 2D grid For H100 gpu with 80GB memory bandwidth 3.35 TB/s, memory bandwidth is the key limitation. Let me try a simpler approach: single kernel, no grid-stride loop, just a single element per thread with large grid. We can safely set gridDim to `(n4 + threads - 1) / threads`. For n4=2,097,152 and threads=256, that's 8,192 blocks, which is well within limits. Wait, we should also consider: 1. Unrolling the grid-stride loop can give better performance 2. Using __ldg for read-only loads 3. Grid launch with enough blocks to saturate all SMs Let me also think about whether we should fuse the GEMM with the elementwise operations. But the problem says we need to produce numerically equivalent outputs at fp32 precision, and the reference uses `nn.Linear`. Fusing the GEMM would be complex and might not give benefits for a single layer unless the GEMM is heavily memory-bound. For (1024, 8192) x (8192, 8192), the GEMM is compute-bound because 1024 * 8192 * 8192 * 2 = 137,438,953,472 ops. At 67 TFLOPS FP32, that's about 2ms. Memory bandwidth of the output is 32MB. So the GEMM is definitely compute-bound. The elementwise kernel is memory-bound: read 32MB + write 32MB = 64MB. At 3.35TB/s, that's about 19 microseconds. This is tiny compared to the GEMM. So the real question is whether the fused kernel helps reduce overall time. The key difference is that in the reference: 1. GEMM outputs a tensor 2. Then sigmoid, scale, and add are done via 3 separate kernels or a single fused CUDA kernel (PyTorch likely fuses some of it) Actually, PyTorch probably fuses the operations after the GEMM since there's only a single backward dependency. Looking at the reference: ```python x = self.gemm(x) original_x = x x = torch.sigmoid(x) x = x * self.scaling_factor x = x + original_x ``` PyTorch might allocate new tensors for each op. By using a custom elementwise kernel, we avoid intermediates and memory traffic from the elementwise ops. Since the GEMM output is already computed, our fused kernel just reads it once and writes the result once. Let me optimize the kernel further by: 1. Using larger blocks (1024 threads) and units of float4 2. Using grid-stride loop with unrolling 3. Maybe use vectorized stores Actually, let me try a much simpler and better approach: Just ensure we launch enough blocks with a grid-stride loop, but each thread processes 4 floats via float4. For 8192 * 1024 = 8,388,608 floats, n4 = 2,097,152 float4 elements. With 1024 threads per block, and say 256 blocks, that's 262,144 threads. Each thread does 8 iterations (2,097,152 / 262,144 = 8). That seems like a good balance. Let me also add __ldg for read-only loads of the input. Actually, for even better performance, I should process multiple float4 per thread per loop iteration. Let me try an optimized version: