The task is to optimize a fused kernel that does: 1. GEMM (Linear): x @ W^T + b 2. Scale by a constant 3. Hardtanh: clamp(min, max) 4. GELU activation Given dimensions: batch_size=2048, in_features=8192, out_features=8192. This is a large GEMM followed by elementwise ops. The best approach is to fuse all elementwise operations into the GEMM output using a custom CUTLASS kernel or a simple CUDA kernel that does the GEMM via cublas and then a fused elementwise kernel. But for max performance, we should try to fuse the epilogue into GEMM using something like cuBLASLt or a custom matmul kernel. However, writing a full custom GEMM kernel with fused epilogue from scratch is complex. Given the H100 has 1979 TFLOPS BF16 and 989 TFLOPS TF32, for a 2048x8192x8192 GEMM: - M=2048, N=8192, K=8192 - FLOPs = 2 * M * N * K = 2 * 2048 * 8192 * 8192 ≈ 274.9 GFLOPS = 2.75e11 FLOPs - At 989 TFLOPS, that's ~0.28 ms. At 67 TFLOPS FP32, that's ~4.1 ms. - Memory bandwidth for output: 2048*8192*4 bytes ~ 64MB. At 3.35 TB/s, memory-bound time is ~19 microseconds. - But weights are 8192*8192*4 = 256MB, inputs 2048*8192*4 = 64MB. Total memory read for GEMM is higher but still compute-bound. Actually, the reference likely uses cuBLAS for the GEMM which is already near-peak. The elementwise ops are bandwidth-bound. If we fuse them into a single kernel after GEMM, we save memory bandwidth. But can we do better by fusing into the GEMM epilogue? Yes. We can use `torch.addmm` followed by a fused kernel? Actually, PyTorch might already fuse some of these if we use torch.compile, but the reference is standard PyTorch, so not fused. The simplest robust approach: 1. Do GEMM using `torch.matmul` or `F.linear` (which calls cuBLAS, extremely fast) 2. Write a fused CUDA kernel for: scale, hardtanh, gelu. But wait: the output of GEMM is 2048*8192 = 16.7M elements. Each element needs: - Read from GEMM output - Multiply by scaling_factor - Hardtanh - GELU - Write back That's essentially bandwidth-bound. Tensor memory bandwidth is 3.35 TB/s, but due to cache, reading/writing FP32 for 16.7M elements is 2 * 16.7M * 4 = ~134MB => ~40 microseconds. That might be negligible compared to GEMM time. But we can do slightly better: if we compute GEMM into a temporary buffer, then run our fused kernel to transform in-place. Or we can use cuBLASLt epilogue fusion. For maximum speedup, using cuBLASLt with epilogue fusion would be ideal because it avoids writing the intermediate GEMM result to global memory entirely: compute GEMM tiles directly into registers, apply scale/hardtanh/gelu, write final output. cuBLASLt supports some epilogues like ReLU, GELU, but does it support Hardtanh + GELU? Probably not custom hardtanh. Alternative: write a custom CUTLASS kernel? That's possible but very complex to do inline in Python. Alternative 2: decompose the GEMM into chunks that fit in shared memory? Not realistic for this size. Alternative 3: Use Triton? But it said "CUDA source" so Triton likely isn't available or we need to use `load_inline` with nvcc. Actually, wait — we can use `torch.addmm` for the GEMM. Then a custom CUDA kernel. But we can also try to write a custom kernel that uses `cublasSgemm` (or cublasGemmEx) and then do a custom elementwise kernel on the output. However, since the reference is just standard PyTorch nn.Linear, the time might be dominated by the GEMM, and the elementwise is negligible. If that's the case, a speedup might be hard if our GEMM is the same speed. But maybe there is room: if reference is not using TensorFloat32 or something? Wait, we are on H100, but PyTorch defaults? The reference uses `nn.Linear` which uses cuBLASLt by default under the hood. It should be very fast. Wait, but the reference does `gelu(hardtanh(x * scale))`. If we fuse the epilogue into a custom kernel, we avoid the overhead of 3 separate PyTorch kernel launches (scale, hardtanh, gelu). Plus Python overhead. That might give a modest speedup. Actually, we can fuse `scale`, `hardtanh`, and `gelu` into the bias addition of the linear layer. If we use `torch.addmm` or `F.linear`, we get output = x @ W^T + b. We can write a single CUDA kernel that takes the GEMM output + bias and computes the final result: `gelu(hardtanh((x @ W^T + b) * scale))` But to do that, we still need to compute x @ W^T. We can do that with `torch.matmul` and then launch our fused kernel. Can we fuse bias application too? Yes, since we have the bias vector. So: 1. Compute `x @ W.T` using `torch.mm` (or `torch.matmul`). But we have bias. 2. We could use `torch.addmm(bias, x, W.T)`. This fuses GEMM + bias addition in cuBLAS. 3. But wait, we want to then apply scaling, hardtanh, gelu. cuBLAS cannot do hardtanh. 4. So: `torch.addmm(bias, x, W.T)` gives us the raw linear output. Then our custom kernel applies `gelu(hardtanh(x * scale))`. Alternatively, we can pre-scale the bias: `bias * scale`, but that doesn't help because we still need to scale the matrix product. Wait, can we scale the weights? If we pre-scale weights and bias during `__init__`, then the GEMM itself computes `x @ (W*scale).T + b*scale`. Then we don't need a separate scaling kernel. But we still need hardtanh and gelu. So then it's GEMM + hardtanh + gelu. We can fuse hardtanh and gelu into a bandwidth-bound kernel. But even better: we can apply hardtanh + gelu in the custom CUDA kernel after GEMM. So the plan: - In `ModelNew.__init__`, store weight and bias. We can pre-scale the weights and bias so that the GEMM computes `scale * (x @ W.T + b)`. - Then forward is: `gemm_output = x @ (scaled_W).T + scaled_b`; then fused `hardtanh_gelu` kernel. - Fused kernel: read element, apply hardtanh, apply gelu, write back. Given that the GEMM with cuBLAS is likely near identical to reference's GEMM, but we saved one multiplicative kernel launch (scale), the speedup might be small. However, Python/framework overhead for launching 4 separate kernels (gemm, mul, hardtanh, gelu) vs 2 kernels (gemm, fused_epilogue) could add up, especially if cuBLAS launch overhead is nontrivial. Actually, wait: can we fuse the epilogue into the GEMM via a custom kernel that loads tiles of x and W and computes the result directly? Since M=2048, N=8192, K=8192, if we write a simple CUDA kernel with shared memory tiling, it might be much slower than cuBLAS. So we should rely on cuBLAS for the GEMM. But maybe we can avoid writing a custom GEMM kernel and use CUTLASS via PyTorch? Not possible via `load_inline` easily. Wait, what about using `cublasLtMatmul` with an epilogue? cublasLt supports `CUBLASLT_EPILOGUE_GELU` and `CUBLASLT_EPILOGUE_GELU_AUX` and `CUBLASLT_EPILOGUE_RELU`. It might support custom epilogues via a callback? Not sure. Let's check: cublasLt epilogue options include `GELU`, `RELU`, `BIAS`, `RELU_BIAS`, `GELU_BIAS`, etc. Hardtanh is not supported. So our best approach is: 1. Use `torch.addmm` for GEMM + bias. Or to be safe, `torch.mm` then add bias? `torch.addmm` is the same as `nn.Linear` forward (without scale). 2. Write a custom CUDA kernel that fuses scale*hardtanh*gelu. Wait, we can even be more clever: pre-scale the weight and bias. Then the `forward` only does `F.linear(x, scaled_weight, scaled_bias)` and then a single fused kernel for hardtanh+gelu. This saves one kernel launch but still does the same cuBLAS GEMM. The speedup will come from: - Fusing scale+hardtanh+gelu into one kernel launch instead of 3. - Plus better memory access patterns (read once, write once). - But the intermediate results in the reference are still in cache on H100? Actually each op reads and writes global memory (or L2). fusing them into 1 read + 1 write vs 3 read + 3 write for the elementwise part saves bandwidth. However, the GEMM output is large (64MB) and might exceed L2? Let's check H100 L2. H100 L2 is 50MB. So 64MB FP32 output exceeds L2. Thus the fused kernel saves significant memory bandwidth. But still the time is likely GEMM-dominated. Wait, is there a way to also fuse the bias addition into our custom kernel? `nn.Linear` does `x @ W.T + b`. If we do `torch.mm(x, W.T)` then our custom kernel does `add_bias + scale + hardtanh + gelu`. But bias addition is bandwidth-bound anyway. Fusing it means we can do: `out = mm(x, W.T)` (no bias) `fused_kernel`: for each element, read GEMM result, add bias, scale, hardtanh, gelu, write. That's still 1 read + 1 write for the elementwise part, plus reading bias (negligible). But compared to using `addmm` (fused GEMM+bias in cuBLAS), which avoids writing the un-biased GEMM output. Wait, `addmm` is actually fusing the bias into the GEMM compute, avoiding a separate kernel. So `addmm` is better than `mm` + bias kernel. So best: - Use `torch.addmm(self.bias, x, self.weight.t())` (which is what `F.linear` does internally via `addmm` when bias is present, but we can call it directly). - Then our fused kernel takes the output and applies scale + hardtanh + gelu. But if we pre-scale weights and bias, then `torch.addmm` computes `scale*(x@W.T + b)`. Then our kernel is just hardtanh + gelu. But `torch.addmm` with pre-scaled weights is identical to doing scaling during GEMM via cublas? Actually scaling weights doesn't change the GEMM performance. Alternatively, we can use `cublasGemmEx` with alpha=scaling_factor to scale the output of GEMM in cuBLAS itself. Then add bias inside cuBLAS via `cublasLtMatmul` with bias and then our kernel only does hardtanh+gelu. That might be even better because the scaling is fused into GEMM. But we can also pre-scale weights which achieves the same effect. Actually, wait: can we use cublasLt epilogue to scale + bias + apply hardtanh + gelu? CublasLt custom epilogue? There is a "custom pointwise" feature in newer versions? Let me think. I know cublasLt 12.x has `cublasLtMatmulDesc`，but custom epilogues are not for arbitrary ops. Let's go with the `addmm` + fused elementwise kernel approach. It should give some speedup. But wait, is there any chance the reference uses `torch.compile`? No, it's standard eager mode. So 3 extra Python-level kernel launches (mul, hardtanh, gelu) have overhead. However, to get a *significant* speedup, we should try to write a kernel that uses Tensor Cores via Warp Matrix Multiply Accumulate (WMMA) or mma.sync? That's extremely complex and error-prone. Alternative: can we use CUTLASS via `torch.utils.cpp_extension` with inline CUDA? Yes, we can write a CUTLASS kernel if we link against the CUTLASS headers (which are header-only). PyTorch includes CUTLASS? Actually, recent PyTorch versions bundle CUTLASS headers in `torch/include`. But writing a complete CUTLASS kernel for this problem is nontrivial but possible. Wait, maybe there is a simpler path: use `torch.matmul` and then a custom kernel. The evaluation compares against the reference. If the reference is already memory-bandwidth bottlenecked on the GEMM? Let's compute more carefully. FP32 GEMM 2048x8192x8192: - M=2048, N=8192, K=8192. - Operations: 2*M*N*K = 2 * 2048 * 8192 * 8192 = 274,877,906,944 ≈ 2.75e11 FLOP. - H100 FP32 non-tensor: 67 TFLOPS = 67e12 FLOP/s. Time = 2.75e11 / 67e12 = 0.0041 s = 4.1 ms. - H100 TF32 Tensor Core: 989 TFLOPS = 989e12. If it uses TF32 automatically (cuBLAS default on Ampere/Hopper), time = 2.75e11 / 989e12 = 0.000278 s = 0.278 ms. - Actually cuBLAS on H100 for this size likely uses TF32 Tensor Cores by default. So GEMM time ~0.3 ms. - Elementwise ops memory: 16.7M elements. Each op reads and writes float (4 bytes). 3 separate ops (mul, hardtanh, gelu) => 6 passes * 16.7M * 4 bytes = ~400MB memory traffic. - At 3.35 TB/s, 400MB takes ~0.12 ms. If it's reading/writing through L2/HBM, maybe 0.2 ms. Plus kernel launch overhead. - Total reference: ~0.3 ms (GEMM) + ~0.15 ms (elementwise) + overhead ≈ 0.45 ms. If we fuse elementwise into one kernel: 1 read + 1 write = 134MB => 0.04 ms. Plus GEMM 0.3 ms = 0.34 ms. That's about ~1.3x speedup. Decent. If we also fuse bias addition and scaling into the elementwise kernel, we could use `torch.mm` instead of `addmm`. But `addmm` fuses bias+GEMM, so it's likely faster than `mm` + elementwise bias add. So we keep `addmm`. Wait, can we use `torch._C._VariableFunctions.addmm`? It's the same. Another idea: since the size is highly divisible, we can use a simple cublas GEMM and then a custom kernel that is vectorized (float4) to maximize bandwidth. Let's write a simple fused kernel: ```cpp __global__ void fused_scale_hardtanh_gelu(float* out, const float* in, float scale, float min_val, float max_val, int n) { int idx = blockIdx.x * blockDim.x + threadIdx.x; if (idx < n) { float x = in[idx]; x = x * scale; x = fmaxf(min_val, fminf(max_val, x)); // GELU approximation float cdf = 0.5f * (1.0f + tanhf(0.7978845608f * (x + 0.044715f * x * x * x))); x = x * cdf; out[idx] = x; } } ``` But to be numerically equivalent to PyTorch's GELU (which uses the exact formula or the tanh approximation?), we need to match PyTorch exactly. PyTorch GELU in FP32 uses the tanh approximation: `0.5 * x * (1 + tanh(sqrt(2/pi) * (x + 0.044715 * x^3)))` More precisely: `GELU(x) = 0.5 * x * (1.0 + torch.tanh(math.sqrt(2.0 / math.pi) * (x + 0.044715 * torch.pow(x, 3.0))))` Actually, PyTorch's GELU uses `approximate='none'` by default in newer versions? Wait, `nn.GELU()` in PyTorch had `approximate='none'` until some version? Let's check. In PyTorch, `nn.GELU(approximate='none')` is the default? Actually, since PyTorch 1.12, `nn.GELU()` default is `approximate='none'`? Let me recall. The documentation says `approximate: 'none' | 'tanh'. Default: 'none'`. But in practice, the CPU/GPU kernels might use different implementations. Wait, the reference implementation in the prompt just does `self.gelu = nn.GELU()`. The default in recent PyTorch versions is still `approximate='none'`. But `nn.GELU()` without argument historically used the tanh approximation on CUDA for performance? Let's verify. Actually, looking at PyTorch source: - `aten/src/ATen/native/cuda/ActivationGELUKernel.cu` has `gelu_kernel` which uses the exact formula with `erf` when `approximate="none"`, and a tanh approximation when `approximate="tanh"`. The default for `nn.GELU` is `approximate='none'`. But is the CUDA kernel exact? Yes, it uses: ``` auto kBeta = M_SQRT2 * M_2_SQRTPI * 0.5; auto KAPPA = 0.044715; // tanh approximation: cdf = 0.5 * (1 + std::tanh(kBeta * (x + KAPPA * x * x * x))); // exact: cdf = 0.5 * (1 + std::erf(x * M_SQRT1_2)); ``` Actually, the default on CUDA for `gelu` might be exact erf? Let's check the code path. In PyTorch's `aten/src/ATen/native/Activation.cpp`, the `gelu` function accepts an `approximate` string. The default is `""` which maps to `"none"`. So it uses exact `erf`. But for numerical tolerance: if we copy the approximation and reference uses exact, we might exceed 1e-4 tolerance, especially for values around ±2 after hardtanh. Let's compute worst-case difference. For x=2, erf(2/sqrt(2))=erf(1.414)=0.9545. Tanh approximation: 0.7978845608*(2 + 0.044715*8) = 0.7978845608*(2 + 0.35772) = 0.7978845608*2.35772 = 1.881. tanh(1.881)=0.9551. Difference in CDF: 0.9545 vs 0.9551 => ~0.0006. Multiplied by x*0.5 => result difference ~0.0006. That's within 1e-4? 0.0006 > 0.0001. So we need to know which one the reference uses. Let's test by writing a small kernel with exact erf and see if it matches. But we can just compile a simple kernel and run correctness to see if approximation passes. Alternatively, we can look at the reference code. `nn.GELU()` without args. In PyTorch 2.x, `nn.GELU()` default is `approximate='none'`. But does the CUDA kernel actually use erf? I think yes. However, `nn.GELU()` on CUDA uses the `gelu` kernel. In `ActivationGELUKernel.cu`, there are two variants: `GeluKernel` and `GeluBackwardKernel`. The forward kernel for `approximate="none"` is actually using the `erf` variant or maybe the tanh variant depending on the dispatch. Let's check typical CUDA code: ```cpp template <typename T> __inline__ __device__ T Gelu(T x) { const T kBeta = M_SQRT2 * M_2_SQRTPI * T(0.5); const T kKappa = 0.044715; T x_cube = x * x * x; T inner = kBeta * (x + kKappa * x_cube); return static_cast<T>(0.5) * x * (static_cast<T>(1) + tanh(inner)); } ``` Wait, no, that's the tanh approximation. The exact one uses `erf`. In PyTorch source file `aten/src/ATen/native/cuda/ActivationGELUKernel.cu`, the `gelu_kernel` is defined with a template parameter `bool approximate`. For `approximate=false`, it uses: ```cpp cdf = scalar_t(0.5) * (scalar_t(1) + ::erf(x * scalar_t(M_SQRT1_2))); ``` For `approximate=true`, it uses tanh. Since default is `approximate='none'` (false), we should use exact `erf` to match. But `erf` in CUDA standard library (`::erf`) is available. We can use `erff` for float precision. So our fused kernel: ```cpp float x = in[idx]; x = x * scale; x = fmaxf(min_val, fminf(max_val, x)); float cdf = 0.5f * (1.0f + erff(x * 0.7071067811865475f)); x = x * cdf; out[idx] = x; ``` This should match exactly (or very close). Using `erff` in CUDA should match the host `erf` used by PyTorch CPU? But PyTorch CUDA likely uses `erf` or custom polynomial approximation. To be safe, we can mimic the exact formula and hope for tolerance. Actually, I can write a quick kernel with the exact GELU formula and see if it passes. Let's do that. Also, for maximum memory bandwidth, we should process 4 floats per thread using `float4` loads/stores. That is a common optimization. But we might not need it if the kernel is simple. However, since the time could be dominated by memory bandwidth, vectorized loads help. But `float4` requires alignment. `out` from `addmm` is aligned to at least 256 bytes? Actually, PyTorch tensors are allocated with at least 512-byte alignment, so `float4` is safe if size is a multiple of 4. 2048*8192 is divisible by 4 (since both are powers of 2). So yes. We should also consider occupancy: using 256 threads per block is standard. Wait, but can we avoid the separate GEMM output buffer entirely? Using `cublasLtMatmul` with epilogue? No, we can't easily do hardtanh. Another thought: maybe we can use a custom Triton kernel? But the instructions say "CUDA source" and `load_inline`. PyTorch's `load_inline` uses nvcc. Triton isn't part of that pipeline. So we stick to CUDA C++. Let's also check if we can use `torch.mm` for GEMM and then our fused kernel that does bias + scale + hardtanh + gelu. This allows us to fuse `bias` into our elementwise kernel, avoiding `addmm` but requiring a separate bias addition kernel. Wait, `addmm` is much better than `mm` + `add` because it fuses bias addition into the GEMM accumulation, avoiding an extra memory pass. So `addmm` is strictly better. But we can also do the GEMM with `torch.addmm` (fused bias) and then do our elementwise kernel with scale + hardtanh + gelu. That's optimal. Can we fuse scale + hardtanh + gelu into `cublasLt` epilogue? cublasLt supports `C = alpha*(A*B) + beta*C + bias`, and then an optional `GELU` or `ReLU` epilogue. But scaling (`alpha`) is already there. Wait, cublasLt supports `C = activation(alpha*(A*B) + bias)`. If we set alpha = scaling_factor, and bias = self.bias, then the output is `scale*GEMM + scale*bias`? Wait, cublasLt epilogue bias is added after scaling: `D = activation(alpha*(A*B) + beta*C + bias)`. Actually for `CUBLASLT_EPILOGUE_BIAS`, it does `D = alpha*(A*B) + bias` (if beta=0). So if we set alpha=scaling_factor and bias = self.bias * scaling_factor, then the output is `scale*(A*B) + scale*bias = scale*(A*B + bias)`. Then if we apply GELU epilogue, we get `GELU(scale*(A*B + bias))`. But we still need hardtanh before GELU. cublasLt also supports `RELU` but not arbitrary clamping. Could we use `CUBLASLT_EPILOGUE_GELU_AUX`? That's just GELU. So cublasLt won't work for this chain. What about using `torch._custom_op` or `torch.library`? Not relevant. So our plan is solid. But wait, there is an elegant way: because hardtanh limits to [-2, 2], and scale is 0.5. The linear output before scaling could be anything. After scaling by 0.5, it's clamped to [-2, 2]. What if we clamp the bias and weights? Not helpful because x is random. So the fused kernel after `addmm` is simply: `out = gelu(hardtanh(in * scale))` Wait, could we even avoid `addmm` by doing the GEMM with `torch.matmul` and then our kernel does `add + scale + hardtanh + gelu`? That would be: - `torch.matmul(x, W.T)`: one kernel, no bias fused. - Our fused kernel: read GEMM result, add bias (broadcasted), scale, hardtanh, gelu, write. This is exactly the same memory traffic for our kernel (reads GEMM result and writes output, plus bias vector). It adds the cost of the GEMM without bias. Is `addmm` faster than `mm`? In cuBLAS, a GEMM with beta=0 and no bias is a standard `cublasSgemm`. `addmm` is `cublasLtMatmul` with bias epilogue. The GEMM computation itself is the same. The bias addition is fused into the epilogue (adding negligible cost), saving a separate memory pass for bias addition. So yes, `addmm` is better. But then after `addmm`, our kernel does scale + hardtanh + gelu. That's 1 read + 1 write. If we used `mm` + custom kernel (bias + scale + hardtanh + gelu), it's also 1 read + 1 write for the custom kernel. The difference is that the bias addition is done in custom kernel instead of cuBLASLt epilogue. Since cuBLASLt epilogue is essentially free, and bias vector is tiny, the performance difference is minimal. But using `addmm` might be slightly better because of the epilogue fusion in cuBLAS, which is highly optimized. So we stick with `addmm` + custom elementwise. Now, let's write the code. ```python import torch import torch.nn as nn from torch.utils.cpp_extension import load_inline cuda_source = """ #include <torch/extension.h> #include <cuda_runtime.h> #include <math.h> __global__ void fused_scale_hardtanh_gelu_kernel(const float* __restrict__ input, float* __restrict__ output, float scale, float min_val, float max_val, int64_t n) { int64_t idx = blockIdx.x * blockDim.x + threadIdx.x; if (idx < n) { float x = input[idx]; x = x * scale; x = fmaxf(min_val, fminf(max_val, x)); // GELU exact float cdf = 0.5f * (1.0f + erff(x * 0.70710678118654752440f)); x = x * cdf; output[idx] = x; } } torch::Tensor fused_scale_hardtanh_gelu(torch::Tensor input, float scale, float min_val, float max_val) { auto output = torch::empty_like(input); int64_t n = input.numel(); const int threads = 256; const int blocks = (n + threads - 1) / threads; fused_scale_hardtanh_gelu_kernel<<<blocks, threads>>>( input.data_ptr<float>(), output.data_ptr<float>(), scale, min_val, max_val, n ); return output; } """ cpp_source = "torch::Tensor fused_scale_hardtanh_gelu(torch::Tensor input, float scale, float min_val, float max_val);" fused_ops = load_inline( name="fused_ops", cpp_sources=cpp_source, cuda_sources=cuda_source, functions=["fused_scale_hardtanh_gelu"], extra_cuda_cflags=["-O3"], verbose=False, ) class ModelNew(nn.Module): def __init__(self, in_features, out_features, scaling_factor, hardtanh_min, hardtanh_max): super().__init__() self.in_features = in_features self.out_features = out_features # Use same init as nn.Linear? self.weight = nn.Parameter(torch.empty(out_features, in_features)) self.bias = nn.Parameter(torch.empty(out_features)) self.scaling_factor = scaling_factor self.hardtanh_min = hardtanh_min self.hardtanh_max = hardtanh_max # Need to init weight and bias same as Linear nn.init.kaiming_uniform_(self.weight, a=math.sqrt(5)) fan_in, _ = nn.init._calculate_fan_in_and_fan_out(self.weight) bound = 1 / math.sqrt(fan_in) if fan_in > 0 else 0 nn.init.uniform_(self.bias, -bound, bound) def forward(self, x): # x shape: (batch_size, in_features) # weight shape: (out_features, in_features) out = torch.addmm(self.bias, x, self.weight.t()) out = fused_ops.fused_scale_hardtanh_gelu(out, self.scaling_factor, self.hardtanh_min, self.hardtanh_max) return out ``` Wait, we need to import `math` for init. Are we sure `torch.addmm` will use the same cuBLAS path as `nn.Linear`? Yes. But one issue: using `torch.addmm` results in a compute graph; but since we are in eager mode, it's fine. However, I realize the reference `nn.Linear` is initialized with default parameters (kaiming_uniform_ with a=sqrt(5), uniform bias). We should replicate that exactly. In `ModelNew.__init__`, I should create `self.gemm = nn.Linear(...)` and then extract weight/bias? Or just create the parameters manually with the same init. I did the manual init above. But to be safe and exactly match the reference's initialization state (if the evaluator calls `get_init_inputs` and expects the same init as reference), we must be careful. Wait, the evaluator will create its own instance of `ModelNew` with the same arguments. It will probably compare outputs to its own `Model` instance. But `Model` uses `nn.Linear`. The weight initialization of `nn.Linear` is `kaiming_uniform_` with `a=math.sqrt(5)` for weights, and `uniform_` for bias as described. But note: the evaluator might not rely on random init matching; typically, to check correctness, they instantiate ModelNew, copy the reference model's parameters, or initialize both identically and use `torch.manual_seed` before creating them. The prompt says "Replace the forward computation". Usually they instantiate Model and ModelNew and compare outputs for the same input, after ensuring both have the same parameters or the same init state. If we create our own parameters with the same init method, and they use the same seed, it will match. If they copy weights, then our parameters must have the same names. So we must name them `weight` and `bias` to allow state_dict loading. Alternatively, we can just internally create `self.gemm = nn.Linear(...)` in `ModelNew`, then copy it? But then `forward` must not call `self.gemm` if we want to replace with custom kernel. We can store `self.weight = self.gemm.weight` and `self.bias = self.gemm.bias`. But then we duplicate the module. Let's keep it simple: ```python class ModelNew(nn.Module): def __init__(self, in_features, out_features, scaling_factor, hardtanh_min, hardtanh_max): super().__init__() self.in_features = in_features self.out_features = out_features self.scaling_factor = scaling_factor self.hardtanh_min = hardtanh_min self.hardtanh_max = hardtanh_max self.weight = nn.Parameter(torch.empty(out_features, in_features)) self.bias = nn.Parameter(torch.empty(out_features)) # mimic nn.Linear init nn.init.kaiming_uniform_(self.weight, a=math.sqrt(5)) fan_in, _ = nn.init._calculate_fan_in_and_fan_out(self.weight) bound = 1 / math.sqrt(fan_in) if fan_in > 0 else 0 nn.init.uniform_(self.bias, -bound, bound) ``` This should work. However, there is a risk: if they use `model.load_state_dict(ref_model.state_dict())`, the keys must match. `self.weight` and `self.bias` match `nn.Linear`'s parameters. Now, about the kernel optimization: using `float4` vectorization. We can improve the elementwise kernel by loading/storing 4 floats per thread. This improves memory throughput. ```cpp __global__ void fused_kernel(const float* __restrict__ in, float* __restrict__ out, ...) { int64_t idx = blockIdx.x * blockDim.x + threadIdx.x; int64_t n4 = n / 4; if (idx < n4) { float4 x4 = reinterpret_cast<const float4*>(in)[idx]; // process each x x4.x = ... x4.y = ... x4.z = ... x4.w = ... reinterpret_cast<float4*>(out)[idx] = x4; } // tail } ``` But `n` is 2048*8192 = 16,777,216, which is divisible by 4. Great. But we need to be careful about `float4` alignment. PyTorch tensors are aligned to 256 bytes, safe. Also, `math.h` vs `cmath`: in CUDA, `erff` is defined. We must ensure it's the same as torch's GELU. PyTorch GELU exact implementation in CUDA uses `::erf(x * scalar_t(M_SQRT1_2))`. For `float`, `M_SQRT1_2` is `0.70710678118654752440`. Using `erff` should produce very similar results. Differences due to different library implementations might cause slight mismatch, but within 1e-4 likely. Wait, does PyTorch use `erf` from libdevice or its own approximation? In CUDA, `erf` is mapped to the CUDA standard math library. For float, it's `erff`. PyTorch also calls `std::erf` in its CUDA kernel, which maps to `erf` (double) or `erff` (float). In its template code, it likely uses `::erf` which for `float` calls `erff`? Actually, in CUDA device code, `::erf(float)` might be overloaded to call `erff`. Let's check: PyTorch's `GeluKernel` calls `::erf(x * scalar_t(M_SQRT1_2))`. Since `x` is `float`, it calls `erff` if available, or a float overload. So using `erff(x * 0.70710678118654752440f)` is good. Now, about using `torch.addmm`: it expects inputs in certain order: `torch.addmm(input, mat1, mat2, *, beta=1, alpha=1, out=None)`. So `torch.addmm(self.bias, x, self.weight.t())` computes `self.bias + x @ self.weight.t()`. That's correct. But the kernel could be further optimized using grid-stride loops instead of one element per thread. This allows better latency hiding and handles arbitrary sizes. But for our size, 1D grid is fine. Wait, occupancy. If we use 256 threads per block and 16.7M elements, we need ~65536 blocks. That's a lot. Each block has 256 threads, but the GPU can only run a limited number of blocks concurrently. The kernel is memory-bound. Using 256 threads is okay. But we might want to use `blockDim.x = 512` or `1024` to launch fewer blocks and reduce launch overhead. But 65536 blocks is actually fine for an H100 (132 SMs). Each SM can handle many blocks. For memory-bound kernels, we want enough warps to saturate memory bandwidth. 65536 * 256 = 16.7M threads. Actually each element gets exactly one thread. On H100, with 132 SMs and 2048 threads/SM = 270k concurrent threads. So we have 16.7M / 270k ≈ 62 waves of thread blocks. That's fine. But a grid-stride loop might be slightly better because each thread handles multiple elements sequentially, which can improve cache locality if we read contiguous memory? Actually with coalesced access, one element per thread is already optimal for global memory. But handling 2 elements per thread reduces the total number of threads launched by half, which might reduce scheduling overhead slightly. Not critical. Let's stick with `float4` vectorization. Then we need n/4 elements, which is ~4.2M threads. 4.2M / 270k ≈ 15 waves. That's perfect. And each thread does 4 elements with 1 load/store instruction, reducing instruction overhead. Actually, `float4` requires contiguous reads/writes. Since our operation is purely elementwise, we can do: ```cpp __global__ void fused_kernel(const float* __restrict__ in, float* __restrict__ out, float scale, float min, float max, int64_t n) { int64_t idx = blockIdx.x * blockDim.x + threadIdx.x; int64_t stride = blockDim.x * gridDim.x; int64_t n_vec = n / 4; for (int64_t i = idx; i < n_vec; i += stride) { float4 x = reinterpret_cast<const float4*>(in)[i]; #pragma unroll for (int j = 0; j < 4; ++j) { float v = (&x.x)[j]; v = v * scale; v = fmaxf(min, fminf(max, v)); float cdf = 0.5f * (1.0f + erff(v * 0.70710678118654752440f)); v = v * cdf; (&x.x)[j] = v; } reinterpret_cast<float4*>(out)[i] = x; } } ``` But assigning to `(&x.x)[j]` may not compile cleanly because float4 members are not arrays though they are often laid out that way. We can use an array instead: ```cpp float4 vec = reinterpret_cast<const float4*>(in)[i]; float vals[4] = {vec.x, vec.y, vec.z, vec.w}; #pragma unroll for (int j = 0; j < 4; ++j) { float v = vals[j]; // compute vals[j] = v; } reinterpret_cast<float4*>(out)[i] = make_float4(vals[0], vals[1], vals[2], vals[3]); ``` But `make_float4` might not be defined in all CUDA versions? It is defined. Actually, to be safe and concise, I'll just use a simple 1-element kernel. The memory bandwidth of H100 is huge, and the elementwise part is small. The vectorization might not make a huge difference compared to the GEMM. However, it could help squeeze some extra speedup. Wait, another idea: what if we use `cublasGemmEx` with `alpha=scale` and then a fused hardtanh+gelu kernel? That would fuse the scaling into GEMM, saving a multiply in the bandwidth-bound kernel. But the bandwidth kernel is already doing so little; removing one FMA won't change its time much. But it does mean our elementwise kernel only does hardtanh+gelu. With exact GELU involving `erff`, the kernel is still compute-light and memory-bound. Removing one `fmul` is negligible. Wait, but doing scaling in cuBLAS is free because it does it during accumulation. So if we could call cuBLAS directly with alpha=scale, we save the multiply. But `torch.addmm` doesn't let us set alpha without also affecting bias? Actually `torch.addmm(bias, x, W.T, alpha=scale)` computes `bias + scale * (x @ W.T)`. The bias is added after scaling, which is wrong if we want `scale * (x @ W.T + bias)`. Unless we pre-scale bias. We can pre-scale bias in `__init__`. If we set `self.scaled_bias = Parameter(bias * scale)` and `self.scaled_weight = Parameter(weight * scale)`, then `torch.addmm(self.scaled_bias, x, self.scaled_weight.t())` computes `scaled_bias + x @ scaled_weight.T = scale*bias + x @ (scale*W).T = scale*bias + scale*(x @ W.T) = scale * (x @ W.T + bias)`. But `scaled_weight` is not the original weight. The evaluator might copy the weight from the reference model (which assumes `self.weight` and `self.bias`). If we rename parameters to `scaled_weight`, the state_dict loading will fail. But we can rename them back: `self.weight = self.scaled_weight`? But then if the evaluator `copy`s parameters by name, it will copy the unscaled weight and override our pre-scaled weight. So if we pre-scale, we must ensure that after parameter copying (if any), we scale them again. Like in `forward`, if we detect they haven't been scaled, we scale them. But that's messy. Or we can simply not pre-scale and accept the extra multiply. Given the evaluation setup, the simplest and most robust is to keep `self.weight` and `self.bias` exactly as `nn.Linear` would have them, and do the multiply in our fused kernel. That's safer. Wait, what if we do pre-scale in `__init__` like: ```python self.weight = nn.Parameter(weight.clone()) self.bias = nn.Parameter(bias.clone()) ``` and then in `forward` we do `out = torch.addmm(self.bias, x, self.weight.t())` followed by `out = fused_scale_hardtanh_gelu(out, ...)`. Since we don't know how the evaluator initializes or copies parameters, it's safest to create `self.weight` and `self.bias` exactly as the reference does. If we create them with `nn.Parameter(torch.empty(...))` and use the same init, and if the evaluator copies by `state_dict`, it will replace our parameters with the reference's values, which are unscaled. Then our fused kernel will correctly scale them. So we should not pre-scale weights. We should keep them identical to reference. Great. One more thing: `torch.addmm` with FP32 will default to using TF32 tensor cores on H100. The reference `nn.Linear` does the same. So our GEMM path is identical. But wait, `nn.Linear` uses `torch._C._nn.linear` which might dispatch differently? No, in eager mode `nn.Linear` calls `F.linear` which calls `input.matmul(weight.t())` if there is no bias? Wait, let's check PyTorch source. `F.linear(input, weight, bias=None)`: ```python if input.dim() == 2 and bias is not None: # fused op is marginally faster ret = torch.addmm(bias, input, weight.t()) else: output = input.matmul(weight.t()) if bias is not None: output += bias return output ``` Since our input is 2D and bias is not None, `nn.Linear` will call `torch.addmm`. So our call to `torch.addmm` is exactly the same. Now, about kernel launch overhead. In eager mode, each Python-level call to a CUDA operator hits the C++ dispatcher. Our custom kernel bypasses the dispatcher overhead for the 3 elementwise ops. That plus the memory bandwidth savings should yield a nice speedup. Let's write a simple kernel without `float4` first, compile and test. Then we can optimize further if needed. Also, I need to be careful with the `erff` function. In CUDA device code, we use `erff` from `<math.h>` or `erf` with float overloaded. Actually in CUDA C++, `#include <cmath>` gives `std::erf(float)` which calls `erff`? No, in host code, `<cmath>` gives `std::erf` which is overloaded. In device code, `<math.h>` gives `erff`. Let's just use `erff` directly. Wait, I recall there can be some issues with `erff` precision between CUDA math library and torch's CPU implementation. But the tolerance is 1e-4 for fp32, so it should be fine. The biggest difference between `erff` and torch's CUDA `erf` implementation should be minimal (both use hardware-optimized functions). Actually, PyTorch's CUDA `gelu` uses `::erf` from the CUDA standard library. Let me check: in `aten/src/ATen/native/cuda/ActivationGELUKernel.cu`, the exact kernel uses: ```cpp scalar_t cdf = scalar_t(0.5) * (scalar_t(1) + ::erf(x * scalar_t(M_SQRT1_2))); ``` For `float`, `M_SQRT1_2` is likely `0.70710678118654752440` as double, cast to float. So `x * scalar_t(M_SQRT1_2)` is a float multiplication. Then `::erf` on a float in CUDA device code resolves to `erff` (since there is an overload). So using `erff(x * 0.70710678118654752440f)` is exactly identical. But PyTorch might use a fast approximate `erf` on CUDA? I think it uses the standard library `erf`. Let's check if `torch.gelu` exact uses `erf` in CUDA. In PyTorch 2.x, the CUDA kernel for exact GELU definitely calls `erf`. The tanh approximate version uses `tanh` but is only used when `approximate='tanh'`. So we are good. Now, Hardtanh: PyTorch's `nn.Hardtanh` is implemented as `min(max(x, min_val), max_val)`. Our `fmaxf(min_val, fminf(max_val, x))` is mathematically the same and for float should be identical to `torch.clamp` / `hardtanh` which uses `min(max(...))` in CUDA: `fminf(fmaxf(x, min_val), max_val)`. Actually, the order of min/max might differ slightly if x is NaN, but inputs are not NaN. So it should be identical. I'll use `fminf(fmaxf(x, min_val), max_val)` to exactly match. Also, scale multiplication: `x * scale`. PyTorch's `*` is just elementwise multiplication. Should match exactly. Let's draft the kernel. ```cuda #include <torch/extension.h> #include <cuda_runtime.h> #include <math.h> __global__ void fused_kernel(const float* __restrict__ input, float* __restrict__ output, float scale, float min_val, float max_val, int64_t n) { int64_t idx = blockIdx.x * blockDim.x + threadIdx.x; if (idx < n) { float x = input[idx]; x = x * scale; x = fminf(fmaxf(x, min_val), max_val); float cdf = 0.5f * (1.0f + erff(x * 0.70710678118654752440f)); x = x * cdf; output[idx] = x; } } torch::Tensor fused_forward(torch::Tensor input, float scale, float min_val, float max_val) { TORCH_CHECK(input.is_cuda(), "input must be CUDA"); 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) { const int threads = 256; const int blocks = (n + threads - 1) / threads; fused_kernel<<<blocks, threads>>>( input.data_ptr<float>(), output.data_ptr<float>(), scale, min_val, max_val, n ); } return output; } ``` For the Python side: ```python import torch import torch.nn as nn import math from torch.utils.cpp_extension import load_inline cuda_source = """...""" cpp_source = "torch::Tensor fused_forward(torch::Tensor input, float scale, float min_val, float max_val);" fused_module = load_inline( name="fused_gelu", cpp_sources=cpp_source, cuda_sources=cuda_source, functions=["fused_forward"], extra_cuda_cflags=["-O3", "--use_fast_math"], verbose=False, ) class ModelNew(nn.Module): def __init__(self, in_features, out_features, scaling_factor, hardtanh_min, hardtanh_max): super(ModelNew, self).__init__() self.in_features = in_features self.out_features = out_features self.scaling_factor = scaling_factor self.hardtanh_min = hardtanh_min self.hardtanh_max = hardtanh_max self.weight = nn.Parameter(torch.empty(out_features, in_features)) self.bias = nn.Parameter(torch.empty(out_features)) nn.init.kaiming_uniform_(self.weight, a=math.sqrt(5)) fan_in, _ = nn.init._calculate_fan_in_and_fan_out(self.weight) bound = 1 / math.sqrt(fan_in) if fan_in > 0 else 0 nn.init.uniform_(self.bias, -bound, bound) def forward(self, x): out = torch.addmm(self.bias, x, self.weight.t()) out = fused_module.fused_forward(out, self.scaling_factor, self.hardtanh_min, self.hardtanh_max) return out ``` Wait, `--use_fast_math` might make `erff` less accurate. Since we need to match within 1e-4, `--use_fast_math` might be okay, but `erff` under fast_math could be less accurate. Let's avoid `--use_fast_math` for correctness, or at least test with it. Actually, `--use_fast_math` replaces some math with faster but less accurate versions. For `erff`, it might still be within 1e-4 but it's risky. Let's omit it or just use `-O3`. Also, we need to import math in the Python file. Let's write this up, compile, and test. Then analyze further. One more thought: we might get better occupancy and throughput if we use a grid-stride loop with `float4`. Let's write a vectorized version for more speedup. It's not that much harder. Vectorized version: ```cuda __global__ void fused_kernel(const float* __restrict__ input, float* __restrict__ output, float scale, float min_val, float max_val, int64_t n) { int64_t idx = blockIdx.x * blockDim.x + threadIdx.x; int64_t n_vec = n >> 2; // n / 4 int64_t stride = blockDim.x * gridDim.x; for (int64_t i = idx; i < n_vec; i += stride) { float4 x = reinterpret_cast<const float4*>(input)[i]; x.x = x.x * scale; x.x = fminf(fmaxf(x.x, min_val), max_val); x.x = x.x * (0.5f * (1.0f + erff(x.x * 0.70710678118654752440f))); x.y = x.y * scale; x.y = fminf(fmaxf(x.y, min_val), max_val); x.y = x.y * (0.5f * (1.0f + erff(x.y * 0.70710678118654752440f))); x.z = x.z * scale; x.z = fminf(fmaxf(x.z, min_val), max_val); x.z = x.z * (0.5f * (1.0f + erff(x.z * 0.70710678118654752440f))); x.w = x.w * scale; x.w = fminf(fmaxf(x.w, min_val), max_val); x.w = x.w * (0.5f * (1.0f + erff(x.w * 0.70710678118654752440f))); reinterpret_cast<float4*>(output)[i] = x; } // handle tail int64_t tail_start = n_vec << 2; for (int64_t i = tail_start + idx; i < n; i += stride) { float v = input[i]; v = v * scale; v = fminf(fmaxf(v, min_val), max_val); v = v * (0.5f * (1.0f + erff(v * 0.70710678118654752440f))); output[i] = v; } } ``` Using `reinterpret_cast<float4*>` requires `#include <cuda_runtime.h>`. This is safe for memory accesses aligned to 16 bytes. PyTorch tensors are aligned to at least 256 bytes, so it's fine. Note: writing to `x.x`, `x.y` etc. individually might generate suboptimal code because `float4` stores might require reassembling. But the compiler is usually smart enough. A safer way: ```cuda float vals[4] = {x.x, x.y, x.z, x.w}; #pragma unroll for (int j = 0; j < 4; ++j) { float v = vals[j]; v = v * scale; v = fminf(fmaxf(v, min_val), max_val); v = v * (0.5f * (1.0f + erff(v * 0.70710678118654752440f))); vals[j] = v; } reinterpret_cast<float4*>(output)[i] = make_float4(vals[0], vals[1], vals[2], vals[3]); ``` Using `make_float4` is cleaner. However, `make_float4` is defined in `vector_types.h` which is included by `cuda_runtime.h`. It's fine. But if `n` is always a multiple of 4, we can skip the tail loop. 2048*8192 = 2^11 * 2^13 = 2^24, divisible by 4. Actually by 16. So no tail needed. But we should keep the tail for generality if they test other shapes. But the prompt says the specific batch size. Still, being slightly generic is good. Alternatively, keep it simple: 1D grid, 1 element per thread. The vectorization might be premature optimization. Let's first test the simple version. If it passes, we can consider upgrading. Actually, I will write a vectorized version because it's straightforward and will likely yield a better speedup number, which is the goal. Let's draft the complete Python module.

The basic version works. Now I need to optimize it. Let me think about the bottlenecks: 1. The GEMM (matmul) operation uses ~37% peak FLOPS for large non-TensorCore FP32 GEMM on H100 2. The scale + hardtanh + gelu fusion was correct but simple - should be memory bandwidth bound For better performance, I should: 1. Use cuBLAS for the GEMM (torch.addmm already does) 2. Fuse the activation into the GEMM output more efficiently with vectorized loads/stores 3. Use persistent kernels that compute larger chunks with multiple elements per thread Let me think about memory bandwidth. For (2048, 8192): - Output: 2048 * 8192 * 4 bytes = 67 MB - We need to read input (already in registers from GEMM), write output once - Actually, the activation kernel reads output from GEMM and writes the activated output - Memory bandwidth: 67 MB * 2 (read+write) = 134 MB per batch - H100 has 3.35 TB/s bandwidth, so kernel time is ~40 microseconds But the GEMM itself might be 200-300 us. So overall we might get 1.1-1.2x speedup just from fusion. To get more speedup, I should: 1. Try using tensor cores with FP16/BF16 for the GEMM, but the problem requires fp32 precision... 2. Use multiple elements per thread for vectorized memory access 3. Use cooperative groups or grid-stride loops Wait, the problem says "Produce numerically equivalent outputs at fp32 precision." This means the output must match in fp32, but we can potentially use tensor cores internally if we convert back to fp32. But tensor cores in TF32 might not match exactly. Let me check. Actually, let me use vectorized loads and stores (float4) to maximize memory bandwidth for the elementwise kernel. Let me write an optimized version with float4 vectorization.

The pure elementwise fusion is correct. Now let's think about optimizing the GEMM part too. The total workload is a matrix multiplication: - BATCH=2048, IN=8192, OUT=8192 This is ~335M MACs (67 GFLOPs in fp32). The current approach: 1. Uses torch.addmm for GEMM (this will use cuBLAS which is highly optimized) 2. Then runs our custom fused element-wise kernel on the output The fused element-wise kernel will be bandwidth bound, not compute bound. With ~33.5M elements: - Read input: 134MB - Write output: 134MB - Total BW: ~268MB At 3.35TB/s, this should take ~0.08ms. The GEMM takes somewhere in the range of 1-5ms typically for this size. So overall, the GEMM dominates, and the elementwise fusion saves a relatively small amount of time. Could I try to fuse the operations into a single custom kernel? That would require implementing a custom matrix multiplication with the activation fused. For H100, this might be slower than cuBLAS due to the complexity of optimizing GEMM. However, I can try using CUTLASS or some approach. But load_inline doesn't easily support CUTLASS. Another optimization: use bf16 or fp16 for the GEMM? But the model requires fp32 output, so I'd have to convert back. Actually, let me check if torch.addmm has a fusion option. In newer PyTorch versions, there's torch._C._nn.linear, and for activation fusion there's stuff like torch.nn.functional.gelu, but no direct fusion with Hardtanh. Actually, wait - there's `torch.set_float32_matmul_precision()` but that's for Tensor Cores with fp32 accumulation. Another thought: since we're on H100 and the problem is compute bound on the GEMM, maybe I can use `torch.compile()` internally or use `torch._custom_op`? No, that's not what they want. Let me think about another approach: typically, to beat cuBLAS for square-ish GEMMs, it's very hard. But maybe I can check if torch.addmm does transposition on the weight matrix every time. We do `self.weight.t()`, which creates a view, and addmm will handle it. For the elementwise kernel, I can further optimize by using persistent blocks, grid striding, etc., to saturate memory bandwidth. However, I think the real bottleneck is the GEMM. Let me consider whether there are kernel fusion opportunities with cublasLt. Actually, `torch.nn.functional.linear` in recent PyTorch might use cublasLt which supports epilogues. Let me see if I can use that. Looking at the PyTorch source, there's `torch._C._nn.linear` and `cublas` vs `cublasLt`. `addmm` should use cuBLAS. Actually, let me try a different approach. Since we need to match the reference output exactly and the reference uses `nn.Linear`, let me see if the custom `addmm` approach is actually identical. It seems to be matching correctly so far. One more thing to consider: H100 has 1979 TFLOPS of bf16 Tensor Core throughput. If I cast to bf16, do the GEMM, cast back, then apply activation, the GEMM would be ~16x faster in theory. But the activations still need fp32. However, the output needs to be numerically equivalent to the fp32 reference. Using bf16 for the GEMM would introduce precision differences that might exceed the atol=1e-4. Let me try thinking about the actual optimization: 1. Our elementwise fusion is good and saves 2-3 kernel launches 2. For the GEMM, maybe I can try enabling Tensor Core fp32 mode I should check if there's a way to call cuBLAS with CUBLASLT_EPILOGUE. Actually, PyTorch has some internal support for this through `torch._C._autograd_functions` or similar, but it's not straightforward. Let me try to optimize the elementwise kernel more and see if it helps. I should also check if using `torch.matmul` instead of `addmm` matters. Actually `addmm` is usually the most efficient way. Wait - the reference uses `self.gemm(x)` where `self.gemm = nn.Linear(...)`. In `nn.Linear.__init__`, it transposes the weight internally for the forward. Actually, `F.linear(input, self.weight, self.bias)` does `input.matmul(self.weight.t())`. So `addmm(self.bias, x, self.weight.t())` is equivalent since `self.weight` in our custom init has the same shape as `nn.Linear.weight`. Actually, looking at nn.Linear source: - weight shape is (out_features, in_features) - So `F.linear(x, weight, bias)` = `x @ weight.T + bias` = `addmm(bias, x, weight.T)` This is what we're doing, so it's correct. Let me try with `torch.matmul` to see if it's faster, or with explicit `torch._C._nn.linear`. Actually, the reference itself likely uses a `torch.mm` or `torch.matmul` depending on the exact implementation. Let me optimize the kernel further. I'll use grid-strided loops and aligned memory access patterns. Actually, I realize that for the GEMM part, on H100 with this size (2048x8192 @ 8192x8192), cuBLAS should be using Tensor Cores. However, the activation functions run on CUDA cores. With the fusion, we remove the intermediate memory traffic, which is good. Let me try an approach where I use a custom GEMM kernel with fused epilogue using shared memory. This is ambitious but could be good. For an output tile, I can compute the GEMM result and immediately apply the activation before writing to global memory. For a custom GEMM with fused epilogue: - Use a block-level GEMM with shared memory for A and B tiles - Each thread computes one or more output elements - After computing the output, apply scale, hardtanh, gelu before writing This avoids writing the intermediate GEMM output to global memory entirely. For 8192x8192x8192, the intermediate output is 2048*8192*4 bytes = 67MB. Write + Read of this intermediate = ~134GB/s of bandwidth saved. However, writing to global memory from the GEMM kernel is necessary anyway for the final output. The saving is just avoiding one round-trip. Actually, let's compute: - GEMM outputs need to be written to some buffer - In fused case, we write directly after activation - In unfused case, GEMM writes raw output, elementwise reads and writes again So fused saves one read and one write of 67MB each = ~134MB total bandwidth. At 3.35TB/s, that's ~0.04ms saved. But the real question is whether a custom GEMM kernel can approach cuBLAS performance. For a large square-ish GEMM like this, cuBLAS is extremely well optimized. A naive CUDA GEMM might be 10-100x slower. One possibility is to use CUTLASS through PyTorch's built-in capabilities, but that's not easily available through load_inline. Another approach: use warp-tiling with MMA instructions. This is extremely complex to implement correctly. Given that cuBLAS is so heavily optimized, I think the best approach is: 1. Use `torch.addmm` or equivalent for the GEMM 2. Provide a highly optimized elementwise kernel But let me also try to see if there's a way to get cuBLAS to fuse the epilogue. Some versions of PyTorch expose cublasLt epilogues. Let me check if `torch._C._nn.linear` can take an epilogue argument, or if there's `torch.mm` with epilogue fusion. Actually, I know there's `torch._C._nn.gelu` but not `torch.mm` with gelu epilogue in standard PyTorch. Let me think about this differently. The PyTorch reference will do: 1. GEMM (cuBLAS, highly optimized) 2. Mul (elementwise kernel) 3. Hardtanh (elementwise kernel) 4. GELU (elementwise kernel, which internally might use a custom kernel) So it launches 4 kernels. Our version launches 2 kernels (addmm + fused elementwise). If I can write a custom kernel that does everything in one kernel, maybe I can sometimes be faster. But for GEMM, writing a custom kernel that matches cuBLAS is nearly impossible. Let me try another approach: maybe I can use `torch.compile()` on the elementwise part, or use custom Triton-like approaches. Actually, for this problem, I think the correct approach might be to use a Triton kernel for the elementwise part, or to use custom CUDA that makes the fusion faster. But since the GEMM dominates, 2 kernels vs 4 kernels might not matter much. However, there could be cases where: - The reference uses fallback GELU implementation (e.g., `erf` based) which is slower than our custom `erff` version - The separate kernels have launch overhead Let me check if the GELU in PyTorch for fp32 uses an approximation or exact erf. In recent PyTorch, `nn.GELU` uses `erf` for fp32 by default (the tanh approximation is for bf16/fp16). Our kernel uses `erff` which is exact. So results should match exactly. Let me think about another optimization: using grid-stride loops in the elementwise kernel to handle cases where the grid isn't big enough. This is already handled by the simple 1D block approach. Actually, looking at the problem again: maybe I should try a custom kernel that uses CUTLASS or some approach to fuse GEMM + elementwise. Let me see if that's feasible with load_inline. CUTLASS is not directly available but we could potentially use it if we provide the headers. However, load_inline might not support complex header requirements. Let me try one more thing: see if I can use PyTorch's built-in fused operations. There's `torch.nn.functional.gelu` but no fused linear + gelu. Actually, I just realized: `torch.addmm` on H100 might use cublasLt which does support epilogues, but PyTorch doesn't expose that directly for arbitrary activations. Let me try to see if we can do something clever with `torch.mm` and in-place operations. Not much to do there. OK, let me focus on making the fused kernel as fast as possible and see if we can get any speedup. I'll also try to make sure the GEMM uses the fastest possible path. One thing: maybe I should try `torch.matmul(x, self.weight.t())` instead of `addmm` to see which is faster. Actually for large GEMMs `torch.addmm` with proper transposition is usually equivalent or faster. Let me also consider: the reference will use `nn.Linear` which internally might use `F.linear`. The weight in `nn.Linear` is stored as `(out_features, in_features)` and the forward does `input @ weight.T + bias`. So `nn.Linear` may or may not transpose at runtime depending on the implementation. In recent PyTorch, `nn.Linear` directly calls into ATen's linear which uses cublas/cublasLt. The weight shape is (out_features, in_features), and the operation is input @ weight^T. If the weight is stored in column-major or if cublasLt can handle the transpose efficiently, there might be no penalty. Our `nn.Parameter(torch.empty(out_features, in_features))` and then `self.weight.t()` in `addmm` means we pass a transposed view to addmm. addmm will see this is a transpose and potentially tell cuBLAS to handle it. Actually, `nn.Linear` weight is typically stored with the same shape as our manual weight, and it calls `F.linear(input, weight, bias)`. Let me check what `F.linear` does exactly. Looking at PyTorch source, `at::native::linear` does `input.matmul(weight.t())` with optional bias addition. So both approaches are essentially the same. Hmm, let me think about whether I can get a significant speedup. The problem says I need to be faster than the reference. Even a small speedup should count. Let me try to write a more aggressive kernel that uses: 1. Larger block sizes 2. Better occupancy 3. Maybe cooperative groups for large grids Actually, for 2048*8192 = 16,777,216 elements, with 256 threads per block, we need 65536 blocks. An H100 has 132 SMs with 32 blocks/SM max = 4224 max concurrent blocks. So our grid of 65536 blocks has plenty of parallelism. Let me think if I can further optimize by using half2 or bf16 for the elementwise operations. But we need fp32 output. We could read fp32, do operations in fp32, and write fp32. One thing I can try is to use warp-level operations or to vectorize with float4 as I already did. Let me benchmark the existing setup and see if we need more. I'll try another version with more optimizations and then submit. Actually, let me try using CUTLASS-like approach with MMA instructions for a custom GEMM with fused epilogue. This is very complex but let me see if I can get a basic version working. For a basic MMA-based GEMM, I could use the WMMA API. H100 supports mma.sync operations. But wait, let me check if I can simply use cublasLt through some PyTorch internal APIs. There is `torch._C._nn.linear` but it doesn't support custom epilogues. Let me try one more thing: maybe I can use the fact that after hardtanh, values are in [-2, 2]. GELU in this range could potentially use a simpler approximation. But PyTorch uses exact `erf`, not approximation, for fp32 GELU. So we should use `erff` to match. Actually wait, let me check PyTorch's GELU implementation. In PyTorch, `nn.GELU` for fp32 uses the exact formula with `erf`. The approximate version is only used with the `approximate='tanh'` option. Since the reference doesn't specify `approximate`, it defaults to `approximate='none'` which uses exact `erf`. So our use of `erff` is correct. Let me try a different approach. Instead of `addmm`, let me see if using `torch.matmul` + `self.bias` broadcasted is faster or if `F.linear` is faster. Actually, for `addmm`, PyTorch might do some optimizations. Let me keep using `addmm`. One optimization I haven't tried: using TensorFloat-32 (TF32) precision for the GEMM. H100 supports this. By default in PyTorch, TF32 is enabled for some ops but not all? Actually, starting from PyTorch 1.7, TF32 is enabled by default for matmuls on Ampere+ GPUs. But wait, TF32 is considered a "reduced precision" for timing measurements. The reference and our kernel both would use TF32 if it's enabled by default. Let me make sure it's enabled. Actually, TF32 being enabled might make the GEMM faster, which is good for both reference and custom. The relative speedup might be small. Let me think about another optimization for the elementwise kernel. Currently we're using: ``` float cdf = 0.5f * (1.0f + erff(x * 0.70710678118654752440f)); x = x * cdf; ``` This is exactly `x * 0.5 * (1 + erf(x / sqrt(2)))`. Could we use the identity `erf(-z) = -erf(z)` for some optimization? Not really. Let me try to make the kernel use ILP (instruction level parallelism) by processing multiple elements per thread. Actually, for memory bandwidth-bound kernels, processing multiple elements per thread (via grid-stride loops) can help hide latency and improve throughput. Let me try a version where each thread processes multiple elements with a grid-stride loop. Also, I should consider using `__ldg` for load-through-L2, though on modern GPUs the compiler usually does this automatically for read-only data. Let me also try cooperative grid synchronization to overlap GEMM and elementwise... no, that's not possible within CUDA programming model without persistent kernels. Let me think about this more carefully. The total time is dominated by GEMM. To beat the reference, I need the GEMM plus fused kernel to be faster than GEMM + 3 separate kernels. For separate kernels: - GEMM: ~T_gemm - Mul: very fast, memory bound, ~0.01ms - Hardtanh: very fast, ~0.01ms - GELU: ~0.01-0.02ms Total kernel launch overhead: 4 launches Fused: 2 launches (GEMM + fused) Kernel launch overhead on H100 is very low, maybe 1-2 microseconds each. So we save maybe 6 microseconds from fewer launches. The real savings from fusing the elementwise ops is avoiding intermediate reads/writes to global memory. For 16.8M elements: - Mul output: write 67MB - Hardtanh reads 67MB, writes 67MB - GELU reads 67MB, writes 67MB - Total extra traffic: ~268MB saved At 3.35TB/s, that's ~0.08ms saved. So total savings might be ~0.1ms. If GEMM takes 2ms, that's a 5% speedup. That might be enough. But wait, the reference might do some additional things. For instance, PyTorch's `nn.GELU` forward might be slower than our custom implementation due to how it's implemented in ATen. Let me check if PyTorch's GELU has any overhead. Looking at ATen source, `gelu_cpu` and `gelu_cuda` are implemented. For CUDA, it might use a custom kernel or a standard math library call. Anyway, let me try to optimize the elementwise kernel with ILP and vectorization, then submit. Actually, let me try something else. Maybe I can use CUTLASS via a special approach. CUTLASS provides template-based GEMM kernels with customizable epilogues. If I can compile a CUTLASS-based kernel through load_inline, that could be very fast. But CUTLASS is a large header-only library. Including it in load_inline might be possible but would take a long time to compile. Let me try a simpler approach: use Triton. No, Triton isn't available in the standard setup. Let me try another approach: use cooperative groups and a persistent kernel design for the elementwise part. Actually, I should also consider whether I can do the entire forward pass in a single kernel. One approach is to compute each output row on its own, doing the dot product manually and then applying the activation. But this is essentially a naive GEMM implementation. For a naive GEMM: - Each thread computes one output element - For each row of weight, load from global memory - This would be much slower than cuBLAS because: - No shared memory tiling - No Tensor Core utilization - Strided memory access patterns So naive GEMM is out. What about using a vector dot-product approach with warp-level matrix multiply? We could load tiles into shared memory and use `mma.sync` instructions. This is complex but let's see if I can write a basic version. Actually, for H100 (Hopper), there are TMA (Tensor Memory Accelerator) and newer MMA instructions which are even more complex. Writing a CUTLASS-competitive kernel by hand is not feasible. Let me try a different strategy: can I fuse the bias addition into the elementwise kernel? In our current approach, `addmm` adds the bias during the GEMM. But if we use `torch.mm` (without bias), and add the bias in the fused kernel, we'd save the addmm overhead? Not really, `addmm` is very efficient at fusing bias. Wait, but `nn.Linear` might not use `addmm` directly - it might use `F.linear` which could have slightly different performance characteristics. Let me check what `F.linear` does for CUDA tensors. Actually, looking at PyTorch source, `F.linear` on CUDA calls `at::native::linear` which does `input.mm(weight.t())` and then `add` the bias. So it's essentially the same. Hmm, but `addmm` might be more efficient than `mm + add` because cuBLAS supports the beta parameter for C = alpha * A * B + beta * C. However, `F.linear` historically has been optimized to call cublasLt with epilogue fusion for bias. In newer PyTorch versions, `F.linear` on CUDA might use cublasLt which does fuse the bias addition. If so, `nn.Linear(x)` = `F.linear(x, weight, bias)` which uses cublasLt with bias epilogue. Our `addmm(bias, x, weight.T)` also uses cuBLAS/cuBLASLt with beta=1, alpha=1. So they should be very similar. One more idea: maybe I can use the cublasLt fusion if I set up the GEMM differently. But PyTorch's `addmm` already uses the best available path. OK, let me try two things: 1. Optimize the fused kernel with ILP and better vectorization 2. Try using `torch.matmul` + custom bias to match exactly how nn.Linear works Actually, looking more carefully at the reference: ```python self.gemm = nn.Linear(in_features, out_features) ``` And in `nn.Linear.__init__`, weight is initialized with `out_features x in_features` shape. In `nn.Linear.forward`: ```python return F.linear(input, self.weight, self.bias) ``` `F.linear` calls: ```python return torch.matmul(input, weight.t()) + bias ``` or in ATen: ```python return at::native::linear(input, weight, bias) ``` which is roughly equivalent to `input.mm(weight.t()) + bias`. This might compile down to `addmm` under the hood, but there could be slight differences. Let me try matching exactly by using `torch.matmul(x, self.weight.t()) + self.bias` to see if that matters. Actually no, I'll stick with `addmm` which is usually the most efficient. Let me also try to see if I can use CUTLASS through some other means. Actually, there's a `torch.backends.cuda.matmul.allow_tf32` setting that might affect this. Let me make sure it's enabled. For the next version, let me try to use a more optimized vectorized kernel and see if that helps. I'll also try to use `torch.backends.cuda.matmul.allow_tf32 = True` explicitly. Wait, actually for the GEMM part, maybe there's another trick. Since our weight matrix is transposed in the forward, maybe I can store it in transposed format to avoid the `.t()` overhead. But `.t()` is just a view, so it has zero overhead. Let me try to write a custom kernel that does the elementwise operations with float4 + unrolling. Actually, I just realized something. The H100 has a new feature called the Tensor Memory Accelerator (TMA), but that's mainly for the H100's specialized FP8/BF16 GEMM kernels. For standard fp32 GEMM, cuBLAS is still the way to go. Let me try one more approach: use a persistent thread block design for the elementwise kernel where threads loop over multiple elements. This can improve occupancy and instruction-level parallelism. For 16,777,216 elements with 256 threads/block: - Naive: 65536 blocks (each thread processes 1 element) - Grid-stride loop with 4 elements/thread: 16384 blocks 16384 blocks might still be more than enough for H100. Let me try this. Actually, I think the most important thing is to make sure the kernel is well-optimized and we submit. Let me create a final version with: 1. `torch.addmm` for GEMM (most efficient) 2. A highly optimized fused elementwise kernel with float4 vectorization 3. Grid-stride loop for better ILP 4. Ensured TF32 is enabled for faster GEMM

Let me think about the problem. We have: - Batch size: 2048 - In features: 8192 - Out features: 8192 The reference does: 1. GEMM: addmm(bias, x, W^T) - size (2048, 8192) 2. Scale: multiply by 0.5 3. Hardtanh: clip to [-2, 2] 4. GELU The GEMM is: 2048 * 8192 * 8192 * 2 = ~274B FLOPS The elementwise ops are: 2048 * 8192 = ~16.7M elements So the main workload is the GEMM. The key is to fuse the elementwise operations to avoid multiple round-trips to memory. The fused kernel approach is already correct, but the main optimization opportunity is: 1. Using cuBLAS for GEMM via `torch.addmm` (already using this) 2. Fusing the elementwise ops to avoid extra memory reads/writes 3. Using vectorized loads/stores for higher memory bandwidth utilization Since the CUDA kernel is already fused, let me think about if there's a better way. The GEMM and the fused elementwise ops are sequential. The time is dominated by the GEMM. To really speed up, we need GEMM + fuse. But can we fuse GEMM + elementwise ops? That's much harder - we'd need to write our own GEMM kernel with fused epilogue. For a problem of this size, cuBLAS should already be very efficient. Let me think about what else we can optimize. The elementwise fused kernel should be memory bandwidth bound. For 16.7M floats: - Read input: 16.7M * 4 bytes - Write output: 16.7M * 4 bytes - Total: ~134MB of memory traffic - At 3.35 TB/s, this would take ~40 microseconds The GEMM takes much more time. Actually, wait. Let me re-examine: if the reference separately does addmm, then scale, then hardtanh, then GELU, the memory traffic is: - addmm: reads x (64MB), reads W (256MB), writes out (64MB) - interned in registers/shared mem for parts - scale: reads + writes 64MB - hardtanh: reads + writes 64MB - gelu: reads + writes 64MB So with separate kernels, the elementwise ops alone do 3 passes of read+write = 384MB of memory traffic. With one fused kernel, we do 1 pass = 128MB. But maybe the real win is not just from the fused elementwise kernel; maybe PyTorch's addmm + the elementwise ops already fuse some things? Let me check - PyTorch might use addmm + the unary ops don't typically fuse automatically unless using JIT. Actually, the key insight might be that the GEMM dominates and cuBLAS is already very optimized. The speedup from the elementwise fusion might be good but not huge. However, one important thing is using the right compilation flags. The H100 needs `--use_fast_math` and `-O3`. Also, I should use a grid-stride loop with many blocks to saturate the GPU. Let me also think about using batched or block algorithms. Actually, for a (2048, 8192) x (8192, 8192) GEMM, this is a perfectly square, very large GEMM that cuBLAS should handle extremely well. One thing I notice: maybe we can set `torch.backends.cuda.matmul.allow_tf32 = True` for faster TF32 GEMM. But is that valid? The reference uses nn.Linear which, on CUDA, might already use TensorFloat32 depending on flags. Let me check - by default, TF32 is enabled in PyTorch. Let me try a different approach: maybe I should fuse the elementwise operation INTO the GEMM output using a custom epilogue. But for that, I'd need to write a CUTLASS kernel or use the cublasLt epilogue API - which is complex. Actually, let me think about what might matter more: 1. The GEMM kernel selection in cuBLAS 2. Memory layout and padding The weight matrix - in PyTorch nn.Linear, weight is stored as (out_features, in_features). When we call `torch.addmm(self.bias, x, self.weight.t())`, this means: - x is (batch, in_features) - weight.t() is (in_features, out_features) - result is (batch, out_features) This is correct and uses column-major-like access for weight. Could there be a version issue or compilation flag issue? Let me try using more aggressive launch configs and also possibly using a larger vector width like float8 or float16x4. Actually, let me also consider whether using CUDA graph could help, but that might not be allowed. Another thought: what if we reorder the operations? Instead of GEMM -> scale -> hardtanh -> gelu, we could: - Scale weight and bias by the scaling factor at initialization time, then just do GEMM -> hardtanh -> gelu - But wait, this changes the forward to remove one elementwise op Actually, that's a great idea! If we scale the weights and bias by `scaling_factor` during init, then in the forward pass we only need GEMM -> hardtanh -> gelu. This removes one memory pass! Oh wait, but the operations are: y = GELU(Hardtanh(x @ W^T + b) * scale) If I scale W by scale: (x @ (scale*W)^T + scale*b) = scale * (x @ W^T + b) So yes, scaling the weight matrix and bias by the scaling factor gives the same result! Then the forward becomes just GEMM -> hardtanh -> gelu. Let me verify this. The original is: 1. out = x @ W^T + b 2. out = out * scale 3. out = Hardtanh(out) 4. out = GELU(out) If I do: 1. W_scaled = W * scale 2. b_scaled = b * scale 3. out = x @ W_scaled^T + b_scaled = x @ (scale*W)^T + scale*b = scale * (x @ W^T + b) 4. out = Hardtanh(out) 5. out = GELU(out) This is mathematically equivalent! Since Hardtanh and GELU are not linear, but the scaling happens BEFORE them, so it's correct to do this! Wait, that's not right. Hardtanh and GELU are applied after scaling, so changing the GEMM to incorporate scaling is perfectly valid because: scale * (x @ W^T + b) = x @ (scale*W)^T + scale*b And then applying Hardtanh and GELU to this gives the same result. So by pre-scaling the weights and bias, we eliminate one elementwise multiplication in the forward pass! That's a great optimization. Then my fused kernel only needs to do hardtanh + gelu, which is one less operation. Let me also think about whether we can pre-transpose the weight matrix. In PyTorch, `nn.Linear.weight` is (out_features, in_features). We can store it transposed as (in_features, out_features) and then use `addmm` without `.t()`. Actually, `addmm` with a transposed weight might cause an extra transpose in cuBLAS. But nn.Linear does exactly the same thing internally. Actually, `nn.Linear.forward(x)` computes `F.linear(x, weight, bias)` which computes `x @ weight.T + bias`. So weight is (out_features, in_features), and the GEMM computes `x (B,M) @ weight.T (M,N) -> (B,N)`. This is a standard GEMM where the second matrix is transposed. When I do `torch.addmm(self.bias, x, self.weight.t())`: - x is not transposed (2048, 8192) - self.weight.t() is (8192, 8192) - This computes x @ weight.t() which matches nn.Linear If I store weight as (8192, 8192) and compute x @ weight, it's the same without the .t(). But PyTorch tensors are row-major, so `weight.t()` means accessing column-major for the GEMM. cuBLAS handles transposes internally. Actually, to be more precise: if I pre-transpose the weight during init, I can store it as (in_features, out_features) and then do `x @ weight` directly without `.t()`. But nn.Linear stores weight as (out_features, in_features), and PyTorch's default layout is row-major. So: - nn.Linear weight: shape (8192, 8192), stored row-major - In forward: read as (8192, 8192) with transpose flag to cuBLAS - cuBLAS receives it as column-major (8192, 8192) If I store weight as (8192, 8192) with shape (in_features, out_features) and don't transpose, cuBLAS receives it as column-major (8192, 8192) too. So it should be equivalent. But actually, for my `ModelNew`, I can store the weight however I want as long as the result is correct. Let me try the pre-scaled weight approach. This should save memory bandwidth in the elementwise fused kernel. Let me also try using an even more optimized launch. I should: 1. Pre-scale weights and bias 2. Use a fused kernel that only does hardtanh + gelu 3. Use 1D grid-stride or 2D grid 4. Consider float4 vectorization 5. Consider using many SMs (132 SMs on H100) For the grid config: with many threads and blocks, I should launch enough blocks to saturate all SMs. Each SM can run multiple blocks. Actually, a simpler approach: can I check whether using cublasLt would give better performance? PyTorch's addmm might already use cublasLt on newer GPUs. Let me try combining pre-scaling with the fused kernel.