reasoning
Let me analyze this problem.
We're replacing `F.elu(x, alpha=self.alpha)` with a custom CUDA kernel.
- Input: `x` is shape (4096, 393216) = 1,610,612,736 elements of fp32
- That's about 6.44 GB of data (1.61B * 4 bytes)
- ELU formula: `x if x > 0 else alpha * (exp(x) - 1)`
- Memory bandwidth: 3.35 TB/s = 3350 GB/s
- Memory-bound operation (elementwise), so peak throughput ~3350/6.44 = ~520 GB batches per second. Each batch reads 6.44GB input and writes 6.44GB output = 12.88GB. At 3350 GB/s, that's 3350/12.88 = 260 batches/sec, or about 3.84ms.
Actually, let's be more precise.
- Total elements: 4096 * 393216 = 1,610,612,736
- Input size: 1,610,612,736 * 4 bytes = 6,442,450,944 bytes ≈ 6.0 GiB
- A simple kernel reads 4 bytes per element and writes 4 bytes per element, total 8 bytes per element.
- Total memory traffic: 1,610,612,736 * 8 = 12,884,901,888 bytes ≈ 12 GiB
- Time = 12.88 GB / 3350 GB/s = 3.84 ms
Wait, but we can fuse this in-place or do it more efficiently. In fact, the pytorch reference might run fairly quickly already. Let's see what PyTorch does.
Actually, wait - the input is (4096, 393216). Let's write a simple fused kernel and see how fast it is.
For memory-bound kernels on H100, the key optimizations are:
1. Use vectorized loads/stores (float4 or float2)
2. Use enough threads and blocks to saturate memory bandwidth
3. Keep the kernel simple to minimize register usage and maximize occupancy
Let me write an optimized ELU kernel using float4 vectorization.
Actually, 393216 = 384 * 1024. Let me check: 393216 / 4 = 98304, which is divisible by 4, so we can use float4.
Float4: each thread processes 4 elements.
Total threads needed = 1,610,612,736 / 4 = 402,653,184
Each block with 256 threads handles 1024 elements.
Number of blocks = 402,653,184 / 256 = 1,572,864
Actually, 402,653,184 / 256 = 1,572,864. That seems like a lot of blocks. But H100 has many SMs and can handle it.
Alternatively, we can have:
- grid_size = (num_elements + block_size * vec_size - 1) / (block_size * vec_size)
- block_size = 256 or 512
- vec_size = 4 (float4)
Let me write a vectorized kernel.
Wait, I should also check if `alpha` is always 1.0 or configurable. Looking at the model, it's configurable in the constructor.
Let me write a straightforward vectorized kernel. The H100 has very high memory bandwidth, so for a simple elementwise op, vectorized memory access is the key.
```python
import torch
from torch.utils.cpp_extension import load_inline
cuda_source = '''
#include <torch/extension.h>
#include <cuda_runtime.h>
template <typename scalar_t>
__global__ void elu_kernel(const scalar_t* __restrict__ input,
scalar_t* __restrict__ output,
int64_t n,
float alpha) {
int64_t idx = blockIdx.x * blockDim.x + threadIdx.x;
if (idx < n) {
scalar_t x = input[idx];
output[idx] = (x > (scalar_t)0) ? x : alpha * (exp(x) - (scalar_t)1);
}
}
torch::Tensor elu_cuda(torch::Tensor input, float alpha) {
auto output = torch::empty_like(input);
int64_t n = input.numel();
AT_DISPATCH_FLOATING_TYPES(input.scalar_type(), "elu_cuda", [&] {
int threads = 512;
int blocks = (n + threads - 1) / threads;
elu_kernel<scalar_t><<<blocks, threads>>>(
input.data_ptr<scalar_t>(),
output.data_ptr<scalar_t>(),
n,
alpha
);
});
return output;
}
'''
cpp_source = '''
torch::Tensor elu_cuda(torch::Tensor input, float alpha);
'''
```
Let me try with vectorized float4 first for more bandwidth:
Actually, I can make it simpler and just use a basic approach, compile, and see. Then optimize from there.
Let me write a vectorized kernel using float4. For H100, vectorized loads/stores on fp32 are critical for achieving peak bandwidth.
Actually, wait - PyTorch uses its own highly optimized elementwise kernels via the TensorIterator. PyTorch's performance should already be quite good. But maybe we can beat it with better vectorization and fewer overheads.
Wait, looking more carefully: PyTorch's ELU is likely dispatched through ATen's native implementation, which may not be perfectly optimized for this exact size. But with 1.6B elements, it should be memory-bandwidth limited.
For H100, the theoretical fp32 memory bandwidth is 3.35 TB/s. For an elementwise kernel with read+write, that's 12.88 GB / 3.35 TB/s = 3.84 ms.
Actually, H100 has much higher effective bandwidth than A100. With float4 vectorization on contiguous memory, we might get very close to theoretical peak.
But wait - I should also consider doing an in-place operation or reusing memory. The reference does `F.elu(x)` which allocates a new tensor. We're doing the same.
Actually, PyTorch uses TensorIterator which automatically vectorizes. So we'd need to be quite clever to beat it.
One optimization: PyTorch's ELU goes through the ATen dispatch, which has some overhead. For very large tensors like this, that overhead is negligible. The kernel itself is what matters.
But maybe we can use a custom kernel that's better optimized. On H100, there are a few things to consider:
1. Thread block size - 512 or 1024
2. Grid size - enough to saturate all SMs
3. Vector loads
4. Occupancy
Let me try a vectorized float4 kernel first.
Actually, a key question is: can we use `__nv_bfloat16` or `__half`? The reference is fp32, and the tolerance for fp32 is 1e-4. ELU involves exp(), which might need high precision. But the problem says "produce numerically equivalent outputs at fp32 precision". This likely means we need to compute in fp32. We can't just cast to fp16 and back.
Wait, we can try using `__expf` intrinsic instead of `expf` for a small speedup. The `__expf` intrinsic is faster but less accurate. With fp32 and tolerance 1e-4, `__expf` should be fine for most inputs since x <= 0 in the path where exp is used. For x in [-1, 0], `__expf` error is small. For x << 0, the absolute error matters less.
Actually let me be more careful. `__expf` has maximum error around 2-3 ULPs. For fp32, that's around 1e-7 relative. So absolute error could be around 1e-7 * exp(x). For x near 0, that's about 1e-7. The tolerance is 1e-4, so __expf should be fine.
Also, in the reference, PyTorch might use `std::exp` which could map to `expf` in libdevice. Using CUDA intrinsic `__expf` might give a small speedup.
But the real win is probably in vectorization and thread configuration.
Let me write a vectorized kernel that handles 4 float elements per thread using float4.
Wait, actually, I should check if 393216 * 4096 is divisible by 4... 393216 is divisible by 4 (393216 / 4 = 98304). 4096 is divisible by 4. So the total number of elements is divisible by 4. This means we don't need to handle tails for float4! That's great.
So we can write:
```cuda
__global__ void elu_kernel_vectorized(const float* __restrict__ input,
float* __restrict__ output,
int64_t n,
float alpha) {
int64_t idx = (blockIdx.x * blockDim.x + threadIdx.x) * 4;
if (idx < n) {
float4 in = reinterpret_cast<const float4*>(input)[(blockIdx.x * blockDim.x + threadIdx.x)];
float4 out;
out.x = (in.x > 0.0f) ? in.x : alpha * (__expf(in.x) - 1.0f);
out.y = (in.y > 0.0f) ? in.y : alpha * (__expf(in.y) - 1.0f);
out.z = (in.z > 0.0f) ? in.z : alpha * (__expf(in.z) - 1.0f);
out.w = (in.w > 0.0f) ? in.w : alpha * (__expf(in.w) - 1.0f);
reinterpret_cast<float4*>(output)[(blockIdx.x * blockDim.x + threadIdx.x)] = out;
}
}
```
This requires input and output to be aligned to 16 bytes. Since they're freshly allocated tensors from PyTorch, they should be aligned. And total elements divisible by 4.
Wait, PyTorch tensor allocation is typically aligned to 512 bytes or so. So float4 alignment should be fine.
Let me also make sure we handle n correctly. `n / 4` is the number of float4 elements.
One more thing: using `__expf` instead of `expf` (or `std::exp`). PyTorch probably doesn't use the intrinsic, so this could be faster.
Actually, I want to check if PyTorch does in-place vs allocating. The reference returns `F.elu(x, alpha=self.alpha)` which allocates a new output tensor. Our kernel does the same.
Let me also consider unrolling - processing 8 or more elements per thread. On H100, more elements per thread can improve instruction-level parallelism and reduce block count.
Actually, for such a large tensor with 1.6B elements:
- With 4 elements/thread and 512 threads/block: 1.6B / 4 / 512 = ~786,432 blocks
- That might be a lot, but H100 can handle it with persistent thread blocks.
Wait, each SM can run at most 32 blocks. With ~132 SMs on H100 (actually H100 SXM has 132 SMs), max concurrent blocks is 132 * 32 = 4224 blocks. But blocks execute very quickly on memory-bound kernels. With 786K blocks, each block processes only a small amount of data.
Actually for bandwidth-bound kernels, having more blocks is fine because the GPU can quickly cycle through them. But there's launch overhead. Let me calculate:
- Each block does 512 * 4 * 8 bytes = 16,384 bytes of I/O
- At 3.35 TB/s, each block takes ~4.9 nanoseconds
- 786K blocks * 4.9ns = 3.84ms total
So the math checks out. But the block launch overhead might be significant.
Actually let me reconsider. Maybe using float4 with 256 threads per block would be good:
- 256 threads * 4 = 1024 elements per block
- Total blocks = 1.6B / 1024 = 1,572,864
Hmm, that's a lot of blocks. Maybe processing 8 elements per thread is better.
Or we could use a grid-stride loop: have each thread process multiple elements with a stride across the grid. This is often recommended for elementwise kernels to handle arbitrary sizes and reduce overhead.
```cuda
__global__ void elu_kernel(const float* __restrict__ input,
float* __restrict__ output,
int64_t n,
float alpha) {
int64_t idx = blockIdx.x * blockDim.x + threadIdx.x;
int64_t grid_size = blockDim.x * gridDim.x;
for (; idx < n / 4; idx += grid_size) {
float4 in = reinterpret_cast<const float4*>(input)[idx];
float4 out;
out.x = (in.x > 0.0f) ? in.x : alpha * (__expf(in.x) - 1.0f);
// ...
reinterpret_cast<float4*>(output)[idx] = out;
}
}
```
With a grid-stride loop, we can launch fewer blocks. For instance, launch enough blocks to saturate SMs and let each thread process multiple elements.
How many blocks? For H100, we'd want at least as many warps as can be concurrently executing. With 132 SMs and maybe 16 warps per SM (for high occupancy), that's 2112 warps. With 32 threads per warp, that's 67,584 threads. If each block is 256 threads, that's ~264 blocks minimum. But we'd want more for good latency hiding.
A common choice is 256 or 512 blocks. Let's say 576 blocks (a multiple of SMs, ~4-5 blocks per SM). With 512 threads/block, that's 294,912 threads total.
Each thread processes: 1.6B / 294,912 ≈ 5464 elements. With float4, that's ~1366 iterations per thread in the grid-stride loop.
Actually, for a memory-bound kernel with very large data, the grid-stride loop is efficient because the compiled code is small and the memory access pattern is fully coalesced.
Let me also try launching more blocks with smaller thread counts. H100 has high sustained memory throughput even with fewer iterations per thread.
Actually, let me think about this differently. PyTorch's TensorIterator dispatches to a kernel that likely uses vectorized loads and a similar grid-stride pattern. The main difference we can make is:
1. Using `__expf` intrinsic
2. Optimizing block/grid sizes specifically for this GPU
3. Avoiding any branching/conditionals that TensorIterator might have
Actually, PyTorch compiles its kernels ahead of time and they are very well optimized. The `__expf` intrinsic might be the main win. Also, we can fuse alpha * (exp(x) - 1) more efficiently.
Let me just write the kernel, compile it, and see if it passes correctness. Then we can iterate.
Wait, I want to test if alpha is a constant at compile time. Since it comes from Python to the CUDA extension, it's passed as a runtime float. But maybe we can use a template? Actually no, we need it to be configurable.
One more optimization: we can compile with `-O3` and `--use_fast_math`. `load_inline` has a `extra_cuda_cflags` parameter. `--use_fast_math` would make `expf` map to `__expf` automatically, but might affect precision. Let me just use `__expf` explicitly to have more control.
Actually, PyTorch by default compiles its kernels without `--use_fast_math` (I think). So using `__expf` explicitly while keeping the compiler flags standard might be a win.
Let me write the code:
```python
import torch
from torch.utils.cpp_extension import load_inline
CUDA_SOURCE = '''
#include <torch/extension.h>
__global__ void elu_kernel(const float* __restrict__ input,
float* __restrict__ output,
int64_t n,
float alpha) {
int64_t idx = blockIdx.x * blockDim.x + threadIdx.x;
int64_t grid_size = gridDim.x * blockDim.x;
#pragma unroll 4
for (int64_t i = idx; i < n / 4; i += grid_size) {
float4 in = reinterpret_cast<const float4*>(input)[i];
float4 out;
out.x = in.x > 0.0f ? in.x : alpha * (__expf(in.x) - 1.0f);
out.y = in.y > 0.0f ? in.y : alpha * (__expf(in.y) - 1.0f);
out.z = in.z > 0.0f ? in.z : alpha * (__expf(in.z) - 1.0f);
out.w = in.w > 0.0f ? in.w : alpha * (__expf(in.w) - 1.0f);
reinterpret_cast<float4*>(output)[i] = out;
}
// Handle remainder (shouldn't happen for this specific size, but for safety)
int64_t remainder_start = (n / 4) * 4;
int64_t r_idx = remainder_start + threadIdx.x + blockIdx.x * blockDim.x;
if (r_idx < n) {
float x = input[r_idx];
output[r_idx] = x > 0.0f ? x : alpha * (__expf(x) - 1.0f);
}
}
torch::Tensor elu_cuda(torch::Tensor input, float alpha) {
auto output = torch::empty_like(input);
int64_t n = input.numel();
const int threads = 512;
const int blocks = 1152; // about 8-9 blocks per SM on H100
elu_kernel<<<blocks, threads>>>(
input.data_ptr<float>(),
output.data_ptr<float>(),
n,
alpha
);
return output;
}
'''
CPP_SOURCE = '''
torch::Tensor elu_cuda(torch::Tensor input, float alpha);
'''
mod = load_inline(
name='elu_cuda',
cpp_sources=CPP_SOURCE,
cuda_sources=CUDA_SOURCE,
functions=['elu_cuda'],
extra_cuda_cflags=['-O3'],
verbose=False
)
```
Wait, there's an issue with the remainder handling. The grid-stride loop for float4 and the remainder loop might have conflicting thread assignments. Let me fix this.
Actually, if `n` is divisible by 4, the remainder section won't execute for any thread in the main grid-stride loop, but the condition `r_idx < n` could still be satisfied by some threads that didn't execute the float4 loop. However, with the grid-stride loop, each thread might execute the float4 loop 0 or more times. The remainder handling should be separate and only run once.
A cleaner approach: only have threads handle the remainder if they would be in the "last" set. But with grid-stride, it's simpler to just write two separate kernels or handle it more carefully.
Actually, let me simplify. Since we know `batch_size * dim = 4096 * 393216 = 1,610,612,736` which is divisible by 4 (393216 is divisible by 4), we don't need remainder handling at all for this specific problem. But for safety, let's include it properly.
A cleaner approach:
```cuda
__global__ void elu_kernel(const float* __restrict__ input,
float* __restrict__ output,
int64_t n,
float alpha) {
int64_t idx = (blockIdx.x * blockDim.x + threadIdx.x);
int64_t n_vec = n / 4;
int64_t grid_size = gridDim.x * blockDim.x;
for (int64_t i = idx; i < n_vec; i += grid_size) {
float4 in = reinterpret_cast<const float4*>(input)[i];
float4 out;
out.x = in.x > 0.0f ? in.x : alpha * (__expf(in.x) - 1.0f);
out.y = in.y > 0.0f ? in.y : alpha * (__expf(in.y) - 1.0f);
out.z = in.z > 0.0f ? in.z : alpha * (__expf(in.z) - 1.0f);
out.w = in.w > 0.0f ? in.w : alpha * (__expf(in.w) - 1.0f);
reinterpret_cast<float4*>(output)[i] = out;
}
// Handle tail
int64_t tail_start = n_vec * 4;
int64_t tail_idx = tail_start + idx;
if (tail_idx < n) {
float x = input[tail_idx];
output[tail_idx] = x > 0.0f ? x : alpha * (__expf(x) - 1.0f);
}
}
```
This is clean. Even if n is divisible by 4, the tail code won't execute because `tail_start = n` so `tail_idx >= n`.
Let me also consider: should we use AT_DISPATCH? The input is guaranteed to be float32, but it's good practice. For simplicity and since the reference is fp32, we can check the dtype and dispatch. But for max performance on this specific input, we can just handle float.
Actually, looking at the reference code, the input is created with `torch.rand` which defaults to fp32. So we can write a float-specific kernel, but it's safer to dispatch. However, dispatch adds a tiny bit of overhead which is negligible at this scale.
Let me use a type dispatch to be safe but primarily target float. Actually, for max simplicity and since the inputs are fp32, let me just write it for float. We can add dispatch for correctness on different dtypes.
Actually, looking at the evaluation, they might test with different shapes of fp32. The dtype is fixed as fp32. So a float-specific kernel is fine.
Wait, is it? Let me check the model forward: `x: torch.Tensor` - no dtype specified. The `get_inputs()` returns `torch.rand(...)` which is fp32. The tolerance mentioned is "fp32 uses atol=rtol=1e-4". So yes, inputs are fp32.
Let me write the kernel more carefully.
Actually, I realize I should also consider whether launching 1152 blocks is optimal. H100 has 132 SMs (maybe 114 for H100 NVL or 132 for SXM5). Let me check. The spec just says "H100" with 80GB HBM3. H100 SXM5 has 132 SMs.
For a memory-bound kernel, we want to saturate memory bandwidth. The number of blocks isn't super critical as long as there's enough thread-level parallelism. 1152 blocks of 512 threads = 589,824 threads. With 132 SMs, that's ~4464 threads per SM, or ~140 warps per SM. Max warps per SM on H100 is 64 (actually with 2048 threads / 32 = 64 warps). So we can have more concurrent warps than physically simultaneous, but that's fine - the GPU will time-slice.
Actually, maximum threads per SM on Hopper is 2048, which is 64 warps. So 140 warps per SM means each SM has 140 warps to schedule across all resident thread blocks. As blocks complete, new ones are launched. This is fine.
But maybe we want more blocks to reduce the work per block and allow better load balancing. Or fewer blocks with larger thread counts?
With memory-bound kernels, occupancy isn't usually the bottleneck - memory bandwidth is. But having enough warps to hide latency is important.
Let me just write the code and see what happens.
One more thought: should I try `__launch_bounds__`? With a simple kernel using few registers, we don't need it. But it might help the compiler. Let's skip it for now.
Another idea: using `fmaf` for `alpha * (__expf(x) - 1.0f)`. Actually it's `alpha * __expf(x) - alpha`. That's `fmaf(alpha, __expf(x), -alpha)`. But wait, `fmaf(a, b, c)` computes `a*b + c`. So `alpha * __expf(x) - alpha` is `fmaf(alpha, __expf(x), -alpha)`. This might be slightly faster and more accurate.
Let me just write a clean version and test.
I should also consider using `torch.float32` explicitly in the kernel to avoid any dtype issues.
Here's my plan:
1. Write a vectorized float4 ELU kernel
2. Use `__expf` for fast exponential
3. Use grid-stride loop with 1024 blocks and 512 threads
4. Compile and test
5. If needed, adjust block/thread counts
One thing: `load_inline` requires `cpp_sources` and `cuda_sources`. Let me write the complete Python module.
Let me draft the code:
```python
import torch
import torch.nn as nn
from torch.utils.cpp_extension import load_inline
# CUDA C++ source
CUDA_SOURCE = r'''
#include <torch/extension.h>
__global__ void elu_kernel(const float* __restrict__ input,
float* __restrict__ output,
int64_t n,
float alpha) {
int64_t tid = blockIdx.x * blockDim.x + threadIdx.x;
int64_t stride = gridDim.x * blockDim.x;
int64_t n_vec = n >> 2; // n / 4
for (int64_t i = tid; i < n_vec; i += stride) {
float4 in = reinterpret_cast<const float4*>(input)[i];
float4 out;
out.x = in.x > 0.0f ? in.x : alpha * (__expf(in.x) - 1.0f);
out.y = in.y > 0.0f ? in.y : alpha * (__expf(in.y) - 1.0f);
out.z = in.z > 0.0f ? in.z : alpha * (__expf(in.z) - 1.0f);
out.w = in.w > 0.0f ? in.w : alpha * (__expf(in.w) - 1.0f);
reinterpret_cast<float4*>(output)[i] = out;
}
// Handle tail elements
int64_t tail_start = n_vec << 2;
for (int64_t i = tail_start + tid; i < n; i += stride) {
float x = input[i];
output[i] = x > 0.0f ? x : alpha * (__expf(x) - 1.0f);
}
}
torch::Tensor elu_cuda(torch::Tensor input, float alpha) {
auto output = torch::empty_like(input);
int64_t n = input.numel();
int threads = 512;
int blocks = 1152;
elu_kernel<<<blocks, threads>>>(
input.data_ptr<float>(),
output.data_ptr<float>(),
n,
alpha
);
return output;
}
'''
CPP_SOURCE = r'''
torch::Tensor elu_cuda(torch::Tensor input, float alpha);
'''
elu_ext = load_inline(
name='elu_ext',
cpp_sources=CPP_SOURCE,
cuda_sources=CUDA_SOURCE,
functions=['elu_cuda'],
extra_cuda_cflags=['-O3', '--use_fast_math'],
verbose=False,
)
class ModelNew(nn.Module):
def __init__(self, alpha: float = 1.0):
super(ModelNew, self).__init__()
self.alpha = alpha
def forward(self, x: torch.Tensor) -> torch.Tensor:
return elu_ext.elu_cuda(x, self.alpha)
```
Wait, `--use_fast_math` includes faster but less accurate intrinsics. But the tolerance is 1e-4 for fp32. Let me check if `--use_fast_math` is safe.
With `--use_fast_math`:
- `expf` becomes `__expf`
- Division might be approximate
- Other operations might be faster
For our kernel, `--use_fast_math` will affect accuracy. The `__expf` intrinsic has max error ~2-3 ULPs. For x near 0, exp(x) ≈ 1 + x + x²/2, so `alpha * (exp(x) - 1)` ≈ `alpha * x`. The relative error in exp near 0 is about 1e-7, and since we subtract 1, the absolute error could be amplified. Let's see: if x = -1e-4, exp(x) ≈ 0.9999, error in exp could be ~1e-7, after subtracting 1 we get ~-1e-4 with absolute error ~1e-7. That's well within 1e-4.
For x = -1, exp(-1) ≈ 0.3679. Absolute error in __expf might be ~1e-7. After subtracting 1: -0.6321 with error ~1e-7. Also fine.
For x = -10, exp(-10) ≈ 4.54e-5. Absolute error could be ~1e-7. After subtracting 1: ~-1.0 with error ~1e-7. Fine.
So `--use_fast_math` should be safe.
Actually, I want to try a slightly different approach. Let me see if we can do better by processing 8 elements per thread (two float4s) to increase instruction-level parallelism and reduce loop overhead.
Or even better: unroll the loop. But the compiler should handle that.
Another optimization: since PyTorch uses its own internal allocator which provides aligned memory, `float4` loads will be fully aligned and very fast.
Let me also think about whether we should use `restrict` on the pointers. Yes, `__restrict__` tells the compiler there's no aliasing, which helps optimization.
Actually, I realize I might want to check the alignment more carefully. PyTorch tensors from `torch.empty_like` should be at least 256-byte aligned, which is sufficient for float4 (16-byte aligned). And since `n` is divisible by 4, everything should work.
Let me compile this and see.
Actually, I should also try different block counts. Let me start with 1152 blocks of 512 threads and see.
Wait, I want to double check one thing. The expression `alpha * (__expf(in.x) - 1.0f)` uses a multiply and a subtract. This is two FMA-eligible operations but not actually using FMA. We can rewrite as `fmaf(alpha, __expf(in.x), -alpha)`. Let me see if that helps.
Let me stick with the simpler expression for now. The compiler might optimize it.
Let me compile and test.
One more thing - I need to be careful about `r` strings for the CUDA source to avoid Python backslash escape issues. I'll use raw strings for the CUDA source.
Let me write the initial code and test.