reasoning
Let me analyze the problem. We need to implement a custom CUDA kernel that:
1. Does a linear layer (matrix multiplication + bias)
2. Applies Mish activation twice
The dimensions are:
- batch_size = 1024
- in_features = 8192
- out_features = 8192
Reference: linear(x) → mish → mish
The H100 has massive compute (1979 TFLOPS bf16/fp16, 67 TFLOPS fp32) and 3.35 TB/s memory bandwidth.
For fp32 matrix multiply 1024 x 8192 x 8192:
- Compute = 2 * 1024 * 8192 * 8192 ≈ 137 GFLOPS
- Time at 67 TFLOPS = 2 microseconds
- Time at 3.35 TB/s for memory-bound = ?
Actually, let's think about this more carefully. The reference implementation does:
1. `linear(x)`: GEMM 1024x8192 times 8192x8192, plus bias add
2. `mish(x)` on the result
3. `mish(x)` on the result again
PyTorch's mish on fp32 might not be extremely optimized. We can fuse everything into a single kernel to reduce memory bandwidth.
For fusing, we need to:
1. Compute GEMM output C = A * B + bias
2. Apply mish(mish(C))
To maximize performance, we should use CUTLASS or cuBLAS for the GEMM part, then apply a custom kernel for the fused mish operations. But since we need a custom CUDA kernel for the whole forward pass, we could:
Option A: Call cuBLAS for GEMM, then call a custom kernel for double mish
Option B: Use CUTLASS for everything fused together
Actually, the simplest approach is:
1. Allocate output buffer
2. Call cuBLAS for GEMM (A * B^T + bias)
3. Launch a custom kernel that applies mish twice in-place
Wait, looking at the task again: "Replace the forward computation ... with a custom CUDA kernel". It says to use `torch.utils.cpp_extension.load_inline` and call the compiled extension from `ModelNew.forward`.
So the approach should be:
1. Write a custom CUDA extension that calls cuBLAS for GEMM and then a custom CUDA kernel for mish(mish(x))
2. Use the custom CUDA kernel from Python
Actually, we might even be able to just call `torch.matmul` and then our custom `double_mish` kernel. But the question is whether we can do better.
Let me think about what's fastest. The H100 has:
- 3.35 TB/s memory bandwidth
- 67 TFLOPS FP32
For the mish activation:
mish(x) = x * tanh(softplus(x))
softplus(x) = ln(1 + exp(x))
So mish(x) = x * tanh(ln(1 + exp(x)))
This is compute-intensive. For 1024 * 8192 = 8.3M elements, applying mish twice would need several operations per element.
Actually, we can use the memory bandwidth and compute optimally if we fuse. But wait - can we use torch.matmul inside our custom kernel? No, we'd need to use cuBLAS.
Wait, there's an important consideration: The reference might already be using cuBLAS, and PyTorch's mish implementation might already be fused or not. Let me think about whether we should:
Option 1: Call `torch.matmul` + custom double-mish kernel
Option 2: Call cuBLAS from CUDA + custom double-mish kernel
Actually, the best approach might be to use `torch.mm` or `torch.matmul` for the GEMM part, and write a custom kernel for the fused double mish. But maybe we can also just write a custom kernel for mish + mish to reduce bandwidth.
Wait, the task says "Your submission is a complete Python file that defines a class called ModelNew" and "call the compiled extension from ModelNew.forward". So we don't have to write the GEMM part in CUDA - we could call `torch.matmul` from Python and then call our custom CUDA kernel for the elementwise operations.
But maybe we can get better performance by loading weights, doing GEMM, and then applying double mish all in one custom kernel? That's hard because GEMM is best done by cuBLAS/CUTLASS.
Let me consider: The forward pass is `linear(x)` then `mish(mish(x))`. The linear layer is a GEMM followed by bias addition.
If we use PyTorch's linear, it handles the GEMM well. The inefficiency comes from:
1. The intermediate output from linear being written to global memory
2. The first mish reading from global memory and writing back
3. The second mish reading from global memory and writing back
If we use our custom kernel for double mish, we can reduce this to:
1. Linear output written to global memory
2. Double mish kernel reads once, processes twice, writes once
This saves one read and one write compared to two separate mish calls.
Actually, we can do even better: After the GEMM (using cuBLAS or torch.mm), we can write a kernel that does `bias_add` + `mish` + `mish`. But wait, bias addition might already be fused into the GEMM by PyTorch if we call `F.linear`. Actually, for `nn.Linear`, if cuDNN or cuBLAS supports it, bias is fused.
Hmm, let me think about whether to implement GEMM ourselves or use PyTorch. Using PyTorch's linear is probably best for the GEMM part. But we need to be careful - can we use torch operations inside our custom forward and still be competitive? Yes, as long as our custom kernel is efficient for the elementwise part.
Wait, but could we get even more speedup by calling cuBLAS from our C++/CUDA code? That would still require us to manage the tensors.
Actually, let me reconsider. The reference does:
1. `self.linear(x)` - cuBLAS GEMM + bias (probably fused already)
2. `F.mish(x)` - reads all elements, computes mish, writes back
3. `F.mish(x)` - reads all elements, computes mish, writes back
If our approach is:
1. `self.linear(x)` - same as reference
2. Custom fused kernel: reads elements, computes mish(mish(x)), writes back
Then we save one full memory pass. For 1024 * 8192 * 4 bytes = 33.5 MB, at 3.35 TB/s that's about 20 microseconds for a full read+write. So we might save 20 microseconds. Is that significant? Let me think about the total time.
For GEMM 1024x8192x8192 in fp32:
- Using TF32 tensor cores or FP32: The problem size is relatively small for an H100.
- 1024 * 8192 * 8192 * 2 = 137 billion ops = 137 GFLOPS
- At 67 TFLOPS theoretical peak, that's 2ms, but realistic GEMM efficiency might be lower for this size.
- Actually wait, 1024 * 8192 * 8192 = ~68 billion multiplications, and with the multiply-add, ~137 GFLOPS.
Wait, no: 1024 * 8192 * 8192 ≈ 68.7M (not billion). Times 2 for MAC = 137M FLOPs = 0.137 GFLOPs.
At 67 TFLOPS, this would take ~2 microseconds theoretically. In practice, small GEMMs have overhead so maybe 50-200 microseconds.
Let me recalculate: 1024 × 8192 × 8192 = 68,719,476,736 ≈ 68.7B multiply-adds = 137.4 GFLOPs.
At 100 TFLOPS (using tensor cores or good utilization), that's 1.37 milliseconds. At 10 TFLOPS, it's 13.7 ms.
Hmm, for H100, a 1024×8192 @ 8192×8192 GEMM in FP32... actually maybe it's not that fast. Let me think about using cuBLAS.
But for mish: 1024 * 8192 = 8,388,608 elements.
Two separate mish calls:
- Each: read 33.5MB, compute, write 33.5MB
- Total memory traffic: 134MB
- At 3.35 TB/s: ~40 microseconds
Wait, 134MB at 3.35 TB/s = 40 microseconds. That seems reasonable.
But we need to consider that the GEMM output is already in GPU memory. For a fused double-mish kernel:
- Read 33.5MB + Write 33.5MB = 67MB total
- Time: ~20 microseconds
So the saving is ~20 microseconds. If the GEMM takes several hundred microseconds, this might be a small percentage improvement. But we can also consider that PyTorch's mish might not be optimally vectorized or might have overhead.
Actually wait, I should think about whether we can call the GEMM from C++ using cuBLAS and then immediately apply the fused elementwise kernel. This would be the same as calling torch.matmul from Python followed by our custom kernel.
Hmm, but there's another consideration: PyTorch 2.0+ compiles models and might already fuse some operations. Since the reference is standard PyTorch, it might not use torch.compile. So we have an opportunity to fuse.
Let me think about a different approach: What if we do the GEMM using PyTorch, but treat `linear` as `matmul` + `bias`? Then we can fuse `bias_add + mish + mish` into a single kernel? But the best approach might be to let cuBLAS handle the GEMM + bias fusion if possible.
Actually, here's an idea: What if we don't use `nn.Linear` at all, but instead use `torch.matmul` with the transposed weight, and then call our custom kernel that does `add_bias + double_mish`? This way:
1. `torch.matmul(x, weight.T)` uses cuBLAS
2. Our custom kernel adds bias and applies double mish
This is similar to what `nn.Linear` does, but we have explicit control over the fusion.
Actually, PyTorch's `F.linear` already calls into `addmm` which fuses bias. But if we do `torch.matmul` then our custom bias+double_mish kernel, we don't gain anything over `F.linear` + our double_mish kernel, because `F.linear` probably already fuses the bias into the GEMM. Though I should check.
Let me just write a kernel that computes `fused_linear_mish_mish`. We can do this by:
1. Using cuBLAS in our C++ extension for GEMM + bias
2. Then applying our custom double-mish kernel
OR, we can just do the GEMM in Python using torch.matmul, and write a CUDA kernel for just the elementwise operations. Let me think about which is easier and as fast.
Actually, for maximum speedup, we should use CUTLASS or cuBLAS from C++ to do the GEMM, and then immediately call our elementwise kernel without returning to Python.
But `load_inline` makes this easy. We can write C++ code that calls `at::matmul_out` or `at::addmm_out`, and then call our CUDA kernel. Actually, we can even use PyTorch's C++ tensor operations in our `cpp_sources` code!
Wait, in `cpp_sources`, we have access to the full ATen/C10 library. So we can write:
```cpp
torch::Tensor custom_forward(torch::Tensor x, torch::Tensor weight, torch::Tensor bias) {
auto output = at::addmm(bias, x, weight.t());
// call our CUDA kernel on output
return output;
}
```
And the CUDA kernel can be defined in `cuda_sources`. This is essentially the same as doing `F.linear(x, weight, bias)` in Python followed by a custom kernel, but with less Python overhead.
But actually, we might even be able to do better by using warp-specialized kernels or vectorized elementwise operations.
Let me think about the kernel design:
The double mish kernel should:
- Read the GEMM output elements
- Add bias (if not already added by GEMM)
- Apply mish twice
- Write back
Wait, if we use `at::addmm` for GEMM + bias, then the bias is already added. Our kernel just needs to read, apply mish twice, write.
For the kernel, we need an efficient implementation of mish:
```python
mish(x) = x * tanh(softplus(x))
softplus(x) = log(1 + exp(x))
```
For numerical stability, softplus is usually computed as:
- For x >= 0: log(1 + exp(x)) (or use exp(x) for small x, x + exp(-x) for large x... wait)
- Actually, the standard stable implementation is:
- For x >= threshold: x
- For x < threshold: log(1 + exp(x))
But PyTorch uses `log1p(exp(x))` with special handling. Let me think about what PyTorch's mish actually computes.
Actually, `mish(x) = x * tanh(softplus(x))`. For large positive x, softplus(x) ≈ x, so mish(x) ≈ x * tanh(x) ≈ x. For large negative x, softplus(x) ≈ exp(x), so mish(x) ≈ x * tanh(exp(x)) ≈ x * exp(x) ≈ 0.
For numerical stability, we need to ensure we match PyTorch's outputs within 1e-4.
Let me look at what functions are available in CUDA:
- `expf`, `log1pf`, `tanhf` are available
A numerically stable softplus:
```cpp
__device__ float softplus(float x) {
return x > 20.0f ? x : log1pf(expf(x));
}
```
Wait, for large negative x, `expf(x)` underflows to 0, and `log1pf(0)` = 0, which is correct.
For large positive x, `expf(x)` overflows, so we need the branch.
Actually, a more standard formulation:
```cpp
__device__ float softplus(float x) {
return x > 0 ? x + log1pf(expf(-x)) : log1pf(expf(x));
}
```
This is stable for both positive and negative large values.
And then:
```cpp
__device__ float mish(float x) {
float sp = softplus(x);
return x * tanhf(sp);
}
```
And for double mish, we just call it twice.
Now, for the kernel itself. Since we're on H100, we should use vectorized loads/stores for fp32. We can process 4 floats per thread (float4).
The tensor size is 1024 * 8192 = 8,388,608 elements.
At 128 threads per block and 4 floats per thread, we need 8,388,608 / (128*4) = 16,384 blocks. That's a lot, but H100 has many SMs.
Actually, we could use a grid-stride loop pattern. Or we could compute how many blocks we need.
For better occupancy and memory coalescing:
- Block size: 256 threads
- Each thread processes 4 elements (1 float4)
- Total threads needed: 8,388,608 / 4 = 2,097,152
- Threads per block: 256
- Blocks: 8192
This should saturate the GPU nicely.
Actually, let me think about whether to use `float4` or whether to let the compiler handle it. `float4` loads/stores are usually the most efficient for memory bandwidth.
Let me draft the CUDA code:
```cpp
#include <cuda_runtime.h>
#include <torch/extension.h>
__device__ __forceinline__ float softplus_scalar(float x) {
return x > 0.0f ? x + log1pf(expf(-x)) : log1pf(expf(x));
}
__device__ __forceinline__ float mish_scalar(float x) {
return x * tanhf(softplus_scalar(x));
}
__global__ void double_mish_kernel(float* __restrict__ out, const float* __restrict__ in, int64_t n) {
int64_t idx = blockIdx.x * blockDim.x + threadIdx.x;
int64_t stride = gridDim.x * blockDim.x;
for (int64_t i = idx; i < n; i += stride) {
float val = in[i];
val = mish_scalar(val);
val = mish_scalar(val);
out[i] = val;
}
}
```
But for in-place operation, we can do:
```cpp
__global__ void double_mish_kernel(float* __restrict__ data, int64_t n) {
int64_t idx = blockIdx.x * blockDim.x + threadIdx.x;
int64_t stride = gridDim.x * blockDim.x;
for (int64_t i = idx; i < n; i += stride) {
float val = data[i];
val = mish_scalar(val);
val = mish_scalar(val);
data[i] = val;
}
}
```
Wait, but we should consider using vectorized loads. For the simplest approach, let's first try a scalar version and see if it's correct and reasonably fast. Then we can optimize.
Actually, let me think about the host code. We want to:
1. Do GEMM with bias
2. Apply double mish
If we're writing C++ code, we can use `at::addmm` for the GEMM. But we need to be careful about `load_inline` and what we're allowed to do.
Let me write the first version:
```python
import torch
import torch.nn as nn
from torch.utils.cpp_extension import load_inline
cuda_source = '''
#include <cuda_runtime.h>
#include <torch/extension.h>
__device__ __forceinline__ float softplus_scalar(float x) {
return x > 0.0f ? x + log1pf(expf(-x)) : log1pf(expf(x));
}
__device__ __forceinline__ float mish_scalar(float x) {
return x * tanhf(softplus_scalar(x));
}
__global__ void double_mish_kernel(float* __restrict__ data, int64_t n) {
int64_t idx = blockIdx.x * blockDim.x + threadIdx.x;
for (int64_t i = idx; i < n; i += gridDim.x * blockDim.x) {
float val = data[i];
val = mish_scalar(val);
val = mish_scalar(val);
data[i] = val;
}
}
torch::Tensor double_mish(torch::Tensor x) {
int64_t n = x.numel();
// Make sure we're operating on contiguous memory
auto x_contig = x.contiguous();
const int threads = 256;
const int blocks = std::min((int)((n + threads - 1) / threads), 65536);
double_mish_kernel<<<blocks, threads>>>(x_contig.data_ptr<float>(), n);
return x_contig;
}
torch::Tensor custom_forward(torch::Tensor x, torch::Tensor weight, torch::Tensor bias) {
// GEMM with bias: y = x @ weight.T + bias
auto output = at::addmm(bias, x, weight.t());
// Apply double mish in-place
int64_t n = output.numel();
auto output_contig = output.contiguous();
const int threads = 256;
const int blocks = std::min((int)((n + threads - 1) / threads), 65536);
double_mish_kernel<<<blocks, threads>>>(output_contig.data_ptr<float>(), n);
return output_contig;
}
'''
cpp_source = '' # or maybe we need some includes
functions = ['custom_forward', 'double_mish']
```
Wait, but there's a subtlety: if we use `at::addmm` inside our C++ extension, the tensor operations use the ATen library which is already linked. That should be fine.
But actually, we also need to think about whether `at::addmm` creates a non-contiguous output. For standard GEMM, the output should be contiguous.
Let me also think about whether to use `float4` vectorization. For H100 with 3.35 TB/s bandwidth, vectorized loads should help.
Actually, there's another optimization we can try: Since we're applying mish twice, and mish is compute-bound, maybe we should use more threads per block or ensure good occupancy.
For mish:
- expf
- log1pf
- tanhf
These are all transcendental functions that take many clock cycles. But the H100 has fast approximations and good throughput for these operations.
Let me think about whether we can use `__nv_expf` etc. No, `expf` is fine.
Let me also think about using multi-stage warps or whether we need to worry about register pressure. Mish uses a few registers per thread. For 256 threads per block with a few dozen registers each, we should get good occupancy.
Now, a key question: Should we do the GEMM in C++ or in Python?
If we do GEMM in Python:
```python
def forward(self, x):
x = torch.nn.functional.linear(x, self.weight, self.bias)
return custom_cuda.double_mish(x)
```
If we do GEMM in C++:
```python
def forward(self, x):
return custom_cuda.custom_forward(x, self.weight, self.bias)
```
The C++ approach might have slightly less overhead since we're staying in C++ land for the entire forward pass. But PyTorch's Python overhead for `F.linear` is probably negligible given the problem size.
Actually, there's another consideration with `load_inline`: the compilation might fail if we use ATen functions that aren't properly available or if we have incorrect headers. Let me think about whether `at::addmm` is available.
In `cpp_sources` of `load_inline`, we have access to `<torch/extension.h>` which includes ATen. And the `load_inline` builds against the PyTorch headers. So `at::addmm` should be available.
But wait, there's a potential issue: `at::addmm` might dispatch to different backends, and the resulting tensor might be on the same device. We should ensure `x` is contiguous before calling `matmul`, but `addmm` should handle that.
Actually, I want to reconsider: maybe the fastest approach is to do everything in one kernel using CUTLASS. But that would require linking against CUTLASS or writing a lot more code. For simplicity and reliability, let me first try the approach with `at::addmm` + custom elementwise kernel. If that's not fast enough, I can optimize further.
But wait - there's a more important point. The task says "Replace the forward computation ... with a custom CUDA kernel." It doesn't say we can't use PyTorch operations alongside our custom kernel. So using `at::addmm` is fine.
Let me think about another optimization:融合 bias addition into the double mish kernel. We can do:
1. `torch.matmul(x, weight.t())` (no bias)
2. Custom kernel: `add bias + double mish`
This might not be faster than `at::addmm` + double mish, because `at::addmm` presumably fuses the bias into the GEMM. But let me verify this.
Actually, if we use `at::linear`, it does `x @ weight.T + bias`. In PyTorch, this calls `addmm` which passes bias to cuBLAS, and cuBLAS may or may not fuse it depending on the implementation and sizes. But for these sizes (1024x8192 @ 8192x8192), cuBLAS should handle it well.
I think the most reliable first attempt is:
1. Use `at::addmm` in C++ for GEMM + bias
2. Use custom CUDA kernel for in-place double mish
Let me write the code.
Actually, another thought: Can we amortize the cost of mish computation by batching it across warps more efficiently? For compute-bound kernels, sometimes using fewer blocks with more work per thread is better.
For 8.4M elements and H100 (~132 SMs), if we launch 8192 blocks of 256 threads = 2M threads, that's 15K threads per SM, which is way more than the hardware can execute concurrently. But with a grid-stride loop, each thread processes about 4 elements, which is fine.
Actually, we can use a different strategy: launch exactly enough blocks to saturate the GPU, and have each thread process multiple elements using vectorized loads.
For H100, we want about 132 SMs * 16 warps/SM * 32 threads/warp = ~67K threads concurrently. But since the kernel is compute-heavy (with exp and tanh), we want good occupancy. For 256 threads/block, we could launch maybe 132*4 = 528 blocks and have each thread process many elements.
Or we could just let CUDA handle it with the grid-stride loop and a reasonable number of blocks. Let's say 8192 blocks.
Hmm, let me think more carefully about launch configuration. For memory bandwidth kernels, we want to saturate all SMs. For compute kernels like exp/tanh, we want enough warps to hide latency.
For elementwise kernels, a common strategy is:
```cpp
const int threads = 256;
const int blocks = min(65536, max(1, (int)(n / threads)));
```
Or on newer GPUs, sometimes fewer blocks with more work per thread is better. But let's start with the standard approach and see.
Actually, I just realized: Does `at::addmm` work in `load_inline`? Let me check if there might be linker issues. `load_inline` compiles a shared library that links against `libc10.so` and `libtorch.so`, so ATen symbols should be available.
Let me draft the complete Python code for ModelNew:
```python
import torch
import torch.nn as nn
from torch.utils.cpp_extension import load_inline
CUDA_SOURCE = r'''
#include <torch/extension.h>
#include <cuda_runtime.h>
__device__ __forceinline__ float stable_softplus(float x) {
return x > 0.0f ? x + log1pf(expf(-x)) : log1pf(expf(x));
}
__device__ __forceinline__ float mish(float x) {
return x * tanhf(stable_softplus(x));
}
__global__ void double_mish_kernel(float* __restrict__ data, int64_t n) {
int64_t idx = blockIdx.x * blockDim.x + threadIdx.x;
int64_t stride = gridDim.x * blockDim.x;
#pragma unroll 4
for (int64_t i = idx; i < n; i += stride) {
float val = data[i];
val = mish(val);
val = mish(val);
data[i] = val;
}
}
torch::Tensor custom_forward_impl(torch::Tensor x, torch::Tensor weight, torch::Tensor bias) {
// Ensure inputs are contiguous and on CUDA
x = x.contiguous();
weight = weight.contiguous();
bias = bias.contiguous();
auto output = at::addmm(bias, x, weight.t());
int64_t n = output.numel();
const int threads = 256;
const int blocks = std::min(65536, std::max(1, (int)((n + threads - 1) / threads)));
double_mish_kernel<<<blocks, threads>>>(output.data_ptr<float>(), n);
return output;
}
'''
CPP_SOURCE = r'''
#include <torch/extension.h>
torch::Tensor custom_forward_impl(torch::Tensor x, torch::Tensor weight, torch::Tensor bias);
'''
custom_cuda = load_inline(
name="custom_mish",
cpp_sources=CPP_SOURCE,
cuda_sources=CUDA_SOURCE,
functions=["custom_forward_impl"],
extra_cuda_cflags=["-O3", "--use_fast_math"],
verbose=True,
)
class ModelNew(nn.Module):
def __init__(self, in_features, out_features):
super().__init__()
self.weight = nn.Parameter(torch.randn(out_features, in_features))
self.bias = nn.Parameter(torch.zeros(out_features))
# Initialize like nn.Linear would
nn.init.kaiming_uniform_(self.weight, a=5**0.5)
nn.init.zeros_(self.bias)
def forward(self, x):
return custom_cuda.custom_forward_impl(x, self.weight, self.bias)
```
Wait, but the reference uses `nn.Linear`, which initializes weights differently. I need to make sure my initialization matches. Actually, we're not expected to match initialization since the evaluation will use our model with the same inputs but probably sets weights before running. Or they might instantiate our model and then copy state_dict? The problem says "accept the same constructor arguments" so I should initialize the same way.
For `nn.Linear`, the weight is transposed compared to what I wrote. In `nn.Linear(in_features, out_features)`, the weight shape is `(out_features, in_features)`, and `forward` does `F.linear(x, self.weight, self.bias)` which computes `x @ weight.T + bias`.
So my `self.weight` with shape `(out_features, in_features)` is correct.
For initialization, `nn.Linear` uses `kaiming_uniform_` with `a=math.sqrt(5)` for the weight and uniform initialization for the bias that depends on `fan_in` and `bound`. Actually, looking at PyTorch's nn.Linear:
```python
def reset_parameters(self):
init.kaiming_uniform_(self.weight, a=math.sqrt(5))
if self.bias is not None:
fan_in, _ = init._calculate_fan_in_and_fan_out(self.weight)
bound = 1 / math.sqrt(fan_in) if fan_in > 0 else 0
init.uniform_(self.bias, -bound, bound)
```
But for evaluation, they likely don't care about initialization values as long as the computation is equivalent for any given weights.
Actually, there's another issue: I used `torch.randn` in my draft, but the reference uses `nn.Linear` which uses `kaiming_uniform_`. I should match that.
But wait, there's a simpler approach: We can keep `nn.Linear` inside our ModelNew and just replace the forward pass!
```python
class ModelNew(nn.Module):
def __init__(self, in_features, out_features):
super().__init__()
self.linear = nn.Linear(in_features, out_features)
def forward(self, x):
x = self.linear(x)
# Call our custom double mish kernel
return custom_cuda.double_mish(x)
```
This is much simpler! We don't have to worry about weight storage, initialization, or parameter management. We just use the built-in `nn.Linear` and apply our custom fused kernel afterwards.
The question is: is this as fast as doing GEMM + bias in C++? Probably yes, because PyTorch's `nn.Linear` is well-optimized for the GEMM part, and the extra Python overhead of one function call is negligible for this problem size.
But wait, the task says "Replace the forward computation... with a custom CUDA kernel". Using `nn.Linear` in Python and then calling our custom CUDA kernel for the rest might be acceptable. However, if we want the best performance, doing the GEMM in C++ might save a tiny bit of overhead.
Actually, let me think about what happens with the reference implementation:
1. `self.linear(x)` returns a tensor
2. `F.mish(x)` reads the tensor and returns a new one
3. `F.mish(x)` reads the new one and returns another
With our approach using `nn.Linear` + custom kernel:
1. `self.linear(x)` returns a tensor (same as reference)
2. `custom_cuda.double_mish(x)` reads the tensor, modifies in-place (or returns new), writes back
This saves one read and one write compared to the reference.
But we can save even more by doing GEMM + double mish in a single C++ call. However, the benefit is marginal since the GEMM output has to be written to memory anyway before the elementwise kernel can read it (unless we tile and fuse, which requires CUTLASS).
Wait, can we fuse GEMM and elementwise without CUTLASS? In theory, cuBLAS has an API for epilogue fusion (cublasLt), but it's complex and might not be available through PyTorch's wrappers easily.
Let me reconsider using cublasLt from our custom extension. Actually, we can use `at::linear` from C++ and it will use whatever PyTorch uses under the hood, which is probably cuBLAS or cuBLASLt.
I think the best trade-off is:
1. Use `nn.Linear` in ModelNew (simple, correct initialization)
2. Write a custom CUDA kernel for double mish
But wait, there's another approach: We can call `torch.matmul` and then our custom `bias_add + double_mish` kernel, which fuses the bias addition with the mish operations. Is this better?
`nn.Linear` might or might not fuse bias into GEMM. In PyTorch, for these large dimensions, it typically does. But even if it doesn't, we're talking about 8M elements for bias addition, which is very fast.
Actually, for maximum performance, let me consider doing ALL operations in a single custom C++ function that:
1. Calls `at::addmm` for GEMM
2. Launches our double mish kernel
This avoids any Python overhead between GEMM and double mish, and we can ensure the output is contiguous.
But honestly, for this problem size, Python overhead is probably negligible. Let me think about what would make the evaluator happiest.
Actually, looking at this from the perspective of a coding challenge: They want to see a custom CUDA kernel that improves performance. The simplest, most maintainable solution is to use `nn.Linear` for the well-optimized GEMM part and write a custom kernel for the elementwise part.
However, there's a potential issue: If we use `nn.Linear`, then the custom kernel is only doing elementwise ops. Some evaluators might consider this "less custom" than doing everything in C++. But the instructions specifically say to replace the forward computation with a custom CUDA kernel, and using `load_inline` to bind CUDA source.
Let me try the simplest approach first: `nn.Linear` + custom `double_mish` kernel. Let me get this working and then optimize if needed.
Wait, I want to make sure I handle the shapes correctly. The output of linear is `[batch_size, out_features] = [1024, 8192]`, total 8,388,608 elements.
For the double mish kernel, we can try vectorized float4 processing:
```cpp
__global__ void double_mish_kernel(float4* __restrict__ data, int64_t n4) {
int64_t idx = blockIdx.x * blockDim.x + threadIdx.x;
int64_t stride = gridDim.x * blockDim.x;
for (int64_t i = idx; i < n4; i += stride) {
float4 val4 = data[i];
val4.x = mish(mish(val4.x));
val4.y = mish(mish(val4.y));
val4.z = mish(mish(val4.z));
val4.w = mish(mish(val4.w));
data[i] = val4;
}
}
```
This would process 4 elements per thread with vectorized memory accesses.
Hmm, but for correctness and simplicity, let me first try the scalar version and make sure it compiles and passes correctness.
Actually, I realize I should think about whether we should allocate a new output or do in-place. If we do in-place on the GEMM output, we might modify a tensor that `nn.Linear` has cached for gradient computation if `requires_grad=True`. Wait, in the forward pass, the input tensor to our custom kernel is just a temporary output from linear. The autograd graph records the operation. If we modify it in-place, will that break anything?
In PyTorch, doing in-place operations on the output of `nn.Linear` during forward pass is generally okay as long as autograd doesn't need the original value. But for `mish`, the autograd graph needs the input to `mish` for backward propagation.
Wait, that's an important point! If we modify the tensor in-place, and PyTorch's autograd needs the pre-mish value for backward, we'll have a problem. But the reference uses `F.mish(x)` which creates a new tensor, so autograd stores the pre-mish value correctly.
If we write our custom `double_mish` as an in-place operation on the linear output, autograd won't know about it. We'd need to ensure our custom kernel is registered as an autograd function if gradients are needed.
Hmm, but looking at the problem statement: It says to replace the forward computation. It doesn't explicitly mention training/backprop. Usually in these challenges, we only need to match the forward pass output for inference. Let me check if the evaluation might test backprop.
The instructions say "Correctness: Outputs must match the reference within tolerance" and "Replace the forward computation". They don't mention backward. So I think we only need to ensure numerical equivalence of the forward pass. If there were backprop tests, we'd need `torch::autograd::Function`.
For safety and simplicity, let me write the custom kernel to allocate and return a NEW tensor rather than doing in-place. This also lets us handle non-contiguous inputs safely.
But wait, allocating a new tensor uses extra memory bandwidth. We could do in-place if we're careful. Actually, for maximum performance:
If we do GEMM in C++ with `at::addmm`, and then apply our kernel in-place on the result, that's fine because the GEMM output is freshly allocated, and no autograd graph depends on its original value yet (since our entire custom_forward is outside autograd unless we wrap it).
Actually, if we just use `load_inline` to compile a regular function (not an autograd Function), and we call it during forward, PyTorch won't track gradients through it. This means if we're in `forward()` of our `nn.Module`, and we call the custom function, the output won't have `grad_fn`. But for inference/evaluation that just checks forward outputs, this is fine.
However, if the evaluation does a backward pass test (which is possible), we'd fail. Let me think about whether to implement an autograd Function.
Actually, the safest approach is to implement a custom autograd Function:
```python
class DoubleMishFunction(torch.autograd.Function):
@staticmethod
def forward(ctx, x):
ctx.save_for_backward(x)
return custom_cuda.double_mish(x)
@staticmethod
def backward(ctx, grad_output):
x, = ctx.saved_tensors
# Compute grad_x
...
```
But computing the backward pass for mish is complex. The derivative of mish is non-trivial, and implementing it in CUDA would be a lot of work. Looking at the problem, it specifically says "Replace the forward computation". This strongly suggests we only need to worry about the forward pass.
If they do test backprop with the reference, our model should produce the same gradients. But that's much harder. Let me look at the instructions more carefully.
The instructions say: "Replace the forward computation of the PyTorch model below with a custom CUDA kernel" and "Produce numerically equivalent outputs at fp32 precision." The "outputs" most naturally means forward outputs.
I'll proceed with a simple forward-only implementation, but I'll make sure to keep `nn.Linear` so that gradients through the linear layer are tracked (since `nn.Linear` is standard PyTorch and handles its own backward). For the mish part, if we use a plain function, autograd won't track it. But wait - the reference has two `F.mish` calls, so autograd does track through them for backward. If evaluation includes a backward test, we need to handle that.
Actually, let me think about this more. We have two options:
1. Don't worry about backward (simplest, most likely correct for evaluation)
2. Implement a custom autograd Function with backward pass
For option 2, the backward of mish is:
mish(x) = x * tanh(softplus(x))
Let sp = softplus(x), th = tanh(sp)
mish(x) = x * th
dmish/dx = th + x * (1 - th^2) * dsp/dx
dsp/dx = sigmoid(x)
So dmish/dx = tanh(softplus(x)) + x * tanh(softplus(x)) * (1 - tanh(softplus(x))) * sigmoid(x)
Wait, let me verify:
d/dx tanh(u) = sech^2(u) * du/dx = (1 - tanh^2(u)) * du/dx
d/dx softplus(x) = sigmoid(x)
So:
d/dx [x * tanh(softplus(x))] = tanh(softplus(x)) + x * (1 - tanh^2(softplus(x))) * sigmoid(x)
For double mish: y = mish(mish(x))
dy/dx = dmish(mish(x))/dmish(x) * dmish(x)/dx
This is getting complicated to implement. Given the problem focus on "forward computation", I think backward tests are unlikely. Let me proceed with a simple forward implementation.
Actually, there's a nice compromise: We can keep `nn.Linear` from PyTorch, which handles its own backward. And for the double mish part, since we're doing a custom kernel, if we just need forward equivalence, we can do whatever.
Let me write the complete code.
Wait, I just realized something important. If I use `nn.Linear` and then call a custom function that PyTorch can't backprop through, and if the evaluation does `output.sum().backward()`, it would fail for our model but work for the reference. This is a serious concern.
Let me re-read the problem: "Replace the forward computation". In typical ML coding challenges, if they say "forward computation", they usually only check the forward pass. And since we're replacing an `nn.Module`'s forward method, as long as forward outputs match, it should be fine.
But to be safe, let me think about whether there's a way to wrap our custom kernel in an autograd function easily. The backward of our custom function is complex but doable.
Actually, looking at this differently: Maybe we can use `torch.utils.cpp_extension.load_inline` and also define the backward in CUDA? That's a lot of code.
Or, we can use a PyTorch autograd function and implement the backward in Python using PyTorch ops? But then our double mish would need a backward implementation.
Let me compute the backward:
For mish(x):
```python
sp = F.softplus(x)
th = torch.tanh(sp)
mish_x = x * th
grad_mish = grad_output * (th + x * (1 - th**2) * torch.sigmoid(x))
```
For double mish:
```python
m1 = F.mish(x)
y = F.mish(m1)
grad_m1 = grad_output * (torch.tanh(F.softplus(m1)) + m1 * (1 - torch.tanh(F.softplus(m1))**2) * torch.sigmoid(m1))
grad_x = grad_m1 * (torch.tanh(F.softplus(x)) + x * (1 - torch.tanh(F.softplus(x))**2) * torch.sigmoid(x))
```
This is quite involved but manageable in Python. If we're concerned about backward tests, we could implement this.
However, implementing backward in CUDA for speed is also possible but would require more code.
For now, let me focus on making the forward pass fast and correct. If we need backward support, we can add a Python autograd function wrapper later.
Actually, you know what, to be completely safe, let me implement a custom autograd Function with the backward pass in Python. This way, if there are backward tests, we'll pass. And the backward pass of mish can be computed using PyTorch operations (which will be slower than a CUDA kernel but correctness is more important for backward).
Wait, but the problem says "Run faster than the PyTorch reference." This usually means forward pass timing. If backward pass is tested, it's typically for correctness, not speed.
Actually, looking at the tools: `run_correctness` checks correctness and `submit_kernel` does correctness AND timing. The timing is likely just the forward pass.
I'll implement a simple forward-only custom kernel using `nn.Linear` + custom double mish kernel. If I need backward later, I can add it.
Let me start writing the kernel 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>
__device__ __forceinline__ float softplus_impl(float x) {
return x > 0.0f ? x + log1pf(expf(-x)) : log1pf(expf(x));
}
__device__ __forceinline__ float mish_impl(float x) {
return x * tanhf(softplus_impl(x));
}
__global__ void double_mish_kernel(const float* __restrict__ input, float* __restrict__ output, int64_t n) {
int64_t idx = blockIdx.x * blockDim.x + threadIdx.x;
int64_t stride = gridDim.x * blockDim.x;
for (int64_t i = idx; i < n; i += stride) {
float v = input[i];
v = mish_impl(v);
v = mish_impl(v);
output[i] = v;
}
}
torch::Tensor double_mish(torch::Tensor input) {
input = input.contiguous();
auto output = torch::empty_like(input);
int64_t n = input.numel();
const int threads = 256;
const int blocks = std::min((int64_t)65536, (n + threads - 1) / threads);
double_mish_kernel<<<blocks, threads>>>(input.data_ptr<float>(), output.data_ptr<float>(), n);
return output;
}
"""
cpp_source = "" # No extra cpp needed since cuda_source has torch::Tensor function
custom_module = load_inline(
name="double_mish_cuda",
cuda_sources=cuda_source,
functions=["double_mish"],
extra_cuda_cflags=["-O3", "--use_fast_math"],
)
```
Hmm, wait. `load_inline` requires `cpp_sources` or `cuda_sources`. For functions that use `torch::Tensor`, they should be in the C++ or CUDA source. The CUDA source with `torch::Tensor` return type is compiled as CUDA+C++ code so it should be fine.
Actually, I realize I should double check the syntax for `load_inline`. The CUDA source string is passed as `cuda_sources` and should contain the device and host code. Let me make sure.
One more thing: Using `--use_fast_math` might affect precision. `log1pf`, `expf`, and `tanhf` with `--use_fast_math` could introduce errors larger than 1e-4 relative to PyTorch's implementation. PyTorch typically uses standard math libraries, not fast math. So I should be careful.
Actually, for FP32, the tolerance is `atol=rtol=1e-4`. The difference between fast math and regular math for these transcendental functions can be around 1-2 ULPs. For reasonable input ranges, this should be well within 1e-4. But let me not use `--use_fast_math` initially and see if it passes. If it's too slow, I can add it.
Wait, do I need `-O3`? The compiler usually defaults to `-O2` or `-O3` for release builds. I'll include `-O3` for safety.
Let me think about vectorized loads. For float4:
```cpp
__global__ void double_mish_kernel_float4(const float4* __restrict__ input, float4* __restrict__ output, int64_t n) {
int64_t idx = blockIdx.x * blockDim.x + threadIdx.x;
int64_t n4 = n / 4;
int64_t stride = gridDim.x * blockDim.x;
for (int64_t i = idx; i < n4; i += stride) {
float4 v4 = input[i];
v4.x = mish_impl(mish_impl(v4.x));
v4.y = mish_impl(mish_impl(v4.y));
v4.z = mish_impl(mish_impl(v4.z));
v4.w = mish_impl(mish_impl(v4.w));
output[i] = v4;
}
}
```
For non-multiple-of-4 elements, we could handle the tail with a scalar loop:
```cpp
for (int64_t i = idx + n4 * 4; i < n; i += stride) { ... }
```
Or we could just assert that n is divisible by 4 (which it is: 1024*8192 = 8388608, divisible by 4).
Using float4 should significantly improve memory bandwidth utilization.
But for now, let me start with a simple scalar implementation to verify correctness, and then optimize if needed.
Actually, I want to try the float4 version from the start, since the dimensions are nice and divisible.
Wait, I also need to think about whether to use `__restrict__` and how the compiler handles it. `__restrict__` is good because it tells the compiler that pointers don't alias.
Let me write the full kernel with float4:
```cpp
#include <torch/extension.h>
#include <cuda_runtime.h>
__device__ __forceinline__ float softplus_impl(float x) {
return x > 0.0f ? x + log1pf(expf(-x)) : log1pf(expf(x));
}
__device__ __forceinline__ float mish_impl(float x) {
return x * tanhf(softplus_impl(x));
}
__global__ void double_mish_kernel(const float* __restrict__ input, float* __restrict__ output, int64_t n) {
int64_t idx = blockIdx.x * blockDim.x + threadIdx.x;
int64_t n4 = n >> 2;
int64_t stride = gridDim.x * blockDim.x;
const float4* in4 = reinterpret_cast<const float4*>(input);
float4* out4 = reinterpret_cast<float4*>(output);
#pragma unroll 4
for (int64_t i = idx; i < n4; i += stride) {
float4 v = in4[i];
v.x = mish_impl(mish_impl(v.x));
v.y = mish_impl(mish_impl(v.y));
v.z = mish_impl(mish_impl(v.z));
v.w = mish_impl(mish_impl(v.w));
out4[i] = v;
}
}
torch::Tensor double_mish(torch::Tensor input) {
input = input.contiguous();
auto output = torch::empty_like(input);
int64_t n = input.numel();
const int threads = 256;
const int blocks = std::min((int64_t)65536, (n >> 2) > 0 ? ((n >> 2) + threads - 1) / threads : 1);
double_mish_kernel<<<blocks, threads>>>(input.data_ptr<float>(), output.data_ptr<float>(), n);
return output;
}
```
Wait, the blocks calculation for float4 is a bit different. `n >> 2` = n/4. But the kernel signature takes `n` and computes `n4` internally. The blocks should be based on `n4`, not `n`. But I could also change the kernel to accept `n` and have it operate on floats, with each thread handling 4 elements via float4.
Actually, if I pass `n` to the kernel as the original number of elements, the kernel computes `n4 = n >> 2`, and the tail is handled automatically if any. Since n is divisible by 4, no tail needed.
I think it's cleaner to pass `n4` to the kernel and have each thread process one float4. Let me adjust:
```cpp
__global__ void double_mish_kernel(const float4* __restrict__ input, float4* __restrict__ output, int64_t n4) {
int64_t idx = blockIdx.x * blockDim.x + threadIdx.x;
int64_t stride = gridDim.x * blockDim.x;
#pragma unroll 4
for (int64_t i = idx; i < n4; i += stride) {
float4 v = input[i];
v.x = mish_impl(mish_impl(v.x));
v.y = mish_impl(mish_impl(v.y));
v.z = mish_impl(mish_impl(v.z));
v.w = mish_impl(mish_impl(v.w));
output[i] = v;
}
}
torch::Tensor double_mish(torch::Tensor input) {
input = input.contiguous();
auto output = torch::empty_like(input);
int64_t n = input.numel();
int64_t n4 = n >> 2;
const int threads = 256;
const int blocks = std::min((int64_t)65536, (n4 + threads - 1) / threads);
double_mish_kernel<<<blocks, threads>>>(
reinterpret_cast<const float4*>(input.data_ptr<float>()),
reinterpret_cast<float4*>(output.data_ptr<float>()),
n4
);
return output;
}
```
This should work. But I need to be careful: the input and output tensors might not be aligned to 16 bytes for float4 access. In PyTorch, for contiguous tensors with float dtype and size divisible by 4, the start address is typically aligned to at least 4 bytes (since each float is 4 bytes), and often aligned to 16 bytes because CUDA memory allocations are aligned to 256 bytes.
Actually, `cudaMalloc` is guaranteed to return 256-byte aligned memory. So accessing the start of the tensor as `float4*` is safe. However, if the tensor has an offset (e.g., it's a slice), the alignment might not be 16 bytes. But we call `.contiguous()` and use `torch::empty_like()`, which creates a fresh contiguous tensor. For a fresh tensor, the data pointer should be 256-byte aligned. So using `float4` is safe.
Now, for the blocks calculation: with n4 = 2,097,152 and threads = 256, blocks = 8192. That's a good amount.
Actually, let me think about whether 8192 blocks is optimal. H100 has 132 SMs. With 8192 blocks, that's ~62 blocks per SM. With 256 threads per block, each block has 8 warps. Each SM can handle up to 32 blocks. With compute-heavy kernels, we might want fewer blocks with more iterations per block to reduce scheduling overhead. But 8192 blocks should be fine since the GPU can execute them concurrently.
Another optimization: we could use a larger block size or have each thread process multiple float4s in a single iteration to better amortize the loop overhead. But this is probably fine for now.
Let me also think about whether the kernel is memory-bound or compute-bound.
For double mish on each element:
- Read 4 bytes
- 2 × (exp, log1p, tanh, several arithmetic ops) ≈ maybe 50-100 FLOPs
- Write 4 bytes
At 3.35 TB/s, the theoretical peak throughput for read+write is 1675M elements/second (4 bytes each way, so 8 bytes total per element).
At 67 TFLOPS and 100 FLOPs per element, the compute limit is 670M elements/second.
So the kernel is likely compute-bound, which means we want to maximize instruction throughput and occupancy.
For compute-bound kernels on H100, we want:
- Good occupancy (many warps per SM to hide latency)
- Avoid register spilling
- Use fast math if precision allows
Let me check: mish involves 2 expf, 2 log1p, 2 tanh per call. For double mish: 4 expf, 4 log1p, 4 tanh. Plus some arithmetic. That's quite a lot.
Actually, there's a potential optimization: We can precompute common terms. Let me think...
mish(x) = x * tanh(softplus(x))
For the second mish, we're computing mish(y) where y = mish(x). There's not much to reuse.
One thing we should definitely do: use `#pragma unroll` for the loop. I already put `#pragma unroll 4` in the kernel.
Another optimization: use `logf` instead of `log1pf` for some cases? No, `log1pf` is more accurate for small inputs.
Wait, I want to think about whether `--use_fast_math` would push us out of the 1e-4 tolerance. Let me check what `--use_fast_math` does:
- `expf` → `__expf` (less accurate)
- `logf` → `__logf` (less accurate)
- `log1pf` might not have a fast math equivalent
- `tanhf` → `__tanhf` (less accurate)
PyTorch typically uses the standard (not fast math) versions of these functions. The error from fast math could be larger for some inputs.
Actually, let me check: PyTorch's CUDA implementation of `tanh` uses `tanhf` from libm. The fast math equivalent `__tanhf` might have errors up to a few ULPs. For inputs in the range we care about (the output of `rand` is [0, 1], but after linear layer it could be larger), the output of mish is close to identity for large positive values, and close to 0 for large negative values. The most sensitive region is where the derivative changes rapidly.
For `rand(1024, 8192)` multiplied by a linear layer with Kaiming initialization, the standard deviation of the output is around 1. So most values are in [-3, 3].
In this range, fast math approximations should be very accurate, likely well within 1e-4. I'll use `--use_fast_math` to get the best performance.
Actually, looking at the PyTorch source for mish:
https://github.com/pytorch/pytorch/blob/main/aten/src/ATen/native/cuda/ActivationMishKernel.cu
Let me think about what formula PyTorch uses. It might use `x * tanh(softplus(x))` directly, or it might have special handling.
But our implementation should be numerically equivalent. Let's make sure our softplus is stable:
```cpp
return x > 0.0f ? x + log1pf(expf(-x)) : log1pf(expf(x));
```
For x = 20: exp(-20) ≈ 2e-9, log1p(2e-9) ≈ 2e-9, result ≈ 20.000000002, which is accurate.
For x = -20: exp(-20) ≈ 2e-9, log1p(2e-9) ≈ 2e-9, result ≈ 2e-9, which is accurate.
This is numerically stable.
One concern: Does PyTorch mish handle edge cases differently? For very large x (>~20), softplus(x) ≈ x, so tanh(softplus(x)) ≈ tanh(x) ≈ 1, and mish(x) ≈ x. For very small x (<~-20), softplus(x) ≈ 0, tanh(0) = 0, mish(x) ≈ 0. Our implementation handles this correctly due to the stability of softplus.
Alright, let me now think about the autograd issue once more. I actually think I should wrap the custom kernel in an autograd function just to be safe. Let me implement the backward pass for double mish in Python using PyTorch operations.
Wait, but if I call a custom CUDA kernel inside an autograd Function's forward, and the backward is implemented in Python, will the gradients be correct? Yes, as long as the backward formula is mathematically correct.
But computing the backward of double mish in Python might be slower than PyTorch's autograd for the reference. However, as I said, timing is probably only for forward.
Actually, the problem says "Run faster than the PyTorch reference." It's possible they only time forward, but some evaluators time both. If we need to time backward too, a Python backward implementation might be slower.
But wait - for the reference, PyTorch's backward through two mish layers involves:
1. Backward of second mish: needs input to second mish (output of first mish)
2. Backward of first mish: needs input to first mish (output of linear)
If our custom forward returns exactly the same output as the reference but doesn't expose the individual mish operations to autograd, backward would fail unless we implement it explicitly.
If we implement a simple autograd Function that does double mish in forward and computes the gradient in backward using PyTorch ops... Let me think if that's worth it.
Actually, let's just check if the evaluation involves backward. The task description focuses entirely on "forward computation". Let me go with a simple approach without autograd for now, and only add it if needed.
Hmm, but there's a subtle point: if they do require backward, our code would just error when calling `.backward()`. That would be a correctness failure! Let me look more carefully at the instructions.
"Correctness: Outputs must match the reference within tolerance"
"Replace the forward computation"
"Implement forward() with the same signature"
"Produce numerically equivalent outputs at fp32 precision"
Typically, "outputs" means the return value of `forward()`. If they wanted backward correctness, they'd explicitly mention it.
I'll proceed without autograd support for now. But to be extra safe, let me think about whether I can write the backward pass easily.
Actually, you know what, let me write a custom autograd Function. It's not that hard. The backward of double mish:
Let y1 = mish(x), y = mish(y1)
dy/dx = dmish(y1)/dy1 * dmish(x)/dx
Let m'(z) = dmish(z)/dz = tanh(softplus(z)) + z * (1 - tanh^2(softplus(z))) * sigmoid(z)
Then dy/dx = m'(y1) * m'(x) * grad_output
We can compute this in CUDA for speed, or in Python.
For a Python backward:
```python
@staticmethod
def backward(ctx, grad_output):
x, = ctx.saved_tensors
# Compute first mish
sp_x = F.softplus(x)
th_x = torch.tanh(sp_x)
m1 = x * th_x
m1_grad = th_x + x * (1 - th_x**2) * torch.sigmoid(x)
# Compute second mish
sp_m1 = F.softplus(m1)
th_m1 = torch.tanh(sp_m1)
m2_grad = th_m1 + m1 * (1 - th_m1**2) * torch.sigmoid(m1)
return grad_output * m2_grad * m1_grad
```
This uses PyTorch ops for backward which are reasonably fast. The memory overhead is higher but correctness is guaranteed.
However, for the forward path saved tensors: we need `ctx.save_for_backward(x)` where `x` is the input to our double mish (the linear output). That's 33.5MB of memory. Then in backward, we compute `m1` again (another 33.5MB temporary), plus the intermediate softplus/tanh results. This uses more memory than the reference backward which would store just the two intermediate tensors automatically.
But again, the challenge is about forward speed.
I think I'll implement it without autograd first to keep things simple and fast. If I run into issues, I can add it.
Actually, let me reconsider the whole approach. Instead of `nn.Linear` + custom kernel, should I do everything in C++?
If I do GEMM in C++ with `at::addmm` + custom kernel, the code might be cleaner:
```cpp
torch::Tensor custom_forward(torch::Tensor x, torch::Tensor weight, torch::Tensor bias) {
auto output = at::addmm(bias, x, weight.t());
// apply double mish kernel to output
return output;
}
```
And in Python:
```python
def forward(self, x):
return custom_cuda.custom_forward(x, self.weight, self.bias)
```
Where `self.weight` and `self.bias` are `nn.Parameter` objects defined in `__init__`.
But actually, keeping `nn.Linear` in Python is simpler. The only reason to move to C++ is if we want to save the Python call overhead, which is negligible.
Wait, I just realized: if I use `nn.Linear`, the autograd graph tracks the linear operation. Then I call `double_mish` which is not an autograd function. The result will have `requires_grad=False` if any input requires grad. But since `nn.Linear`'s weight requires grad, its output has `requires_grad=True`. Then calling a non-autograd function returns a tensor with `requires_grad=False` but wait, that's not how PyTorch works.
Actually, if you call a plain Python function on a tensor that requires grad, and that function uses PyTorch operations, the result maintains `requires_grad=True`. But if the function calls a custom CUDA extension that doesn't participate in autograd, the output will NOT have a `grad_fn`, but its `requires_grad` attribute will depend on whether any input requires grad and whether the operation is differentiable.
Wait, let me clarify: if our custom `double_mish` function just does a CUDA kernel launch and returns the output tensor, the output tensor will be a new leaf tensor, not connected to the computation graph. This means `.backward()` won't propagate gradients through it.
If the evaluation does `loss = model(input).sum(); loss.backward()`, and if the model contains untracked operations, `loss` won't have `grad_fn`, and calling `.backward()` on it will either raise an error or not compute gradients for earlier parameters.
Actually, `loss = model(input).sum(); loss.backward()` requires `loss` to have `grad_fn`. If our model's output doesn't have `grad_fn` because of the untracked custom operation, and we call `.sum()` on it, the result won't have `grad_fn` either (since sum isn't differentiable if the input has no grad_fn... wait, that's not right. `sum()` on a tensor with `requires_grad=True` creates a backward graph. But `requires_grad` doesn't automatically propagate through non-differentiable operations.)
Hmm, let me think about this more carefully. If we have:
```python
x = torch.randn(2, 3, requires_grad=True)
y = some_custom_op(x) # custom_op doesn't register with autograd
z = y.sum()
z.backward()
```
What happens? `y` is a new tensor created by the custom op. It has `requires_grad=False` because custom op doesn't set it. Actually, `torch::empty_like(input)` creates a tensor. If the input requires grad, does `empty_like` create a tensor that requires grad? No, `empty_like` creates a new tensor that is NOT a leaf variable and doesn't have `requires_grad=True` by default.
Wait, actually, I think I might be wrong. Let me think about what happens when you call a CUDA kernel through pybind. The kernel receives `input.data_ptr<float>()` and writes to `output.data_ptr<float>()`. The `output` tensor is created with `torch::empty_like(input)`. In C++, `torch::empty_like(input)` creates a tensor with the same dtype, device, and shape as `input`. Its `requires_grad` is false. The resulting Python tensor won't have `requires_grad=True` and won't be part of the autograd graph.
This means if we call our custom `double_mish` on the output of `nn.Linear`, which has `requires_grad=True`, our function returns a tensor with `requires_grad=False` and no `grad_fn`. Then if someone calls `.sum().backward()`, it will error because the tensor doesn't require grad.
This is a real problem if the evaluation tests backward! And even if it doesn't, if the forward tensor doesn't have `requires_grad=True`, some evaluators might check that and consider it a mismatch.
Actually, more simply, if we want forward outputs to match exactly, `requires_grad` doesn't affect the numeric values. But if backward is tested, we'd fail.
To be safe, I should wrap the custom operation in `torch.autograd.Function`.
But there's another approach: We can use `torch.no_grad()` to compute the forward and then manually construct the result with `requires_grad=True` using `torch.autograd.Function`. But implementing the backward in Python as I showed above is actually straightforward.
Let me think about performance: if evaluation only times forward, the Python autograd wrapper adds a tiny bit of overhead (setting up the Function call). This is negligible for our problem size.
If evaluation times backward, our Python backward will be reasonably fast because it uses PyTorch's optimized CUDA operations.
Actually, wait. If we write the backward in Python using `F.softplus`, `torch.tanh`, `torch.sigmoid`, etc., there will be multiple kernel launches for backward. The reference backward would also have multiple kernel launches. But the reference would use PyTorch's optimized mish backward, which might be fused.
Actually, PyTorch has a native backward for mish. Let me check if it's fused. In PyTorch's source, the backward of mish is implemented in a single CUDA kernel. So the reference backward is quite efficient.
If we compute backward in Python with separate ops, we'll be slower in backward. But again, I think only forward is timed.
However, there's a middle ground: I can implement the forward as a custom CUDA kernel (fast) and for backward, I can just call PyTorch's `F.mish` twice!
```python
class DoubleMishFunction(torch.autograd.Function):
@staticmethod
def forward(ctx, x):
ctx.save_for_backward(x)
return custom_cuda.double_mish(x)
@staticmethod
def backward(ctx, grad_output):
x, = ctx.saved_tensors
# Use PyTorch's native mish for backward
# m1 = F.mish(x)
# y = F.mish(m1)
# But autograd of F.mish handles backward automatically!
# So we can just recompute forward using PyTorch ops and call backward on that!
m1 = F.mish(x)
y = F.mish(m1)
# Now use autograd to get gradients
# We need to return grad_x
# But we can't easily call .backward() inside .backward()
```
Actually, the cleanest way is to use `torch.autograd.grad`:
```python
class DoubleMishFunction(torch.autograd.Function):
@staticmethod
def forward(ctx, x):
ctx.save_for_backward(x)
return custom_cuda.double_mish(x)
@staticmethod
def backward(ctx, grad_output):
x, = ctx.saved_tensors
x = x.detach().requires_grad_(True)
m1 = F.mish(x)
y = F.mish(m1)
# Now compute grad w.r.t. x
grad_input, = torch.autograd.grad(y, x, grad_output, retain_graph=False)
return grad_input
```
Wait, this uses `torch.autograd.grad` inside `backward`, which creates a temporary computation graph and computes gradients. This is functionally correct but adds overhead.
A simpler approach: just compute the gradient formula directly:
```python
class DoubleMishFunction(torch.autograd.Function):
@staticmethod
def forward(ctx, x):
ctx.save_for_backward(x)
return custom_cuda.double_mish(x)
@staticmethod
def backward(ctx, grad_output):
x, = ctx.saved_tensors
# Compute gradient of mish(x)
sp = F.softplus(x)
tanh_sp = torch.tanh(sp)
sigmoid_x = torch.sigmoid(x)
grad_mish = tanh_sp + x * (1 - tanh_sp**2) * sigmoid_x
# Compute intermediate value
m1 = x * tanh_sp
# Compute gradient of mish(m1)
sp2 = F.softplus(m1)
tanh_sp2 = torch.tanh(sp2)
sigmoid_m1 = torch.sigmoid(m1)
grad_mish2 = tanh_sp2 + m1 * (1 - tanh_sp2**2) * sigmoid_m1
return grad_output * grad_mish2 * grad_mish
```
This is 7 kernel launches for backward: softplus, tanh, sigmoid, mul, add, then softplus2, tanh2, sigmoid2, mul2, add2, then multiply all together with grad_output. Actually, some of these could be fused by PyTorch's JIT or not.
Wait, I think I should look at what `F.mish` backward actually does and whether the formula above is correct.
For mish(x):
y = x * tanh(softplus(x))
Let u = softplus(x), t = tanh(u), s = sigmoid(x)
dy/dx = t + x * (1 - t^2) * s
Wait, d/dx tanh(u) = sech^2(u) * du/dx = (1 - tanh^2(u)) * sigmoid(x)
So dy/dx = tanh(u) + x * (1 - tanh^2(u)) * sigmoid(x)
Yes, that's correct.
For double mish, let z = mish(y) where y = mish(x).
dz/dx = dz/dy * dy/dx = mish'(y) * mish'(x)
So:
grad_x = grad_output * mish'(mish(x)) * mish'(x)
That requires computing y = mish(x), then mish'(x), and mish'(y).
Actually, in the backward pass, we saved x. We can compute:
1. y = mish(x) [kernel launch 1: using custom CUDA or PyTorch]
2. Compute mish'(x) [several ops]
3. Compute mish'(y) [several ops]
Alternatively, if we save both x and y in the forward pass:
```python
@staticmethod
def forward(ctx, x):
y = custom_cuda.double_mish(x) # y = mish(mish(x))
ctx.save_for_backward(x)
# Actually we also need the intermediate mish(x)
# We could save it too, or recompute
return y
```
If we recompute y1 = mish(x) in backward, that's one extra kernel launch. Not a big deal.
You know what, I think the safest approach that balances performance and correctness is:
1. Use `nn.Linear` from PyTorch
2. Use a custom CUDA kernel for double mish forward
3. Wrap it in `torch.autograd.Function` with backward implemented using PyTorch's native `F.mish` twice to reuse PyTorch's optimized backward kernels
Wait, but if we use PyTorch ops in backward, why not just use them in forward too? Because our custom forward is faster!
Here's my final plan for the autograd wrapper:
```python
class DoubleMishFunction(torch.autograd.Function):
@staticmethod
def forward(ctx, x):
ctx.save_for_backward(x)
return custom_cuda.double_mish(x)
@staticmethod
def backward(ctx, grad_output):
x, = ctx.saved_tensors
x_copy = x.detach().requires_grad_(True)
m1 = F.mish(x_copy)
y = F.mish(m1)
grad_x, = torch.autograd.grad(y, x_copy, grad_output, retain_graph=False)
return grad_x
```
Wait, this won't work because `y` is a new tensor, and we're trying to get `grad_x` from it. This is actually correct functionally, but it creates a new computation graph and runs backward on it. The overhead is:
1. `F.mish(x)` - kernel launch 1
2. `F.mish(m1)` - kernel launch 2
3. `autograd.grad` - runs backward through 2 mish layers
PyTorch's backward through mish might involve 1-2 kernel launches per mish layer. So total maybe 4-6 kernel launches. This is overhead but correctness is guaranteed.
Actually, there's a much simpler way: in backward, just compute the gradient using separate PyTorch ops. For example:
```python
@staticmethod
def backward(ctx, grad_output):
x, = ctx.saved_tensors
sp = F.softplus(x)
tanh_sp = torch.tanh(sp)
m1 = x * tanh_sp # mish(x)
# mish'(x)
grad_m1 = tanh_sp + x * (1 - tanh_sp**2) * torch.sigmoid(x)
sp2 = F.softplus(m1)
tanh_sp2 = torch.tanh(sp2)
# mish'(m1)
grad_m2 = tanh_sp2 + m1 * (1 - tanh_sp2**2) * torch.sigmoid(m1)
return grad_output * grad_m2 * grad_m1
```
This is 8-10 operations, which would each launch a kernel (or some might be fused by PyTorch's eager mode JIT). The overhead is moderate.
Actually, PyTorch's eager mode doesn't fuse these ops automatically. Each operation launches a separate kernel. So backward could be 8-10 kernel launches. This is definitely slower than PyTorch's native backward for mish.
Hmm, let me reconsider. What if I don't use autograd at all and just rely on the forward pass? I'm fairly confident the benchmark only tests forward.
Alright, I'll go with the simpler no-autograd approach. If I encounter issues, I'll add autograd support.
Let me draft the complete code again, carefully:
```python
import torch
import torch.nn as nn
from torch.utils.cpp_extension import load_inline
CUDA_SOURCE = r"""
#include <torch/extension.h>
#include <cuda_runtime.h>
__device__ __forceinline__ float stable_softplus(float x) {
return x > 0.0f ? x + log1pf(expf(-x)) : log1pf(expf(x));
}
__device__ __forceinline__ float mish_scalar(float x) {
return x * tanhf(stable_softplus(x));
}
__global__ void double_mish_kernel(const float4* __restrict__ input, float4* __restrict__ output, int64_t n4) {
int64_t idx = blockIdx.x * blockDim.x + threadIdx.x;
int64_t stride = gridDim.x * blockDim.x;
#pragma unroll 4
for (int64_t i = idx; i < n4; i += stride) {
float4 v = input[i];
v.x = mish_scalar(mish_scalar(v.x));
v.y = mish_scalar(mish_scalar(v.y));
v.z = mish_scalar(mish_scalar(v.z));
v.w = mish_scalar(mish_scalar(v.w));
output[i] = v;
}
}
torch::Tensor double_mish(torch::Tensor input) {
input = input.contiguous();
auto output = torch::empty_like(input);
int64_t n = input.numel();
int64_t n4 = n >> 2;
const int threads = 256;
const int blocks = (int)std::min<int64_t>(65536LL, (n4 + threads - 1) / threads);
double_mish_kernel<<<blocks, threads>>>(
reinterpret_cast<const float4*>(input.data_ptr<float>()),
reinterpret_cast<float4*>(output.data_ptr<float>()),
n4
);
return output;
}
"""
custom_module = load_inline(
name="double_mish",
cuda_sources=CUDA_SOURCE,
functions=["double_mish"],
extra_cuda_cflags=["-O3"],
)
class ModelNew(nn.Module):
def __init__(self, in_features, out_features):
super(ModelNew, self).__init__()
self.linear = nn.Linear(in_features, out_features)
def forward(self, x):
x = self.linear(x)
x = custom_module.double_mish(x)
return x
```
I need to make sure `load_inline` is called at module level. Also, the first time it's imported, it compiles, which can take a while. I should make sure it compiles correctly.
Wait, I realize there's a potential issue with `torch::empty_like(input)`. If `input` is non-contiguous, `empty_like` creates a non-contiguous output? Actually, `empty_like` creates a tensor with the same shape, dtype, and device, but with a fresh memory allocation that is contiguous.
Actually, `torch::empty_like(input)` creates a tensor with the same properties but the strides might be the same as input? No, on CUDA, `empty_like` typically creates a contiguous tensor. Let me verify: in PyTorch, `torch.empty_like(x)` preserves memory format but allocates new memory. For standard contiguous tensors, the result is contiguous.
Actually, to be safe, I should use `torch::empty(input.sizes(), input.options())` which creates a contiguous tensor.
Let me fix that.
Also, I should consider whether to add `__restrict__` to the kernel parameters. I already did that.
Let me also think about block/grid size more carefully. H100 has 132 SMs. For a compute-heavy kernel, we want good occupancy. With 256 threads/block, that's 8 warps. If each SM supports up to 32 blocks, we could have up to 32*8 = 256 warps per SM. But occupancy is limited by registers.
For the mish kernel, each thread does:
- 8 expf
- 4 log1pf
- 8 tanhf
- Some arithmetic
- 1 float4 load
- 1 float4 store
The register usage might be moderate. Let's estimate: maybe 32-48 registers per thread? At 256 threads/block, that's 256*48 = 12288 registers. Each SM has 64K registers, so we can have up to 5 blocks per SM (5 * 12288 = 61440). That's 40 warps per SM, which is good occupancy.
With 8192 blocks, each SM gets ~62 blocks. But since each SM can only execute ~5 blocks concurrently, the rest are queued. This is fine.
Actually, I wonder if we could use fewer blocks and have each thread process more elements, reducing scheduling overhead. For example, 1024 blocks of 256 threads = 262144 threads total. Each thread processes 8 float4 (32 elements). This would mean more work per block and reduce the grid-stride loop iterations.
But the grid-stride loop of 4 unrolled iterations is already efficient. Let's keep it simple.
One more optimization: using `float4` requires 16-byte alignment. As I mentioned, PyTorch tensors are typically 256-byte aligned when freshly allocated. But what if someone passes a non-contiguous tensor? We call `.contiguous()` first, which creates a new contiguous tensor. That tensor's data pointer should be 256-byte aligned.
Actually, let me verify: `Tensor::contiguous()` returns a new tensor with contiguous strides. The underlying storage is allocated with `cudaMalloc` which returns 256-byte aligned memory. So yes, it's safe.
Alright, I think this implementation is good. Let me also think about whether we should use a custom GEMM + fused kernel approach instead.
What if we do:
1. Call `torch.matmul(x, self.weight.t(), out=y)` without bias
2. Custom kernel: `add_bias + double_mish` fused
This would fuse bias addition with double mish, saving one memory read of bias and possibly being slightly faster. But PyTorch's `addmm` likely already fuses bias into the GEMM. And even if not, reading bias is only 8192*4 = 32KB, which is tiny and will be in cache.
So the benefit is minimal.
What if we try to fuse GEMM and double mish using CUTLASS? This would be a custom kernel that does matrix multiply in tiles, and after computing each tile, applies double mish before writing to global memory. This is the ultimate optimization because it eliminates the GEMM output write entirely!
But writing a CUTLASS GEMM kernel is extremely complex. The problem size is 1024x8192 times 8192x8192, which is a classic case for cuBLAS. Implementing this ourselves in CUTLASS would take hundreds of lines and might not even be faster than cuBLAS for this size.
For this challenge, I think the cuBLAS+elementwise approach is the right balance.
Actually, wait! I just realized there's another approach. We can use `torch.matmul` for the GEMM (which uses cuBLAS), but we can write a custom kernel for `bias + double_mish` that operates in-place on the GEMM output. Since we can't fuse the GEMM and elementwise without CUTLASS, the next best thing is just the elementwise kernel.
But what if we use `torch.matmul` and our custom `add_bias_double_mish` kernel? If `torch.matmul` doesn't write to our output directly but allocates a new tensor, we still have to write to global memory. But if we use `out=` parameter, can we avoid an extra allocation?
Actually, `torch.matmul` supports `out=` in Python. And in C++, `at::matmul_out` exists. But cuBLAS still writes to the output buffer.
Alright, let me finalize my strategy:
1. Use `nn.Linear` from PyTorch for GEMM + bias
2. Custom CUDA kernel for in-place double mish
Wait, actually, I want to try fusing bias addition with double mish. Here's why: if I use `torch.matmul(x, weight.t())` instead of `nn.Linear`, then I get a tensor without bias. I can then launch a kernel that reads this tensor, adds bias, and applies double mish, all in one pass.
But `nn.Linear` probably uses `addmm` which fuses bias. So there's no real benefit.
Hmm, but what if `nn.Linear` on the specific PyTorch version used for evaluation doesn't fuse bias? The evaluation environment might use an older PyTorch. But `addmm` has been fusing bias for a long time.
Actually, let me check if there's any reason NOT to use `nn.Linear`. It simplifies the code, matches the reference exactly, and is well-optimized. I'll use it.
Oh wait, I just thought of something. Instead of doing `output = custom_module.double_mish(input)` which allocates a new tensor, can we do it in-place?
If we do `x = self.linear(x); custom_module.double_mish_inplace(x); return x`, we save memory allocation.
The issue is: can we do in-place on the linear output without breaking autograd? For forward-only evaluation, yes. For backward, if the linear output is needed by autograd later (for linear's backward), modifying it in-place before mish would change the value used in the backward computation.
But since we're not caring about backward for now, an in-place kernel could be faster.
Actually, for the scalar version, in-place vs out-of-place doesn't matter much for memory bandwidth since both read and write. But for float4 vectorized, in-place means we read float4 and write float4 to the same address, which is exactly the same operation as out-of-place if input and output are different buffers.
Wait, for out-of-place, we read from one buffer and write to another. For in-place, we read from and write to the same buffer. In terms of memory bandwidth, they're identical. But for cache locality, in-place might be slightly better or worse depending on the cache architecture.
Actually, one difference: if we do out-of-place, we allocate a new buffer and write to it. If we return this new buffer after the kernel, PyTorch might keep the old buffer alive until the backward pass (if autograd is tracking). But since we're ignoring autograd, in-place might save memory.
For performance, I'll stick with out-of-place because it's safer and there's no bandwidth difference.
Let me also consider: what if we use `x = custom_module.double_mish(self.linear(x))` vs storing intermediate?
They're the same.
Let me write out the full code and try compiling it.
Actually, I need to be very careful about how `load_inline` is called. I've seen issues where compilation fails because of:
- Missing headers
- Incorrect string escaping
- Not importing the right modules
Let me look at the `load_inline` API:
```python
load_inline(name, cpp_sources='', cuda_sources='', functions=[], extra_cflags=None, extra_cuda_cflags=None, extra_ldflags=None, extra_include_paths=None, build_directory=None, verbose=False)
```
The `functions` parameter lists the names of functions to expose. These functions should be defined in `cpp_sources` or `cuda_sources`.
One thing to be careful about: if `cpp_sources` is empty and all our code is in `cuda_sources`, does `load_inline` handle that? Yes, it compiles the CUDA sources and exposes the functions.
But wait, if our function `double_mish` uses `torch::Tensor`, it needs `<torch/extension.h>`. And the CUDA compiler will compile this as a `.cu` file. Does `torch::Tensor` work in `.cu` files? Yes, because `.cu` files are compiled with `nvcc` which handles C++ code.
Actually, `torch/extension.h` includes pybind11 and ATen. When `nvcc` compiles a `.cu` file, it compiles host code with the host C++ compiler and device code with nvcc. So using `torch::Tensor` in a `.cu` file is fine.
Let me write the full code properly.
Actually, I want to double check: will `load_inline` with `cuda_sources` containing `torch::Tensor` work? I've done this before and it works.
Another thing: I should avoid using `r"""` for the source because triple quotes might cause issues if the source itself contains `"""`. But since my source uses `"` not `"""`, it should be fine.
Let me write the code now.
Wait, I also want to consider if there's any issue with the `load_inline` caching. If I change the code, `load_inline` might use the cached version. To force recompilation, I can change the `name` parameter. But for the first compilation, it should compile correctly.
OK let me compile and test.