reasoning
Great! H100 GPU. Let me analyze the problem more carefully.
The operations in forward():
1. delta = softplus(proj_delta(x)) # (B, L, D)
2. B_t = proj_B(x) # (B, L, N)
3. C_t = proj_C(x) # (B, L, N)
4. A = -exp(A_log) # (D, N)
5. dA = exp(delta.unsqueeze(-1) * A.unsqueeze(0).unsqueeze(0)) # (B, L, D, N)
6. dB = delta.unsqueeze(-1) * B_t.unsqueeze(2) # (B, L, D, N)
7. x_db = x.unsqueeze(-1) * dB # (B, L, D, N)
8. Recurrence: for t in range(L): h = dA[:, t] * h + x_db[:, t]; y_t = (h * C_t[:, t].unsqueeze(1)).sum(dim=-1)
9. y = y + x * D # skip connection
The key optimization opportunities:
1. Fuse the projections with parameter computation (dA, dB, x_db)
2. Fuse the recurrence - keep h in registers/shared memory
3. Avoid materializing intermediate tensors (dA, dB, x_db) - compute on the fly
4. Store C_t and use it in the recurrence without extra reads
Let me design a kernel:
1. Each thread block handles one (batch, channel) pair
2. Shared memory for:
- h (N elements per channel - small enough)
- C_t for current timestep (N elements)
- Possibly x_db and dA for current timestep (N elements)
Actually, let me think about the parallelization strategy:
- We have (B, L, D) output
- The recurrence is sequential in L but parallel in B and D
- So each thread block can handle one (b, d) pair and iterate through L
For each (b, d):
- Need to iterate t = 0 to L-1
- Maintain h of size N (in registers)
- For each t:
- Load delta[b,t,d], x[b,t,d]
- Load B_t[b,t,:] (N values)
- Load C_t[b,t,:] (N values)
- Load A[d,:] (N values)
- Compute dA = exp(delta * A), dB = delta * B_t, x_db = x * dB
- Update h = dA * h + x_db
- Compute y = sum(h * C_t) = dot(h, C_t)
Actually, the thread organization:
- We need to compute for each (b, d, t) the output y
- The recurrence means we need to process t sequentially
- But within each t, the computation for different (b, d) pairs is independent
So the parallelization is:
- Block: (b, d) pair - there are B*D blocks
- Each block iterates through L timesteps
- Within a block, we can have multiple threads cooperating on the N-dimensional vectors
Let me think about how to organize threads within a block:
- N = d_state = 16 is small
- We can have each thread handle part of the N dimension
- Or have one thread handle all N (simpler)
For simplicity, let's have each thread handle one (b, d) pair entirely:
- Grid dimension: B * D threads (one per channel per batch element)
- But we need blocks, so: blockDim.x = 256 (or some power of 2), and we have enough blocks
Actually, let's do:
- Each thread handles one (b, d) pair
- blockDim = (256, 1, 1)
- gridDim = ((B*D + 255) / 256, 1, 1)
Each thread:
1. Compute its (b, d) index
2. Load A[d, :] into registers (N values)
3. For t = 0 to L-1:
a. Load delta[b,t,d], x[b,t,d]
b. Load B_t[b,t,:], C_t[b,t,:] (N values each)
c. Compute: dA = exp(delta * A), dB = delta * B_t, x_db = x * dB
d. Update h = dA * h + x_db
e. Compute y = sum(h * C_t)
f. Store y[b,t,d]
4. Add skip connection: y = y + x * D[d]
This should work well. Let me write the kernel.
For the projections, I need to include them in the kernel or do them separately. Let me check the sizes:
- proj_delta: (D, D) = (256, 256)
- proj_B: (D, N) = (256, 16)
- proj_C: (D, N) = (256, 16)
These are small, so we can either:
1. Do them on the CPU and pass the results
2. Fuse them into the main kernel
For maximum performance, let's fuse everything. But this requires handling the projections inside the kernel.
Actually, let me first write a version that assumes the projections are done externally (as PyTorch operations), and only the recurrence and final steps are in CUDA. Then I can optimize further if needed.
Wait, but the problem says to replace the entire forward. Let me think about this differently.
I can have the kernel do:
1. All projections (using weight matrices passed as arguments)
2. Compute delta, dA, dB, x_db
3. Run the recurrence
4. Apply skip connection
The projections are:
- proj_delta(x): output is (B, L, D), weight is (D, D) - this is a matrix multiplication
- proj_B(x): output is (B, L, N), weight is (D, N)
- proj_C(x): output is (B, L, N), weight is (D, N)
For these projections, I can use cuBLAS or write my own small matrix multiplication kernels. But since these are large matrix multiplications, PyTorch's cuBLAS should be efficient. Let me just do the projections with PyTorch and then call a custom kernel for the rest.
Actually, for a proper implementation, let me structure it as:
1. Use PyTorch for the three linear projections
2. Use a custom CUDA kernel for: delta computation, A computation, dA/dB/x_db computation, recurrence, and output
Let me write this:
```python
import torch
import torch.nn as nn
from torch.utils.cpp_extension import load_inline
# Define the CUDA kernel
cuda_source = """
#include <torch/extension.h>
#include <cuda.h>
#include <cuda_runtime.h>
#include <cmath>
__global__ void selective_scan_kernel(
const float* __restrict__ delta_proj_out, // (B, L, D) - output of proj_delta, BEFORE softplus
const float* __restrict__ B_t, // (B, L, N)
const float* __restrict__ C_t, // (B, L, N)
const float* __restrict__ A_log, // (D, N)
const float* __restrict__ x, // (B, L, D)
const float* __restrict__ D_param, // (D,)
float* __restrict__ y, // (B, L, D) output
int B, int L, int D, int N
) {
int bd_idx = blockIdx.x * blockDim.x + threadIdx.x;
if (bd_idx >= B * D) return;
int b = bd_idx / D;
int d = bd_idx % D;
// Load A_log[d, :] and compute A = -exp(A_log)
float A[16]; // Assuming N <= 16
float A_neg_exp[16];
for (int n = 0; n < N; n++) {
A_neg_exp[n] = -expf(A_log[d * N + n]);
}
// Initialize hidden state h
float h[16] = {0.0f};
// Process each timestep
for (int t = 0; t < L; t++) {
// Load delta (softplus of projection output)
float delta_proj = delta_proj_out[(b * L + t) * D + d];
float delta = log1pf(expf(delta_proj)); // softplus
// Load x
float x_val = x[(b * L + t) * D + d];
// Compute dA, dB, x_db and update h
for (int n = 0; n < N; n++) {
float dA = expf(delta * A_neg_exp[n]);
float B_t_val = B_t[(b * L + t) * N + n];
float dB = delta * B_t_val;
float x_db = x_val * dB;
h[n] = dA * h[n] + x_db;
}
// Compute output y_t = sum(h * C_t)
float y_t = 0.0f;
for (int n = 0; n < N; n++) {
float C_t_val = C_t[(b * L + t) * N + n];
y_t += h[n] * C_t_val;
}
// Add skip connection and store
float D_val = D_param[d];
y[(b * L + t) * D + d] = y_t + x_val * D_val;
}
}
torch::Tensor selective_scan_cuda(
torch::Tensor delta_proj_out,
torch::Tensor B_t,
torch::Tensor C_t,
torch::Tensor A_log,
torch::Tensor x,
torch::Tensor D_param
) {
int B = x.size(0);
int L = x.size(1);
int D = x.size(2);
int N = A_log.size(1);
auto y = torch::empty_like(x);
const int threads = 256;
const int blocks = (B * D + threads - 1) / threads;
selective_scan_kernel<<<blocks, threads>>>(
delta_proj_out.data_ptr<float>(),
B_t.data_ptr<float>(),
C_t.data_ptr<float>(),
A_log.data_ptr<float>(),
x.data_ptr<float>(),
D_param.data_ptr<float>(),
y.data_ptr<float>(),
B, L, D, N
);
return y;
}
"""
cpp_source = """
torch::Tensor selective_scan_cuda(
torch::Tensor delta_proj_out,
torch::Tensor B_t,
torch::Tensor C_t,
torch::Tensor A_log,
torch::Tensor x,
torch::Tensor D_param
);
"""
# Load the extension
selective_scan_ext = load_inline(
name='selective_scan_ext',
cpp_sources=cpp_source,
cuda_sources=cuda_source,
functions=['selective_scan_cuda'],
extra_cuda_cflags=['-O3'],
verbose=True
)
class ModelNew(nn.Module):
def __init__(self, d_model: int, d_state: int):
super(ModelNew, self).__init__()
self.d_model = d_model
self.d_state = d_state
self.A_log = nn.Parameter(torch.randn(d_model, d_state))
self.proj_B = nn.Linear(d_model, d_state, bias=False)
self.proj_C = nn.Linear(d_model, d_state, bias=False)
self.proj_delta = nn.Linear(d_model, d_model, bias=True)
self.D = nn.Parameter(torch.ones(d_model))
def forward(self, x: torch.Tensor) -> torch.Tensor:
B, L, D = x.shape
N = self.d_state
# Do projections with PyTorch
delta_proj_out = self.proj_delta(x)
B_t = self.proj_B(x)
C_t = self.proj_C(x)
# Call CUDA kernel for the rest
y = selective_scan_ext.selective_scan_cuda(
delta_proj_out,
B_t,
C_t,
self.A_log,
x,
self.D
)
return y
```
Wait, I realized there's an issue. The reference computes A = -exp(A_log) and then dA = exp(delta * A). But A is already negative, so delta * A is negative, and exp of a negative is < 1. Let me check the math again.
In the reference:
```python
A = -torch.exp(self.A_log) # A is negative
dA = torch.exp(delta.unsqueeze(-1) * A.unsqueeze(0).unsqueeze(0)) # exp(delta * A)
```
So A = -exp(A_log), which makes A negative.
Then dA = exp(delta * A) = exp(delta * (-exp(A_log))) = exp(-delta * exp(A_log)).
This is between 0 and 1 (since delta > 0 and exp(A_log) > 0).
So dA represents a decay factor. This makes sense for the recurrence h = dA * h + x_db.
Let me fix my kernel to match this computation.
Actually wait, in my kernel I'm doing:
```cpp
A_neg_exp[n] = -expf(A_log[d * N + n]);
...
float dA = expf(delta * A_neg_exp[n]);
```
This should be correct: A_neg_exp = -exp(A_log), and then dA = exp(delta * A_neg_exp).
Let me verify by tracing through:
- A_log is some learned parameter
- A = -exp(A_log) (this is negative)
- delta = softplus(proj_delta(x)) (this is positive)
- dA = exp(delta * A) = exp(delta * (-exp(A_log))) (this is < 1)
Yes, my computation is correct.
Now let me also think about memory access patterns:
- delta_proj_out, B_t, C_t, x are all accessed in a coalesced manner (within a warp, threads access adjacent d indices)
- A_log is accessed the same for all timesteps (cached in L1/registers)
- D_param is accessed the same for all timesteps (cached)
The memory access pattern seems reasonable. The main optimization would be to:
1. Fuse the projections into the kernel (avoid storing B_t, C_t, delta_proj_out to global memory)
2. Use shared memory to cache values across threads
Let me first compile and test this version, then optimize further if needed.
Actually, let me think about potential numerical issues:
1. exp(A_log) could overflow if A_log is large. But typically A_log is initialized from randn, so it should be reasonable.
2. exp(delta * A) could underflow if delta is large and A is very negative. This would make dA close to 0.
Let me also double-check the softplus computation:
```python
delta = torch.nn.functional.softplus(self.proj_delta(x))
```
softplus(x) = log(1 + exp(x))
In my kernel I have:
```cpp
float delta = log1pf(expf(delta_proj));
```
This is equivalent to log(1 + exp(delta_proj)) = softplus(delta_proj).
But there could be numerical issues:
- If delta_proj is large, exp(delta_proj) could overflow.
- A stable implementation would be: max(delta_proj, 0) + log(1 + exp(-abs(delta_proj)))
Let me use a more numerically stable version:
```cpp
float delta;
if (delta_proj > 20.0f) {
delta = delta_proj; // For large values, softplus(x) ≈ x
} else {
delta = log1pf(expf(delta_proj));
}
```
Actually, PyTorch's softplus handles this internally, so let me use a similar approach.
Let me write the complete kernel now.
call → compile_kernel
callcompile_kernel
{
"kernel_code": "import torch\nimport torch.nn as nn\nfrom torch.utils.cpp_extension import load_inline\n\ncuda_source = \"\"\"\n#include <torch/extension.h>\n#include <cuda.h>\n#include <cuda_runtime.h>\n#include <cmath>\n\n__device__ __forceinline__ float softplus_stable(float x) {\n // Numerically stable softplus: log(1 + exp(x))\n if (x > 20.0f) {\n return x;\n }\n return log1pf(expf(x));\n}\n\n__global__ void selective_scan_kernel(\n const float* __restrict__ delta_proj_out, // (B, L, D) - output of proj_delta, BEFORE softplus\n const float* __restrict__ B_t, // (B, L, N)\n const float* __restrict__ C_t, // (B, L, N)\n const float* __restrict__ A_log, // (D, N)\n const float* __restrict__ x, // (B, L, D)\n const float* __restrict__ D_param, // (D,)\n float* __restrict__ y, // (B, L, D) output\n int B, int L, int D, int N\n) {\n int bd_idx = blockIdx.x * blockDim.x + threadIdx.x;\n if (bd_idx >= B * D) return;\n \n int b = bd_idx / D;\n int d = bd_idx % D;\n \n // Load A_log[d, :] and compute A = -exp(A_log)\n // N is small (typically 16), use registers\n float A_neg[16];\n for (int n = 0; n < N; n++) {\n A_neg[n] = -expf(A_log[d * N + n]);\n }\n \n // Initialize hidden state h\n float h[16];\n for (int n = 0; n < N; n++) {\n h[n] = 0.0f;\n }\n \n // Process each timestep\n for (int t = 0; t < L; t++) {\n // Load delta (apply softplus to projection output)\n float delta_proj = delta_proj_out[(b * L + t) * D + d];\n float delta = softplus_stable(delta_proj);\n \n // Load x\n float x_val = x[(b * L + t) * D + d];\n \n // Compute dA, dB, x_db and update h\n for (int n = 0; n < N; n++) {\n float dA = expf(delta * A_neg[n]);\n float B_t_val = B_t[(b * L + t) * N + n];\n float dB = delta * B_t_val;\n float x_db = x_val * dB;\n h[n] = dA * h[n] + x_db;\n }\n \n // Compute output y_t = sum(h * C_t)\n float y_t = 0.0f;\n for (int n = 0; n < N; n++) {\n float C_t_val = C_t[(b * L + t) * N + n];\n y_t += h[n] * C_t_val;\n }\n \n // Add skip connection and store\n float D_val = D_param[d];\n y[(b * L + t) * D + d] = y_t + x_val * D_val;\n }\n}\n\ntorch::Tensor selective_scan_cuda(\n torch::Tensor delta_proj_out,\n torch::Tensor B_t,\n torch::Tensor C_t,\n torch::Tensor A_log,\n torch::Tensor x,\n torch::Tensor D_param\n) {\n int B = x.size(0);\n int L = x.size(1);\n int D = x.size(2);\n int N = A_log.size(1);\n \n auto y = torch::empty_like(x);\n \n const int threads = 256;\n const int blocks = (B * D + threads - 1) / threads;\n \n selective_scan_kernel<<<blocks, threads>>>(\n delta_proj_out.data_ptr<float>(),\n B_t.data_ptr<float>(),\n C_t.data_ptr<float>(),\n A_log.data_ptr<float>(),\n x.data_ptr<float>(),\n D_param.data_ptr<float>(),\n y.data_ptr<float>(),\n B, L, D, N\n );\n \n return y;\n}\n\"\"\"\n\ncpp_source = \"\"\"\ntorch::Tensor selective_scan_cuda(\n torch::Tensor delta_proj_out,\n torch::Tensor B_t,\n torch::Tensor C_t,\n torch::Tensor A_log,\n torch::Tensor x,\n torch::Tensor D_param\n);\n\"\"\"\n\nselective_scan_ext = load_inline(\n name='selective_scan_ext_v1',\n cpp_sources=cpp_source,\n cuda_sources=cuda_source,\n functions=['selective_scan_cuda'],\n extra_cuda_cflags=['-O3'],\n verbose=False\n)\n\nclass ModelNew(nn.Module):\n def __init__(self, d_model: int, d_state: int):\n super(ModelNew, self).__init__()\n self.d_model = d_model\n self.d_state = d_state\n self.A_log = nn.Parameter(torch.randn(d_model, d_state))\n self.proj_B = nn.Linear(d_model, d_state, bias=False)\n self.proj_C = nn.Linear(d_model, d_state, bias=False)\n self.proj_delta = nn.Linear(d_model, d_model, bias=True)\n self.D = nn.Parameter(torch.ones(d_model))\n\n def forward(self, x: torch.Tensor) -> torch.Tensor:\n B, L, D = x.shape\n N = self.d_state\n \n # Do projections with PyTorch (these use cuBLAS which is highly optimized)\n delta_proj_out = self.proj_delta(x)\n B_t = self.proj_B(x)\n C_t = self.proj_C(x)\n \n # Call CUDA kernel for the rest (recurrence + skip connection)\n y = selective_scan_ext.selective_scan_cuda(\n delta_proj_out,\n B_t,\n C_t,\n self.A_log,\n x,\n self.D\n )\n \n return y"
}