reasoning
Let me fix the CUDA code. The issue is `AT_CUDA_CHECK` - I should use `C10_CUDA_KERNEL_LAUNCH_CHECK()` instead. Also, I need to rethink the kernel design significantly.
The current kernel design has issues:
1. `AT_CUDA_CHECK` macro not found - use `C10_CUDA_KERNEL_LAUNCH_CHECK()`
2. The kernel uses a block per (b, i, j) with P threads. But P=64, which means we have 2 warps but only some threads doing work for the hidden dimension H=32.
3. For H=32, only half the threads are doing useful work in the einsum part.
4. The layer norm is being done redundantly for each k loop iteration.
Let me completely redesign this kernel more carefully:
**Key insight**: The einsum `bkih,bkjh->bijh` can be computed as a matrix multiplication. If we reshape:
- a: (B, N, N, H) -> (B*N, N, H) where we index by (b, i, k) for a
- Actually reshaping: a[b, k, i, :] and b[b, k, j, :] - so for fixed b, i, j, we dot-product over k.
For each (b, i, j), we compute sum_k a[b,k,i,:] * b[b,k,j,:].
This is essentially a batched GEMM where for each (b, i, j), the vectors a[b,:,i,:] and b[b,:,j,:] are dotted over the k dimension.
For small N=32, a direct CUDA kernel is feasible, but we need to be smart about it.
**New design approach:**
1. First, in a separate kernel or using a PyTorch ops, compute:
- `x = layer_norm(pair_repr)`
- `a = proj_a(x) * sigmoid(gate_a(x)) * mask`
- `b = proj_b(x) * sigmoid(gate_b(x)) * mask`
2. Then compute the einsum `bkih,bkjh->bijh` using a custom kernel or cuBLAS batched GEMM.
3. Then apply layer_norm_out, out_proj, and the output gate.
Actually, for the einsum, we can view it as:
- For each (b, h), we want C[b, :, :, h] = A[b, :, :, h] @ B[b, :, :, h].T
where A[b, k, i, h] and B[b, k, j, h]
This is a batched matrix multiplication over (b, h) dimensions with matrix sizes (N, N) @ (N, N).T -> (N, N)
Wait, A is (B, N, N, H) and B is (B, N, N, H). For each b and h:
- A_slice = A[b, :, :, h] is (N, N)
- B_slice = B[b, :, :, h] is (N, N)
- C_slice = A_slice @ B_slice.T is (N, N)
Actually: `out[b, i, j, h] = sum_k a[b, k, i, h] * b[b, k, j, h]`
This is: C[b, h, i, j] = sum_k A[b, h, i, k] * B[b, h, j, k]
Where A[b, h, i, k] = a[b, k, i, h] and B[b, h, j, k] = b[b, k, j, h]
So C = A @ B.T where A and B are reshaped to (B*H, N, N)
This is exactly `torch.bmm(A, B.transpose(-1, -2))` for tensors of shape (B*H, N, N).
So we can do:
1. Compute a and b with fused kernel
2. Reshape and use `torch.bmm` for the einsum
3. Apply the rest
This should be very fast since bmm uses cuBLAS.
Actually, even simpler: `out = torch.einsum("bkih,bkjh->bijh", a, b)` in PyTorch likely gets optimized to a batched GEMM. But since we're writing a custom kernel, let me try using bmm explicitly.
Let me write a cleaner two-kernel approach:
Kernel 1: Fused preprocessing kernel
- Input: pair_repr (B, N, N, P), pair_mask (B, N, N)
- Output: a (B, N, N, H), b (B, N, N, H)
- For each (b, n1, n2), compute:
- layer_norm_in on pair_repr[b, n1, n2, :]
- a = proj_a(x) * sigmoid(gate_a(x)) * mask
- b = proj_b(x) * sigmoid(gate_b(x)) * mask
Kernel 2: Fused postprocessing kernel
- Input: einsum_out (B, N, N, H), pair_repr_orig (B, N, N, P)
- Output: final (B, N, N, P)
- For each (b, i, j), compute:
- layer_norm_out on einsum_out[b, i, j, :]
- out_proj
- sigmoid(out_gate(pair_repr_orig[b, i, j, :]))
- multiply
And the einsum in between can be done with `torch.bmm`.
Actually, wait. Let me verify: einsum("bkih,bkjh->bijh", a, b)
This is: for each b, i, j, h: sum over k of a[b,k,i,h] * b[b,k,j,h]
If we reshape: a -> (B, N, N, H), permute to (B, H, N, N) -> (B*H, N, N)
Then b -> (B, H, N, N) -> (B*H, N, N)
out[b,i,j,h] = sum_k a_permuted[b,h,i,k] * b_permuted[b,h,j,k]
So out_permuted[b,h,i,j] = sum_k A[b,h,i,k] * B[b,h,j,k]
This is A @ B^T where A and B are (B*H, N, N).
So: `out = torch.bmm(a_reshaped, b_reshaped.transpose(-1, -2))`
where a_reshaped = a.permute(0,3,1,2).reshape(B*H, N, N)
and b_reshaped = b.permute(0,3,1,2).reshape(B*H, N, N)
Let me verify with a small example:
a = tensor([[[[1,2],[3,4]], [[5,6],[7,8]]]]) # B=1, N=2, K=2, H=2
Actually let me use b,k,i,h indexing:
a[b=0, k=0, i=0, h=0] = 1
a[b=0, k=0, i=0, h=1] = 2
a[b=0, k=0, i=1, h=0] = 3
a[b=0, k=0, i=1, h=1] = 4
a[b=0, k=1, i=0, h=0] = 5
a[b=0, k=1, i=0, h=1] = 6
a[b=0, k=1, i=1, h=0] = 7
a[b=0, k=1, i=1, h=1] = 8
b: same values
out[b=0, i=0, j=0, h=0] = a[0,0,0,0]*b[0,0,0,0] + a[0,1,0,0]*b[0,1,0,0] = 1*1 + 5*5 = 26
out[b=0, i=0, j=1, h=0] = a[0,0,0,0]*b[0,0,1,0] + a[0,1,0,0]*b[0,1,1,0] = 1*3 + 5*7 = 38
out[b=0, i=1, j=0, h=0] = a[0,0,1,0]*b[0,0,0,0] + a[0,1,1,0]*b[0,1,0,0] = 3*1 + 7*5 = 38
out[b=0, i=1, j=1, h=0] = a[0,0,1,0]*b[0,0,1,0] + a[0,1,1,0]*b[0,1,1,0] = 3*3 + 7*7 = 58
A (after permute/reshape for h=0): [[1, 3], [5, 7]]
B (after permute/reshape for h=0): [[1, 3], [5, 7]]
A @ B^T = [[1+25, 3+35], [3+35, 9+49]] = [[26, 38], [38, 58]]
Yes! It matches.
OK, so the plan is:
1. Use a CUDA kernel to compute `a` and `b` from pair_repr with all the projections, gates, and masking
2. Use `torch.bmm` for the einsum
3. Use a CUDA kernel for the output projection and gating
This should be quite fast.
Let me also fuse the output bias into the matmul for better performance.
Actually, for even more speedup, I could do everything in a single kernel. Let me think about whether it's worth it for N=32.
For N=32, the einsum involves 32 multiplications and 31 additions per output element. For each (b, i, j, h), we have 32 FMAs. With B=2, N=32, H=32, that's 2*32*32*32 = 65536 output elements, each requiring 32 FMAs = ~2M FMAs. This is tiny. The bottleneck is probably memory bandwidth for reading pair_repr and writing intermediate tensors.
If I use two kernels + bmm:
- Kernel 1 reads pair_repr once, writes a and b: 2 * 64 * 4 bytes read + 2 * 32 * 4 bytes written per element = ~768 bytes per (b,n,n) element
- For B*N*N = 2048 elements, total memory = ~1.5MB read + ~0.5MB written
- bmm reads a and b, writes out: similar memory
- Kernel 2 reads out and pair_repr, writes final: similar memory
So total memory traffic is ~4-5MB, which at 3.35 TB/s would take ~1.5 microseconds. The compute is negligible.
If I fuse everything into one kernel:
- Each (b, i, j) block reads pair_repr[b, :, i, :] and pair_repr[b, :, j, :] for the einsum, plus pair_repr[b, i, j, :] for the output gate
- That's 2*N*P + P floats read per (b, i, j)
- Writes P floats
- For 2048 (b, i, j) elements: 2048 * (2*32*64 + 64) * 4 = ~34MB read + 0.5MB written
Wait, that's more reads because each element reads the same pair_repr values multiple times (pair_repr[b, k, i, :] is read for all j values). With a single kernel per (b, i, j), we lose reuse opportunities.
With the two-kernel approach, pair_repr is read once in kernel 1 and produces a and b. Then in bmm, a and b are consumed efficiently. The output is then post-processed.
Actually, the two-kernel approach might be better for this size because the intermediate a and b tensors fit in cache and the bmm is memory-bandwidth-efficient.
Let me implement the two-kernel approach. Actually, I can combine kernels 1 and 2 into a single kernel call per (b, n1, n2) for preprocessing, and a single kernel call per (b, i, j) for postprocessing.
For kernel 1 (preprocessing):
- One thread block per (b, n1, n2)
- Threads cooperatively load pair_repr[b, n1, n2, :] into shared memory
- Do layer norm
- Compute a and b projections
- Write results
For kernel 2 (postprocessing):
- One thread block per (b, i, j)
- Load einsum result and original pair_repr
- Do layer norm, out_proj, gate, write output
For the bmm in between, we can use torch.bmm.
This is a clean approach. Let me also think about using half precision, but the problem says fp32.
Actually, wait - let me reconsider the single-kernel fused approach. If I assign one block to each (b, i, j):
- The block loads pair_repr[b, k, i, :] for all k and pair_repr[b, k, j, :] for all k
- For N=32, P=64, that's 32*64*2 = 4096 floats = 16KB per (b, i, j) block
- For H=32, the einsum result per h is 32 FMAs
- Then we process the output
The issue is that pair_repr[b, k, i, :] is read by N different j-blocks for the same (b, k, i). So there's a lot of redundant reading. The L2 cache might help since N=32 blocks run concurrently on an H100.
With 132 SMs, we can run many blocks in parallel. The total number of (b, i, j) blocks is B*N*N = 2*32*32 = 2048. With 32 threads per block (H=32), that's 65536 threads total, which the GPU can handle easily.
For memory, pair_repr is B*N*N*P = 2*32*32*64 = 131072 floats = 512KB. This easily fits in L2 cache. So even with redundant reads, the data should stay in L2 and be served quickly.
Let me actually try the single fused kernel again with the AT_CUDA_CHECK fix and a better design.
Actually, let me first try the simpler two-kernel + bmm approach since it's cleaner and likely fast enough.
**Design:**
Kernel 1: `preprocess_kernel`
- Grid: (B, N, N)
- Block: max(P, H) threads, or better yet, enough threads to cover both dimensions
- For each (b, i, j):
- Load pair_repr[b, i, j, :] into shared memory (P=64 elements)
- Layer norm
- Compute a_h and b_h for each h
- Write a[b, i, j, h] and b[b, i, j, h]
Wait, the einsum is `a[k,i] * b[k,j]`. So we need a indexed as (b, k, i, h) and b as (b, k, j, h). In the preprocessing, a and b should be computed for all indices.
Actually, in the original code, a and b both have shape (B, N, N, H). The einsum uses a[b, k, i, h] and b[b, k, j, h]. So for each (b, n1, n2), a[b, n1, n2, :] uses the pair_repr at (b, n1, n2, :), and b[b, n1, n2, :] uses pair_repr at (b, n1, n2, :).
So in kernel 1, for each (b, i, j), we compute both a[b, i, j, :] and b[b, i, j, :]. They use the same input pair_repr[b, i, j, :] but different weights.
Then bmm:
- a_reshaped = a.permute(0, 3, 1, 2).reshape(B*H, N, N) # (B*H, N, N)
- b_reshaped = b.permute(0, 3, 1, 2).reshape(B*H, N, N) # (B*H, N, N)
- c = torch.bmm(a_reshaped, b_reshaped.transpose(-1, -2)) # (B*H, N, N)
- c = c.reshape(B, H, N, N).permute(0, 2, 3, 1) # (B, N, N, H)
Then kernel 2:
- For each (b, i, j):
- Load c[b, i, j, :] and pair_repr[b, i, j, :]
- Layer norm on c
- out_proj(c)
- sigmoid(out_gate(pair_repr))
- multiply and write
This is clean and leverages cuBLAS for the compute-heavy part.
Let me code this up.
For kernel 1, I need to handle both P=64 and H=32. I'll use 64 threads per block. Each thread can compute one or more h values.
Actually, for better efficiency in kernel 1, I could have the block compute:
- First, all threads compute layer_norm for P=64
- Then, threads 0-31 compute a for h=0-31, and compute b for h=0-31 simultaneously
Wait, with 64 threads, threads 0-31 can compute a[h] and threads 32-63 can compute b[h]. That's perfect!
Actually, but then writing to a and b requires all threads to participate. With a as (B, N, N, H) contiguous, each thread writes one h.
Similarly for kernel 2: 64 threads, layer_norm on H=32 (only first 32 threads participate, rest wait), then all 64 threads compute out_proj (each thread handles one output element), and all 64 threads compute gate.
Let me write this more carefully.
For kernel 1 (preprocess):
```
preprocess_kernel(pair_repr, pair_mask,
ln_w, ln_b,
proj_a_w, proj_a_b, gate_a_w, gate_a_b,
proj_b_w, proj_b_b, gate_b_w, gate_b_b,
a_out, b_out,
B, N, P, H)
```
Block = (P) = 64 threads. Grid = (B, N, N).
Shared memory per block:
- input[P] = 64 floats
- scratch for reduction = 2 floats (one warp)
- Actually for layer norm reduction with 64 threads, scratch needs ceil(64/32) = 2 floats
For each (b, i, j):
1. tid < P reads pair_repr[b, i, j, tid] into input[tid]
2. __syncthreads()
3. Compute mean and variance of input (all 64 threads participate in reduction)
4. Apply layer norm
5. __syncthreads()
6. If tid < H:
- Compute a_proj = proj_a_b[tid] + sum_d input[d] * proj_a_w[tid, d]
- Compute a_gate = gate_a_b[tid] + sum_d input[d] * gate_a_w[tid, d]
- a_out = a_proj * sigmoid(a_gate) * mask[b, i, j]
7. If (tid + H) < H, well tid < H so always:
Wait, with 64 threads and H=32, only first 32 compute a and b.
Actually let me think about this. If I have 64 threads, and I want to compute both a and b:
- tid 0-31: compute a_h for h = tid
- tid 0-31: also compute b_h for h = tid
But then I need 32 threads, not 64. I could launch with 32 threads. But then layer norm over 64 elements with 32 threads means each thread handles 2 elements. That's fine.
Or I could launch with 64 threads:
- tid 0-31: compute a_h
- tid 32-63: compute b_{h-32}
But that creates a divergence. And b uses different weights than a.
Hmm, actually since H=32 and I have 64 threads, I can have:
- tid 0-31: compute a_h
- tid 0-31: also compute b_h (serially)
Wait no, each thread can only do one at a time.
Better approach:
- tid 0-31: compute a_h and write to a
- then all threads compute b_h
- but if I have 64 threads and only 32 outputs, this wastes half.
Actually, the simplest is 64 threads, tid < 32 handles a, and all threads participate in b if needed. Or I can just use 64 threads and have each thread handle up to 2 elements if necessary.
Wait, P=64 and H=32. I'll just use 64 threads for the block:
- Step 1: layer norm (all 64 threads)
- Step 2: if tid < 32, compute a; else compute b_{tid-32}
- But then half the threads write to a and half to b
- This is fine! We just need to be careful about synchronization.
Actually, for the einsum we need both a and b. So each block producing a[b,i,j,:] and b[b,i,j,:] should have both computed. Let me just use 32 threads and have each compute both a and b serially.
Block = 32 threads. Each thread handles one h.
For each (b, i, j):
1. All 32 threads load pair_repr[b, i, j, :] into shared memory (each loads 2 elements)
2. Compute layer norm cooperatively
3. Each thread tid computes:
- a_val = proj_a(tid) * sigmoid(gate_a(tid)) * mask
- b_val = proj_b(tid) * sigmoid(gate_b(tid)) * mask
4. Write a and b
This is clean and has good occupancy.
Actually let me just use the simpler approach: call all the PyTorch ops except the einsum, and use a custom kernel for the einsum. Or just write a custom kernel for the einsum since the rest might be fused by PyTorch's compiler.
But the user wants us to write a custom CUDA kernel that replaces the forward computation. We should try to make it faster than PyTorch.
Let me measure what PyTorch does first... but I can't. Let me just write the best kernel I can.
I'll go with the single fused kernel since it avoids intermediate tensor materialization entirely. The key insight is that for small N=32, the compute is tiny and memory bandwidth dominates. A single kernel that reads the input once and writes the output once minimizes memory traffic.
Let me redesign the single kernel properly:
For each (b, i, j) block with 32 threads (one per h):
1. Load pair_repr[b, :, i, :] into a_shared[N][P] - but with 32 threads and N=32, we can have each thread load one row of P elements... that's 64 floats per thread, not efficient.
Actually, if we use 64 threads per block, each thread loads one element of pair_repr[b, k, i, :] per k. For N=32, that's 32 loads per thread, but they're strided.
Hmm, this is getting complex. Let me just use the two-kernel + bmm approach for now.
Actually, let me try yet another approach. Since N=32 is small, I can use a block per (b, i, j, h) with 32 threads summing over k using warp-level reductions.
Block = (k=0..31), one warp. Grid = (B, N, N, H).
For each (b, i, j, h):
- Load a[b, k, i, h] and b[b, k, j, h] for k = tid
- Warp-level reduce sum of a * b
- Write result
This is very simple and H=32, so the grid is B*N*N*H = 2*32*32*32 = 65536 blocks, each with 32 threads. The total threads are 2M. The occupancy should be good.
But then we need a separate kernel for preprocessing (computing a and b) and postprocessing.
OK let me just do this:
1. Kernel to compute a and b from pair_repr
2. bmm for einsum
3. Kernel for postprocessing
This is clean, fast, and correct.
Actually, I just realized - the preprocessing in PyTorch can already be fused quite well. Let me think about what's the actual bottleneck.
For the einsum alone, with a and b of shape (2, 32, 32, 32), the einsum is over 32 elements. PyTorch likely converts this to a batched matmul. If so, it should already be very fast (< 10 microseconds).
The whole model probably takes ~50-100 microseconds. So to beat it, we need to either:
1. Fuse more operations (especially the layer norms with other stuff)
2. Avoid overhead of many small PyTorch ops
Let me write a kernel that computes everything in one go. I'll use a custom block design where each (b, i, j) block has enough threads to handle all the compute.
For my single fused kernel:
- Block = (H) = 32 threads for the einsum + output processing
- But we also need to read P=64 elements for the input
- And do layer norm on P elements
- This means we need thread cooperation
Actually, I'll use block size = max(P, H) = 64 threads. Each block handles one (b, i, j).
Detailed design:
```
extern __shared__ float smem[];
float* row_buffer = smem; // Size N*P for storing rows from global mem
float* a_local = row_buffer + N*P; // Size N*H
float* b_local = a_local + N*H; // Size N*H
float* out_h = b_local + N*H; // Size H
float* pair_ij = out_h + H; // Size P
float* scratch = pair_ij + P; // Size 2
```
Total shared memory: N*P + N*H + N*H + H + P + 2 = 32*64 + 32*32 + 32*32 + 32 + 64 + 2 = 2048 + 1024 + 1024 + 98 = 4194 floats ≈ 16.8KB. This fits easily in shared memory.
Algorithm per block (b, i, j):
1. All 64 threads load pair_repr[b, k, i, :] for all k:
- For k = 0 to 31, for each tid, load pair_repr[b, k, i, tid] if tid < 64, or pair_repr[b, k, i, tid-64] if tid >= 64... wait that's overloading.
- Actually with 64 threads: for each k, thread tid loads pair_repr[b, k, i, tid] if tid < P. So for each k, we need 64 threads to load 64 elements. Over 32 k's, each thread loads 32 elements.
This is:
```
for (int k = 0; k < N; k++) {
if (tid < P) row_buffer[k*P + tid] = pair_repr[((b*N + k)*N + i)*P + tid];
}
```
But this is serialized over k with a syncthreads each time, which is slow.
Better: let each thread load multiple elements:
```
for (int idx = tid; idx < N*P; idx += blockDim.x) {
int k = idx / P;
int d = idx % P;
row_buffer[idx] = pair_repr[((b*N + k)*N + i)*P + d];
}
```
Similarly for pair_repr[b, k, j, :].
Wait, I can read both row_i and row_j data. But they require 2*N*P = 4096 loads. With 64 threads, each thread does 64 loads. That's fine.
Actually, I don't need to store all of row_i and row_j in shared memory. I can process one k at a time:
1. Thread tid loads pair_repr[b, k, i, tid] if tid < P
2. Do layer norm on the row
3. Compute a_kh for all h
4. Thread tid loads pair_repr[b, k, j, tid] if tid < P
5. Do layer norm on the row
6. Compute b_kh for all h
7. Accumulate a_kh * b_kh into out_h[h]
For step 3 and 6, computing a_kh for all h requires each thread to compute one h, but that means reading the normalized row from shared memory. With H=32 and 64 threads, I can have 32 threads compute a and 32 compute b. But then I need to store both results.
Wait, I only need the product a_kh * b_kh summed over k. I don't need to store all a and b.
For each k:
1. Load and normalize row_i (the pair_repr[b, k, i, :])
2. In shared memory, compute all a_kh values
3. Load and normalize row_j (the pair_repr[b, k, j, :])
4. In shared memory, compute all b_kh values
5. Each thread tid < H reads a_kh and b_kh and accumulates
This means for each k, I need to store H a-values and H b-values in shared memory temporarily.
Actually no - each thread can compute its own h's a and b directly:
1. Normalize row_i
2. Thread tid (if tid < H) computes a_k, tid by dot-product over P
3. Normalize row_j
4. Thread tid (if tid < H) computes b_k, tid by dot-product over P
5. Accumulate a_k,tid * b_k,tid into a register
This avoids storing intermediate a and b values in shared memory, but each thread does 2*P FMAs per k (for the projections) and then multiplies. For N=32 k's, that's 2*32*64 = 4096 FMAs per thread for the projections, plus 32 multiplies. Total ~4128 FMAs per thread. With 32 threads, that's ~132K FMAs per block.
At the same time, the block reads 2*N*P = 4096 floats from global memory (the pair_repr rows).
This is actually compute-bound! 132K FMAs vs 4096 floats read. The compute intensity is ~32 FMAs per float read, which is great for GPU.
Wait, but each thread is doing 4096 FMAs, and with 32 threads per block, that's 131K FMAs. At 67 TFLOPS (FP32 on H100), that's 131K / 67T = ~2 nanoseconds. But memory is 3.35 TB/s, so reading 4096*4 bytes = 16KB takes 16KB/3.35TB = ~5 nanoseconds. So memory is the bottleneck.
But if data is in L2 cache (which it should be for N=32, B=2), the effective bandwidth is higher. And with many blocks running concurrently, we can hide latency.
Let me go ahead and implement this design:
```
__global__ void fused_kernel(
const float* pair_repr, const float* pair_mask, float* output,
// all weights and biases...
int B, int N, int P, int H
) {
__shared__ float smem_row[P]; // For layer norm, one row at a time
__shared__ float scratch[2]; // For block reduction
__shared__ float pair_ij[P]; // Original pair_repr[b,i,j,:] for output gate
int b = blockIdx.x;
int i = blockIdx.y;
int j = blockIdx.z;
int tid = threadIdx.x;
// Load pair_repr[b, i, j, :] for the output gate
if (tid < P) pair_ij[tid] = pair_repr[(((b*N + i)*N + j)*P + tid)];
__syncthreads();
float acc_h = 0.0f;
for (int k = 0; k < N; k++) {
// Load row_i = pair_repr[b, k, i, :]
if (tid < P) smem_row[tid] = pair_repr[(((b*N + k)*N + i)*P + tid)];
__syncthreads();
// Layer norm on row_i
...
float a_val = 0.0f;
if (tid < H) {
// Compute a projection and gate
float proj = proj_a_b[tid];
float gate = gate_a_b[tid];
for (int d = 0; d < P; d++) {
float x = smem_row[d];
proj += x * proj_a_w[tid*P + d];
gate += x * gate_a_w[tid*P + d];
}
a_val = proj * sigmoid(gate) * pair_mask[((b*N + k)*N + i)];
}
__syncthreads();
// Load row_j = pair_repr[b, k, j, :]
if (tid < P) smem_row[tid] = pair_repr[(((b*N + k)*N + j)*P + tid)];
__syncthreads();
// Layer norm on row_j
...
float b_val = 0.0f;
if (tid < H) {
float proj = proj_b_b[tid];
float gate = gate_b_b[tid];
for (int d = 0; d < P; d++) {
float x = smem_row[d];
proj += x * proj_b_w[tid*P + d];
gate += x * gate_b_w[tid*P + d];
}
b_val = proj * sigmoid(gate) * pair_mask[((b*N + k)*N + j)];
}
__syncthreads();
if (tid < H) acc_h += a_val * b_val;
}
// Now apply output layer norm, projection, and gate
// Layer norm on acc_h values in shared memory (H elements)
if (tid < H) smem_row[tid] = acc_h;
__syncthreads();
// layer_norm on H elements...
// out_proj: P outputs from H inputs
float out_val = out_proj_b[tid];
for (int h = 0; h < H; h++) out_val += smem_row[h] * out_proj_w[tid*H + h];
// Output gate from pair_ij
float gate_val = out_gate_b[tid];
for (int d = 0; d < P; d++) gate_val += pair_ij[d] * out_gate_w[tid*P + d];
output[(((b*N + i)*N + j)*P + tid)] = sigmoid(gate_val) * out_val;
}
```
The issue with this design is:
1. For each k, we're doing layer norm twice. Layer norm requires block-level reductions which are somewhat expensive.
2. The weights are read from global memory for each k iteration. We could cache them in shared memory or rely on L1/L2 cache.
For H=32, P=64, the weights for proj_a are 32*64 = 2048 floats = 8KB. If all blocks read these weights, they should stay in L2 cache. Actually, with 2048 blocks, each reading the same weights, the L1 cache per SM should help, but it's a lot of memory traffic.
To optimize further, I could store the weight matrices in shared memory, but 4 weight matrices of 8KB each = 32KB, which is too much. I can store one at a time, but I'd need to reload for each k.
Actually, each block only needs a subset of the weights. If tid handles one h, it only needs one row of each weight matrix (P floats). So each thread naturally caches its own row in registers, not shared memory. The dot-product loops over P, reading each weight matrix element once. With 32 iterations and good instruction-level parallelism, the weights should be pipelineable.
One optimization: prefetch the next weight while computing the current one. The compiler might already do this.
Another concern: the layer norm function I wrote uses `blockReduceSum` with `__syncthreads()` calls. In the loop over k, this adds synchronization overhead.
Let me simplify the layer norm to use warp-level reductions only, since P=64 which is exactly 2 warps.
```
__inline__ __device__ void layer_norm_warp(float* data, int size, const float* w, const float* b) {
// size = 64, 2 warps
float x = data[threadIdx.x];
float sum = warpReduceSum(x);
if (threadIdx.x == 0 || threadIdx.x == 32) {
// combine warp sums
}
}
```
Actually, with 64 threads and 2 warps, I can do:
```
float sum = warpReduceSum(x);
__shared__ float warp_sums[2];
if (lane == 0) warp_sums[warp_id] = sum;
__syncthreads();
float total = warp_sums[0] + warp_sums[1];
mean = total / 64;
```
Then broadcast mean to all threads. Similarly for variance.
This is more efficient than my previous blockReduceSum.
OK let me write this properly now.
Also, I need to make sure I use `C10_CUDA_KERNEL_LAUNCH_CHECK()` instead of `AT_CUDA_CHECK`.
Wait, actually `C10_CUDA_KERNEL_LAUNCH_CHECK` might also not be available. Let me just use a simple `TORCH_CHECK(cudaGetLastError() == cudaSuccess, "CUDA error");` or just omit it.
Actually, looking at PyTorch source, `C10_CUDA_KERNEL_LAUNCH_CHECK()` should be available in `<c10/cuda/CUDAGuard.h>` or similar. But the safest is to not use any explicit error check after the kernel launch - PyTorch will catch async errors later. Or I can use `cudaDeviceSynchronize()` and check, but that's slow.
Actually, for correctness, I can just check `cudaGetLastError()` synchronously:
```cpp
cudaError_t err = cudaGetLastError();
if (err != cudaSuccess) {
AT_ERROR("CUDA kernel failed: ", cudaGetErrorString(err));
}
```
Or simpler, `TORCH_CHECK(cudaGetLastError() == cudaSuccess, "Kernel launch failed");`
Let me go with the simpler approach and not do explicit kernel error checking inside the wrapper function. Instead just rely on PyTorch's error handling.
None of that matters for compilation though. Let me fix the compilation issue.
Actually, wait - looking at the error again: `AT_CUDA_CHECK` is undefined. Let me just use `TORCH_CHECK_CUDA(cudaGetLastError() == cudaSuccess, "Kernel failed")` or simply omit the check.
Let me write the kernel step by step, keeping it clean.
For P=64 and varying H (given as hidden_dim, could be 32 or other values), I should make the code robust. The block size can be set to max(P, H) or just P since P >= H in most cases.
Actually, looking at the problem: pair_dim=64, hidden_dim=32. So P=64, H=32.
Let me write the kernel with block size = 64 threads (warp size * 2).
For the layer norm over P=64, I can use the two-warp reduction.
For the projections, only threads 0-31 compute (since H=32). But to avoid branch divergence in weight loading, it's better to not have the `if (tid < H)` because the threads still execute the same instructions anyway, just some discard results. Actually, with `if (tid < H)`, the warp is divergent - first 32 threads do the work, next 32 wait. That's fine.
Hmm wait, for P=64 and blockDim.x=64, if only threads 0-31 compute a and b, then 32 threads are idle during the projection compute. If I use blockDim.x=32 threads, then layer norm over 64 elements requires each thread to process 2 elements. This might be better!
Let me think about using 32 threads:
- Layer norm: each thread handles 2 elements (indices tid and tid+32)
- Reduction: need warpReduceSum for both elements, then combine
- This is more complex but uses all threads efficiently.
Actually, for simplicity, let me use 64 threads. During projection computation, threads 0-31 are active and 32-63 are idle. The GPU scheduler handles this fine. The instruction throughput for the active threads is the same.
But wait, what if I use 32 threads and have each compute 2 h values? That is, tid handles h=tid and h=tid+16 (if H=32). Then all 32 threads are active computing 2 outputs each.
For H=32, tid computes h=tid and h=tid+32? No, H=32, so tid computes h=tid.
Actually, with 32 threads and H=32, each thread handles exactly one h.
For P=64, with 32 threads, each thread needs to process the 64-element row. During layer norm:
- Thread tid reads elements tid and tid+32
- Computes local sum = x1 + x2
- Warp reduce sum of local sums
- Mean computed
- Each thread computes diff1 = x1 - mean, diff2 = x2 - mean
- Local sq = diff1^2 + diff2^2
- Warp reduce sum of local sqs
- Variance computed, inv_std
- Each thread writes normalized values back
This uses all threads efficiently! Let me write this version.
Block size = 32 threads
Shared memory:
- row_buffer[64] for layer norm
- pair_ij[64] for output gate
- out_h[32] for accumulated einsum result
- warp temp vars
Total SMEM per block: 64 + 64 + 32 + a few = ~160 floats = 640 bytes. Very small!
Wait, I also need to compute the einsum result and store it temporarily. With H=32 and 32 threads, each thread accumulates one h in a register. Then for output processing, we need all H values in shared memory for layer norm.
So after the k loop:
- Each thread writes its acc_h to out_h[tid]
- __syncthreads()
- Layer norm on out_h[32]
- Each thread tid reads out_h[0..31] and computes out_proj for output element tid
Wait, P=64 outputs but only 32 threads. So each thread needs to compute 2 output elements!
Ah, this is the issue. If block size = 32, and P = 64, then each thread needs to compute 2 outputs for the out_proj and out_gate.
Let me instead use block size = max(P, H) = 64 threads. Then:
- All 64 threads participate in layer norm for P=64
- Threads 0-31 compute a and b for H=32
- After einsum, write H=32 values to shared memory
- All 64 threads participate in layer norm for H=32 (half are idle/redundant)
- Each thread computes one output element for P=64
This is simpler and avoids thread overloading. The idle threads during H-computation don't hurt much.
Let me use 64 threads.
Actually, I realize there's a better approach. Since the grid has B*N*N = 2048 blocks and we have 132 SMs, each SM gets about 15 blocks. With 64 threads per block, that's 960 threads per SM. Occupancy on H100: max 2048 threads per SM, so we're at ~47% thread occupancy. Not great, but also not terrible.
Can we increase occupancy? If we reduce threads per block to 32, each SM can run 2048/32 = 64 blocks (theoretical max is typically 32 blocks per SM). So we'd get ~32 blocks per SM for 100% thread occupancy. Since we have 2048/132 ≈ 15.5 blocks per SM total, we'd achieve ~50% block occupancy but 100% thread occupancy. Wait, 15.5 blocks per SM with 32 threads each = 496 threads per SM, which is only 24% thread occupancy.
Hmm, actually with 32 threads per block, max blocks per SM is typically limited by other factors (registers, shared memory, etc.). If shared memory is small (640 bytes), we can have many blocks.
OK, I think my analysis is getting too complicated. Let me just write a clean kernel with 64 threads and see if it works.
Actually, for maximum simplicity and performance, let me just launch the kernel twice:
1. Kernel to compute a and b from pair_repr
2. bmm for einsum
3. Kernel for output processing
For kernel 1, block size = P = 64 threads.
For kernel 3, block size = P = 64 threads.
This way each kernel is simple and the bmm uses cuBLAS efficiently.
Let me write this version.
Kernel 1 (preprocess):
- Grid: (B, N, N)
- Block: 64 threads
- For each (b, i, j):
- Load pair_repr[b, i, j, :] into smem[64]
- Layer norm
- Thread tid < 32:
- Compute a[b, i, j, tid] and b[b, i, j, tid]
- Write to global memory
Kernel 3 (postprocess):
- Grid: (B, N, N)
- Block: 64 threads
- For each (b, i, j):
- Load einsum_out[b, i, j, :] into smem[32]
- Layer norm on H=32
- Load pair_repr[b, i, j, :] into registers
- Each thread tid computes:
- out_proj[tid] = bias + sum_h smem[h] * weight[tid, h]
- gate = sigmoid(bias + sum_d pair_repr[d] * weight[tid, d])
- output = gate * out_proj
This should be straightforward and fast enough.
Let me estimate performance:
- Kernel 1: reads 512KB (pair_repr), writes 256KB (a) + 256KB (b) = ~1MB traffic, compute is small
- bmm: reads 512KB (a) + 512KB (b), writes 256KB (c) = ~1.3MB traffic
- Kernel 3: reads 256KB (c) + 512KB (pair_repr), writes 512KB (output) = ~1.3MB traffic
- Total: ~3.6MB
- At 3.35 TB/s: ~1.1 microseconds (theoretical min)
- Realistically with overheads: maybe 10-50 microseconds
The PyTorch reference with separate ops likely takes 50-100 microseconds due to kernel launch overheads and multiple memory roundtrips. Our fused approach should be faster.
Let me code this up now.
One more thing: I'll write the kernels to accept transposed weight matrices. PyTorch Linear layers store weights as (out_features, in_features). In CUDA, for the dot-product `sum_d x[d] * w[h, d]`, it's better to have w in row-major with contiguous rows. PyTorch stores nn.Linear weights as (out_features, in_features) which is already row-major for each output feature. So `w[h*in_features + d]` is the correct access pattern.
Actually wait, nn.Linear in PyTorch does `x @ W^T + b`, where W is (out_features, in_features). So for computing manually: `out[h] = sum_d x[d] * W[h, d] + b[h]`. Yes, `W[h*in_features + d]` is correct.
Let me verify: `self.proj_a = nn.Linear(pair_dim, hidden_dim)`. Weight shape is (hidden_dim, pair_dim). Stored as (H, P). Accessing `proj_a_w[h*P + d]` gives W[h, d]. In the linear layer operation: `x @ W^T`. Wait, `nn.Linear` computes `y = xA^T + b`, where A is (out_features, in_features). So the weight matrix is A, and the computation is `sum_j x[j] * A[i, j]`. So yes, `w[h*P + d]` = A[h, d].
My manual dot-product `sum_d x[d] * w[h*P + d]` matches PyTorch exactly.
OK let me write the code now.
For gate computation: `gate_a(x) = x @ gate_a_W^T + gate_a_b`. And `sigmoid(gate_a(x))`. Then `proj_a(x) * sigmoid(gate_a(x))`.
Wait, no. The original code is:
`a = self.proj_a(x) * torch.sigmoid(self.gate_a(x)) * mask`
So first compute the projection, then compute the gate, then multiply elementwise. Yes, my code does this correctly.
One more optimization: Instead of computing two separate dot products (one for proj, one for gate) over the same input x, I can interleave them or compute them together. But the compiler can handle this.
Actually, for better instruction-level parallelism, I can accumulate both proj and gate in the same loop:
```cpp
float proj = proj_b[h];
float gate = gate_b[h];
#pragma unroll
for (int d = 0; d < P; d++) {
float xv = row[d];
proj += xv * proj_w[h*P + d];
gate += xv * gate_w[h*P + d];
}
```
This is what I'll write.
Let me also think about whether to declare weights as `const float* __restrict__` for better cache behavior.
OK, here's the complete code:
```cpp
#include <torch/extension.h>
#include <cuda_runtime.h>
__inline__ __device__ float sigmoidf(float x) {
return 1.0f / (1.0f + expf(-x));
}
// Launch with blockDim.x = max(P, H) threads
__global__ void preprocess_kernel(
const float* __restrict__ pair_repr,
const float* __restrict__ pair_mask,
const float* __restrict__ ln_w,
const float* __restrict__ ln_b,
const float* __restrict__ proj_a_w,
const float* __restrict__ proj_a_b,
const float* __restrict__ gate_a_w,
const float* __restrict__ gate_a_b,
const float* __restrict__ proj_b_w,
const float* __restrict__ proj_b_b,
const float* __restrict__ gate_b_w,
const float* __restrict__ gate_b_b,
float* __restrict__ a_out,
float* __restrict__ b_out,
int B, int N, int P, int H
) {
__shared__ float row[P];
__shared__ float warp_sums[2];
int b = blockIdx.x;
int i = blockIdx.y;
int j = blockIdx.z;
int tid = threadIdx.x;
// Load pair_repr[b, i, j, :] into shared memory
if (tid < P) {
row[tid] = pair_repr[((b * N + i) * N + j) * P + tid];
}
__syncthreads();
// Layer norm over P elements with blockDim.x threads
// For P=64 and blockDim.x=64: each thread handles 1 element
float x = (tid < P) ? row[tid] : 0.0f;
// Warp reduce sum
float sum = x;
#pragma unroll
for (int offset = 16; offset > 0; offset /= 2) {
sum += __shfl_down_sync(0xffffffff, sum, offset);
}
if ((tid & 31) == 0) {
warp_sums[tid >> 5] = sum; // tid >> 5 is warp id (0 or 1)
}
__syncthreads();
float total = warp_sums[0] + warp_sums[1];
float mean = total / P;
float diff = (tid < P) ? (row[tid] - mean) : 0.0f;
// Variance
float sq = diff * diff;
sum = sq;
#pragma unroll
for (int offset = 16; offset > 0; offset /= 2) {
sum += __shfl_down_sync(0xffffffff, sum, offset);
}
if ((tid & 31) == 0) warp_sums[tid >> 5] = sum;
__syncthreads();
float var = (warp_sums[0] + warp_sums[1]) / P;
float inv_std = rsqrtf(var + 1e-5f);
if (tid < P) {
row[tid] = diff * inv_std * ln_w[tid] + ln_b[tid];
}
__syncthreads();
// Compute a and b projections
if (tid < H) {
int h = tid;
float a_proj = proj_a_b[h];
float a_gate = gate_a_b[h];
float b_proj = proj_b_b[h];
float b_gate = gate_b_b[h];
#pragma unroll
for (int d = 0; d < P; d++) {
float xv = row[d];
a_proj += xv * proj_a_w[h * P + d];
a_gate += xv * gate_a_w[h * P + d];
b_proj += xv * proj_b_w[h * P + d];
b_gate += xv * gate_b_w[h * P + d];
}
float mask = pair_mask[(b * N + i) * N + j];
a_out[((b * N + i) * N + j) * H + h] = a_proj * sigmoidf(a_gate) * mask;
b_out[((b * N + i) * N + j) * H + h] = b_proj * sigmoidf(b_gate) * mask;
}
}
```
And the postprocess kernel:
```cpp
__global__ void postprocess_kernel(
const float* __restrict__ einsum_out,
const float* __restrict__ pair_repr,
const float* __restrict__ ln_w,
const float* __restrict__ ln_b,
const float* __restrict__ out_proj_w,
const float* __restrict__ out_proj_b,
const float* __restrict__ out_gate_w,
const float* __restrict__ out_gate_b,
float* __restrict__ output,
int B, int N, int H, int P
) {
__shared__ float einsum_row[32]; // H
__shared__ float pair_row[64]; // P
__shared__ float warp_sums[2];
int b = blockIdx.x;
int i = blockIdx.y;
int j = blockIdx.z;
int tid = threadIdx.x;
// Load einsum result
if (tid < H) {
einsum_row[tid] = einsum_out[((b * N + i) * N + j) * H + tid];
}
__syncthreads();
// Layer norm on H elements
float x = (tid < H) ? einsum_row[tid] : 0.0f;
float sum = x;
#pragma unroll
for (int offset = 16; offset > 0; offset /= 2) {
sum += __shfl_down_sync(0xffffffff, sum, offset);
}
if ((tid & 31) == 0) warp_sums[tid >> 5] = sum;
__syncthreads();
float total = warp_sums[0] + warp_sums[1];
float mean = total / H;
float diff = (tid < H) ? (einsum_row[tid] - mean) : 0.0f;
float sq = diff * diff;
sum = sq;
#pragma unroll
for (int offset = 16; offset > 0; offset /= 2) {
sum += __shfl_down_sync(0xffffffff, sum, offset);
}
if ((tid & 31) == 0) warp_sums[tid >> 5] = sum;
__syncthreads();
float var = (warp_sums[0] + warp_sums[1]) / H;
float inv_std = rsqrtf(var + 1e-5f);
if (tid < H) {
einsum_row[tid] = diff * inv_std * ln_w[tid] + ln_b[tid];
}
__syncthreads();
// Load original pair_repr for gate
if (tid < P) {
pair_row[tid] = pair_repr[((b * N + i) * N + j) * P + tid];
}
__syncthreads();
// Each thread computes one output element
if (tid < P) {
float out_val = out_proj_b[tid];
for (int h = 0; h < H; h++) {
out_val += einsum_row[h] * out_proj_w[tid * H + h];
}
float gate = out_gate_b[tid];
for (int d = 0; d < P; d++) {
gate += pair_row[d] * out_gate_w[tid * P + d];
}
output[((b * N + i) * N + j) * P + tid] = sigmoidf(gate) * out_val;
}
}
```
Wait, I need to handle the case where H and P might have different values. For H=32 and blockDim.x = max(H, P) = 64:
- In the layer norm for H=32, only threads 0-31 participate.
- The warp reduction within threads 0-31 works fine.
- `warp_sums[0]` gets the sum from warp 0. `warp_sums[1]` gets the sum from warp 1, but warp 1 has all zeros (since threads 32-63 evaluate x=0).
- So total = warp_sums[0] + 0 = correct sum.
- Same for variance.
- This works!
But I need to be careful: `warp_sums[1] = sum` for warp 1. Threads 32-63 have x=0, sum=0, so warp_sums[1] = 0. This is correct.
Similarly for writes to einsum_row: only threads 0-31 write. Threads 32-63 don't write, which is fine.
For the output: all 64 threads (tid < P=64) compute one output each.
This design should work. Let me make sure the shared memory layout is correct.
Actually, I realize `warp_sums` should be declared properly:
```cpp
__shared__ float warp_sums[2];
```
For 2 warps. In the layer norm, each warp's lane 0 writes to warp_sums[warp_id].
Wait, there's a potential race condition. Two warps might try to write to warp_sums simultaneously? No, because warp_sums[0] is only written by lane 0 of warp 0, and warp_sums[1] is only written by lane 0 of warp 1. These are different memory locations, so no race.
But what if blockDim.x is not exactly 64? If I launch with 64 threads, there are exactly 2 warps. Good.
What if H or P is not 32/64? I should make the code more general, but for this problem the sizes are fixed.
Let me finalize the code. In the PyTorch wrapper:
```cpp
torch::Tensor forward_cuda(
torch::Tensor pair_repr,
torch::Tensor pair_mask,
...
) {
int B = pair_repr.size(0);
int N = pair_repr.size(1);
int P = pair_repr.size(3);
int H = proj_a_w.size(0);
auto a = torch::empty({B, N, N, H}, pair_repr.options());
auto b = torch::empty({B, N, N, H}, pair_repr.options());
dim3 grid(B, N, N);
dim3 block(64);
preprocess_kernel<<<grid, block>>>(...);
// Einsum via bmm
auto a_r = a.permute({0, 3, 1, 2}).reshape({B * H, N, N});
auto b_r = b.permute({0, 3, 1, 2}).reshape({B * H, N, N});
auto c = torch::bmm(a_r, b_r.transpose(-1, -2));
auto einsum_out = c.reshape({B, H, N, N}).permute({0, 2, 3, 1});
auto output = torch::empty_like(pair_repr);
postprocess_kernel<<<grid, block>>>(einsum_out, pair_repr, ..., output, B, N, H, P);
return output;
}
```
One concern: the permute and reshape operations create views, which are cheap. The `bmm` call uses cuBLAS.
Wait, actually for the reshape and permute, we need contiguous tensors. `permute` doesn't make tensors contiguous. So `a.permute(0,3,1,2)` might not be contiguous for bmm. Let me check:
a shape: (B, N, N, H)
permute(0,3,1,2): (B, H, N, N)
Is the memory layout contiguous for (B, H, N, N)? The strides would be (N*N*H, 1, N*H, H). For bmm, we need the last two dimensions to represent matrices. The innermost dimension (N) has stride H, and the next dimension (N) has stride N*H. So each N×N matrix is contiguous within the H dimension. But bmm expects the innermost dimension to be contiguous in a specific way.
Actually, for `torch.bmm(input, mat2)`, both inputs should be 3D tensors of shape (batch, n, m) and (batch, m, p). The matrices along the batch dimension should be laid out such that the last dimension has stride 1.
For a_r of shape (B*H, N, N):
- Original a has strides: stride[0]=N*N*H, stride[1]=N*H, stride[2]=H, stride[3]=1
- After permute(0,3,1,2), the tensor has shape (B, H, N, N) with strides (N*N*H, 1, H, N*H)
- After reshape to (B*H, N, N), the strides would be (N*N, N*N*H, H)... wait, reshape doesn't work on non-contiguous tensors!
Actually, `reshape` on a non-contiguous tensor might return a copy or fail. Let me check: `reshape` is similar to `view` but can return a copy if needed. For permuted tensors, it typically returns a copy.
A copy would be expensive! Let me use `contiguous()` first, or better yet, write the a and b tensors in the right layout from the kernel.
If I write a as (B, H, N, N) from the kernel, then it would be contiguous for bmm. But the original PyTorch code has a as (B, N, N, H). If we need a as (B, N, N, H) for the einsum, but we want to use bmm which needs (B*H, N, N), we have a layout mismatch.
Options:
1. Write a and b in (B, N, N, H) layout, then use `permute(0,3,1,2).contiguous().reshape(B*H, N, N)` for bmm. This involves a copy.
2. Write a and b in (B, H, N, N) layout in the kernel, then `reshape(B*H, N, N)` directly. But then the einsum output would be in (B, H, N, N) layout, which needs to be permuted back to (B, N, N, H) for the output kernel.
Option 2 is better but requires changing the kernel output layout.
Actually, for bmm, `a` can be strided as long as the matrix dimensions are correct. Let me check: in PyTorch, `bmm` calls into cuBLAS batched GEMM. For batched GEMM, the batch matrices can have arbitrary strides.
From PyTorch docs for `torch.bmm`: "The tensors must be 3-D, containing matrices of the same size. If input is a (b×n×m) tensor, mat2 is a (b×m×p) tensor, out will be a (b×n×p) tensor."
The strides matter: for input of shape (B*H, N, N), the stride between batches should be N*N. But if we permute a (B,N,N,H) tensor to (B,H,N,N) and reshape to (B*H,N,N), the stride between batch elements might not be N*N.
Let me think about this more carefully:
- a: shape (B, N, N, H), strides (N*N*H, N*H, H, 1)
- a.permute(0, 3, 1, 2): shape (B, H, N, N), strides (N*N*H, 1, H, N*H)
- .reshape(B*H, N, N): The last dimension N has stride N*H, and the middle dimension N has stride H. This is NOT a standard row-major layout for (B*H, N, N).
For bmm, each matrix in the batch should be laid out as (N, N) row-major. But here, the innermost dimension has stride N*H, not 1. So bmm would need to handle this strided layout.
Actually, PyTorch's bmm should handle this by using the appropriate leading dimensions for cuBLAS. Let me verify: `a` after permute/reshape has shape (B*H, N, N) and strides (N*N*H, H, N*H). For bmm, it would call `cublasSgemmBatched` with appropriate leading dimensions. The leading dimension for each matrix would be N*H (the stride of the last dimension), and the batch stride would be N*N*H.
Wait, for `cublasSgemmBatched`, each matrix in the batch must be accessible via a pointer. If the matrices are not contiguous in the batch dimension, it would need to construct an array of pointers. But `bmm` in PyTorch handles strided inputs by using `cublasSgemmStridedBatched` with appropriate strides.
For `cublasSgemmStridedBatched`, the requirements are:
- Shape (batch, m, k)
- StrideA = batch_stride
- lda = leading_dimension (stride of rows)
In our case:
- a_r has shape (B*H, N, N)
- The stride between rows: for a matrix at batch index bh, element (i,j) is at offset bh*N*N*H + i*H + j*N*H
Wait, this is getting messy. Let me just check with a concrete example.
For B=2, H=32, N=2:
a has shape (2, 2, 2, 32)
permute(0,3,1,2): shape (2, 32, 2, 2)
Index (b, h, i, j) maps to original (b, i, j, h)
Offset in original: b*2*2*32 + i*2*32 + j*32 + h
Offset in permuted: same as original = b*128 + i*64 + j*32 + h
Strides of permuted: (128, 1, 64, 32)
reshape(64, 2, 2): The first element of batch 0 is at offset 0. First element of batch 1 is at offset 128. But 128 != 2*2 = 4. So batch elements are not contiguous.
For `bmm`, PyTorch should handle this. Let me just test it.
Actually, looking at the PyTorch `bmm` implementation, it calls into `addmm` or specialized path. For 3D inputs, it should handle arbitrary strides. The strided batched GEMM in cuBLAS can handle non-contiguous batch layouts.
Let me just use bmm and see if it works. If there are issues, we'll debug.
I think PyTorch should handle this correctly because:
1. `permute(0,3,1,2)` returns a view that's valid for operations
2. `reshape(B*H, N, N)` creates a new view (or copy if needed)
3. `bmm` on 3D tensors calls the appropriate cuBLAS routine
The one potential issue is whether bmm requires contiguous memory for the matrix dimensions. For the inner 2D matrix, each row should be contiguous (stride 1 for the last dimension). In our case, the last dimension has stride H, not 1. So each "row" in bmm terminology is not contiguous.
Wait, in bmm, the input is (batch, n, m). The "rows" are along dimension -2, and "columns" along dimension -1. For our a_r of shape (B*H, N, N), the stride of the last dimension is N*H, and the stride of the second-to-last dimension is H.
For a standard (B, N, N) tensor with contiguous memory, strides would be (N*N, N, 1). Our tensor has strides (N*N*H, H, N*H). For cuBLAS, the leading dimension (lda) is the stride between columns of the same row. For our tensor, the columns of a row are separated by N*H. This is not 1, but cuBLAS supports arbitrary lda (as long as lda >= the matrix dimension).
So `cublasSgemmStridedBatched` with:
- m = N, n = N, k = N
- A: lda = N*H, batch_stride = N*N*H
- B: ldb = N*H, batch_stride = N*N*H
- C: ldc = N, batch_stride = N*N
Wait, actually for B (the second operand), we transpose it: `b_r.transpose(-1, -2)`. The transposed tensor would have shape (B*H, N, N) with strides (N*N*H, 1, H). Wait, no:
Original b_r (same as a_r): shape (B*H, N, N), strides (N*N*H, H, N*H)
.transpose(-1, -2): shape (B*H, N, N), strides (N*N*H, N*H, H)
For the transpose in cuBLAS, we can pass `CUBLAS_OP_T` with the original strides, or use the transposed tensor directly.
Actually, looking at this more carefully, the transposed b_r has its last dimension with stride H, not 1. So neither a_r nor b_r have contiguous last dimensions.
For `torch.bmm`, does it handle non-contiguous inputs? According to docs and common usage, yes, it does handle non-contiguous inputs by copying them to contiguous temporary tensors if needed.
If PyTorch has to copy the inputs to contiguous memory, that's expensive! For B*H=64, N=32, each tensor is 64*32*32*4 = 256KB. Two copies = 512KB. That's not huge, but it adds memory traffic.
To avoid this, let me write a and b in (B, H, N, N) layout from the kernel. Then for bmm:
- a_r = a.reshape(B*H, N, N) - this should be a proper view since the last two dimensions are contiguous
- b_r = b.reshape(B*H, N, N)
- c = bmm(a_r, b_r.transpose(-1, -2))
Wait, for a in (B, H, N, N) layout, contiguous:
- Strides: (H*N*N, N*N, N, 1)
- reshape(B*H, N, N): strides (N*N, N, 1) - contiguous!
This is much better!
So I should write the kernel to produce a and b in (B, H, N, N) layout instead of (B, N, N, H).
For the output `a[b, i, j, h]` in the original, I'm writing `a[b, h, i, j]` in the new layout.
Then bmm produces c of shape (B*H, N, N), which reshape back to (B, H, N, N).
Then in postprocess, I can read c[b, h, i, j] directly.
And finally, the output should be (B, N, N, P), which is the same as the original.
Let me update the kernel to write in (B, H, N, N) layout:
```
a_out[((b * H + h) * N + i) * N + j] = a_proj * sigmoidf(a_gate) * mask;
```
And in postprocess, read:
```
einsum_row[tid] = einsum_out[((b * H + tid) * N + i) * N + j];
```
This should work with bmm without copies!
OK, let me now write the complete code.
One more optimization: since bmm produces output in (B, H, N, N) layout, and we need to read it in the postprocess kernel, this is perfect. No permute needed for the einsum output!
So the flow is:
1. Kernel produces a, b in (B, H, N, N)
2. bmm(a.reshape(B*H, N, N), b.reshape(B*H, N, N).transpose(-1, -2)) gives c of shape (B*H, N, N)
3. c.reshape(B, H, N, N) is the einsum output
4. Postprocess reads c[b, h, i, j] and produces output[b, i, j, p]
Great! Let me write this.
For the bmm call, I need to make sure `b.reshape(B*H, N, N).transpose(-1, -2)` also doesn't require a copy. Since b is in (B, H, N, N) layout, reshape to (B*H, N, N) gives contiguous (B*H, N, N). Transpose gives (B*H, N, N) with swapped strides. For bmm, PyTorch should handle the transpose directly by telling cuBLAS to transpose the matrix.
Actually, looking at the cuBLAS call: `bmm(A, B)` where A=(B*H,N,N) and B=(B*H,N,N).transpose(-2,-1). PyTorch can pass CUBLAS_OP_N for A and CUBLAS_OP_T for B to cuBLAS without actually transposing B in memory.
Yes, this is optimized in PyTorch. When you do `bmm(A, B.t())`, it uses the transpose flags in the GEMM call rather than physically transposing B.
So `torch.bmm(a_r, b_r.transpose(-2, -1))` should be efficient.
Let me also verify: `a_r = a.reshape(B*H, N, N)`. Since a is (B, H, N, N) contiguous, reshape to (B*H, N, N) is a view (no copy).
And `c = torch.bmm(a_r, b_r.transpose(-2, -1))`. c has shape (B*H, N, N).
Then `c_r = c.reshape(B, H, N, N)`. This is a view since c is contiguous as the output of bmm.
And `einsum_out = c_r.permute(0, 2, 3, 1)`? No wait, `c_r` is already in (B, H, N, N) but we need (B, N, N, H) for the postprocess? No, the postprocess kernel reads `c[b, h, i, j]` which is `((b * H + h) * N + i) * N + j`. And c_r of shape (B, H, N, N) with contiguous memory has exactly this layout.
Actually, wait. If c has shape (B*H, N, N) and is contiguous, then c[bh, i, j] is at offset bh*N*N + i*N + j. After reshape(B, H, N, N), offset is b*H*N*N + h*N*N + i*N + j. This matches `((b * H + h) * N + i) * N + j`.
So the postprocess kernel can read directly from `c.reshape(B, H, N, N)`.
But we can also just use `c.view(B, H, N, N)` in the postprocess call.
Actually, I realize there's another issue. The original PyTorch model stores a as (B, N, N, H). If I write a custom kernel that only replaces `forward()`, the weights are still in the same format. I can create intermediate tensors in any layout I want. The model doesn't expose the intermediate tensors.
So my approach is fine.
Let me code this up now.
Also, I should handle the weight access in the kernel. PyTorch passes tensors to the CUDA function. I need to make sure weights are contiguous. PyTorch nn.Module weights are typically contiguous.
For the preprocess kernel, I'm passing:
- ln_w: shape (P,), stride (1,)
- proj_a_w: shape (H, P), stride (P, 1)
- etc.
These should be contiguous. I'll add `.contiguous()` in the Python wrapper to be safe, though it might not be necessary.
One small thing: `pair_mask` in the original is (B, N, N). In the kernel, `pair_mask[(b * N + i) * N + j]` gives the mask value for the (i, j) element in the current batch.
Let me finalize the complete Python file.
```python
import torch
import torch.nn as nn
from torch.utils.cpp_extension import load_inline
cpp_source = r"""
torch::Tensor forward_cuda(
torch::Tensor pair_repr,
torch::Tensor pair_mask,
torch::Tensor ln_in_w, torch::Tensor ln_in_b,
torch::Tensor proj_a_w, torch::Tensor proj_a_b,
torch::Tensor gate_a_w, torch::Tensor gate_a_b,
torch::Tensor proj_b_w, torch::Tensor proj_b_b,
torch::Tensor gate_b_w, torch::Tensor gate_b_b,
torch::Tensor ln_out_w, torch::Tensor ln_out_b,
torch::Tensor out_proj_w, torch::Tensor out_proj_b,
torch::Tensor out_gate_w, torch::Tensor out_gate_b
);
"""
cuda_source = r"""
#include <torch/extension.h>
#include <cuda_runtime.h>
__inline__ __device__ float sigmoidf(float x) {
return 1.0f / (1.0f + expf(-x));
}
__global__ void preprocess_kernel(
const float* __restrict__ pair_repr,
const float* __restrict__ pair_mask,
const float* __restrict__ ln_w,
const float* __restrict__ ln_b,
const float* __restrict__ proj_a_w,
const float* __restrict__ proj_a_b,
const float* __restrict__ gate_a_w,
const float* __restrict__ gate_a_b,
const float* __restrict__ proj_b_w,
const float* __restrict__ proj_b_b,
const float* __restrict__ gate_b_w,
const float* __restrict__ gate_b_b,
float* __restrict__ a_out,
float* __restrict__ b_out,
int B, int N, int P, int H
) {
__shared__ float row[64];
__shared__ float warp_sums[2];
int b = blockIdx.x;
int i = blockIdx.y;
int j = blockIdx.z;
int tid = threadIdx.x;
if (tid < P) {
row[tid] = pair_repr[((b * N + i) * N + j) * P + tid];
}
__syncthreads();
float x = (tid < P) ? row[tid] : 0.0f;
float sum = x;
#pragma unroll
for (int offset = 16; offset > 0; offset /= 2) {
sum += __shfl_down_sync(0xffffffff, sum, offset);
}
if ((tid & 31) == 0) warp_sums[tid >> 5] = sum;
__syncthreads();
float total = warp_sums[0] + warp_sums[1];
float mean = total / P;
float diff = (tid < P) ? (row[tid] - mean) : 0.0f;
float sq = diff * diff;
sum = sq;
#pragma unroll
for (int offset = 16; offset > 0; offset /= 2) {
sum += __shfl_down_sync(0xffffffff, sum, offset);
}
if ((tid & 31) == 0) warp_sums[tid >> 5] = sum;
__syncthreads();
float var = (warp_sums[0] + warp_sums[1]) / P;
float inv_std = rsqrtf(var + 1e-5f);
if (tid < P) {
row[tid] = diff * inv_std * ln_w[tid] + ln_b[tid];
}
__syncthreads();
if (tid < H) {
int h = tid;
float a_proj = proj_a_b[h];
float a_gate = gate_a_b[h];
float b_proj = proj_b_b[h];
float b_gate = gate_b_b[h];
#pragma unroll
for (int d = 0; d < P; d++) {
float xv = row[d];
a_proj += xv * proj_a_w[h * P + d];
a_gate += xv * gate_a_w[h * P + d];
b_proj += xv * proj_b_w[h * P + d];
b_gate += xv * gate_b_w[h * P + d];
}
float mask_val = pair_mask[(b * N + i) * N + j];
float a_sig = sigmoidf(a_gate);
float b_sig = sigmoidf(b_gate);
a_out[((b * H + h) * N + i) * N + j] = a_proj * a_sig * mask_val;
b_out[((b * H + h) * N + i) * N + j] = b_proj * b_sig * mask_val;
}
}
__global__ void postprocess_kernel(
const float* __restrict__ einsum_out,
const float* __restrict__ pair_repr,
const float* __restrict__ ln_w,
const float* __restrict__ ln_b,
const float* __restrict__ out_proj_w,
const float* __restrict__ out_proj_b,
const float* __restrict__ out_gate_w,
const float* __restrict__ out_gate_b,
float* __restrict__ output,
int B, int N, int H, int P
) {
__shared__ float einsum_row[32];
__shared__ float pair_row[64];
__shared__ float warp_sums[2];
int b = blockIdx.x;
int i = blockIdx.y;
int j = blockIdx.z;
int tid = threadIdx.x;
if (tid < H) {
einsum_row[tid] = einsum_out[((b * H + tid) * N + i) * N + j];
}
__syncthreads();
float x = (tid < H) ? einsum_row[tid] : 0.0f;
float sum = x;
#pragma unroll
for (int offset = 16; offset > 0; offset /= 2) {
sum += __shfl_down_sync(0xffffffff, sum, offset);
}
if ((tid & 31) == 0) warp_sums[tid >> 5] = sum;
__syncthreads();
float total = warp_sums[0] + warp_sums[1];
float mean = total / H;
float diff = (tid < H) ? (einsum_row[tid] - mean) : 0.0f;
float sq = diff * diff;
sum = sq;
#pragma unroll
for (int offset = 16; offset > 0; offset /= 2) {
sum += __shfl_down_sync(0xffffffff, sum, offset);
}
if ((tid & 31) == 0) warp_sums[tid >> 5] = sum;
__syncthreads();
float var = (warp_sums[0] + warp_sums[1]) / H;
float inv_std = rsqrtf(var + 1e-5f);
if (tid < H) {
einsum_row[tid] = diff * inv_std * ln_w[tid] + ln_b[tid];
}
__syncthreads();
if (tid < P) {
pair_row[tid] = pair_repr[((b * N + i) * N + j) * P + tid];
}
__syncthreads();
if (tid < P) {
float out_val = out_proj_b[tid];
for (int h = 0; h < H; h++) {
out_val += einsum_row[h] * out_proj_w[tid * H + h];
}
float gate = out_gate_b[tid];
for (int d = 0; d < P; d++) {
gate += pair_row[d] * out_gate_w[tid * P + d];
}
output[((b * N + i) * N + j) * P + tid] = sigmoidf(gate) * out_val;
}
}
torch::Tensor forward_cuda(
torch::Tensor pair_repr,
torch::Tensor pair_mask,
torch::Tensor ln_in_w, torch::Tensor ln_in_b,
torch::Tensor proj_a_w, torch::Tensor proj_a_b,
torch::Tensor gate_a_w, torch::Tensor gate_a_b,
torch::Tensor proj_b_w, torch::Tensor proj_b_b,
torch::Tensor gate_b_w, torch::Tensor gate_b_b,
torch::Tensor ln_out_w, torch::Tensor ln_out_b,
torch::Tensor out_proj_w, torch::Tensor out_proj_b,
torch::Tensor out_gate_w, torch::Tensor out_gate_b
) {
int B = pair_repr.size(0);
int N = pair_repr.size(1);
int P = pair_repr.size(3);
int H = proj_a_w.size(0);
auto options = pair_repr.options();
auto a = torch::empty({B, H, N, N}, options);
auto b = torch::empty({B, H, N, N}, options);
dim3 grid(B, N, N);
dim3 block(64);
preprocess_kernel<<<grid, block>>>(
pair_repr.data_ptr<float>(),
pair_mask.data_ptr<float>(),
ln_in_w.data_ptr<float>(), ln_in_b.data_ptr<float>(),
proj_a_w.data_ptr<float>(), proj_a_b.data_ptr<float>(),
gate_a_w.data_ptr<float>(), gate_a_b.data_ptr<float>(),
proj_b_w.data_ptr<float>(), proj_b_b.data_ptr<float>(),
gate_b_w.data_ptr<float>(), gate_b_b.data_ptr<float>(),
a.data_ptr<float>(), b.data_ptr<float>(),
B, N, P, H
);
auto a_r = a.reshape({B * H, N, N});
auto b_r = b.reshape({B * H, N, N});
auto c = torch::bmm(a_r, b_r.transpose(-2, -1)); // (B*H, N, N)
auto einsum = c.reshape({B, H, N, N});
auto output = torch::empty_like(pair_repr);
postprocess_kernel<<<grid, block>>>(
einsum.data_ptr<float>(),
pair_repr.data_ptr<float>(),
ln_out_w.data_ptr<float>(), ln_out_b.data_ptr<float>(),
out_proj_w.data_ptr<float>(), out_proj_b.data_ptr<float>(),
out_gate_w.data_ptr<float>(), out_gate_b.data_ptr<float>(),
output.data_ptr<float>(),
B, N, H, P
);
return output;
}
"""
fused = load_inline(
name="fused_tri_mult",
cpp_sources=cpp_source,
cuda_sources=cuda_source,
functions=["forward_cuda"],
extra_cuda_cflags=["-O3"],
verbose=False,
)
class ModelNew(nn.Module):
def __init__(self, pair_dim, hidden_dim):
super().__init__()
self.layer_norm_in = nn.LayerNorm(pair_dim)
self.proj_a = nn.Linear(pair_dim, hidden_dim)
self.gate_a = nn.Linear(pair_dim, hidden_dim)
self.proj_b = nn.Linear(pair_dim, hidden_dim)
self.gate_b = nn.Linear(pair_dim, hidden_dim)
self.layer_norm_out = nn.LayerNorm(hidden_dim)
self.out_proj = nn.Linear(hidden_dim, pair_dim)
self.out_gate = nn.Linear(pair_dim, pair_dim)
def forward(self, pair_repr, pair_mask):
pair_repr = pair_repr.contiguous()
pair_mask = pair_mask.contiguous()
return fused.forward_cuda(
pair_repr, pair_mask,
self.layer_norm_in.weight, self.layer_norm_in.bias,
self.proj_a.weight, self.proj_a.bias,
self.gate_a.weight, self.gate_a.bias,
self.proj_b.weight, self.proj_b.bias,
self.gate_b.weight, self.gate_b.bias,
self.layer_norm_out.weight, self.layer_norm_out.bias,
self.out_proj.weight, self.out_proj.bias,
self.out_gate.weight, self.out_gate.bias,
)
```
Wait, I need to be careful about the `__shared__` arrays. In the code I have `__shared__ float row[64];` which is a fixed-size array. But P might not always be 64. For this specific problem, the sizes are fixed (P=64, H=32), but to be safe and general, I should not hardcode sizes.
However, the kernel is compiled with these sizes. If the test uses different sizes, the kernel might not work correctly. But the problem says the sizes are as given. For safety, let me make the shared memory dynamic or keep the fixed sizes with comments.
Actually, the reference implementation uses `pair_dim = 64, hidden_dim = 32, seq_len = 32, batch_size = 2`. But the class should work for any valid inputs. If we hardcode 64 and 32, and someone passes different values, it could fail.
Let me make the shared memory dynamic where needed. Or I can just pass P and H and use them for bounds checking, while the shared arrays are sized to the maximum expected values.
Actually, dynamic shared memory is:
```cpp
extern __shared__ float smem[];
```
But then I need to calculate the size at kernel launch. For simplicity, and since the sizes in the problem are fixed, let me use fixed-size arrays but make sure the code works for smaller sizes too.
Wait actually, the problem says:
"Your ModelNew class must: 1. Accept the same constructor arguments as Model."
The constructor takes pair_dim and hidden_dim. The forward takes arbitrary pair_repr and pair_mask. We should handle arbitrary sizes within reason.
But for the kernel compilation, if I use `__shared__ float row[64]`, it allocates 64 floats regardless of P. For P < 64, I waste some shared memory but the code still works (since I only read/write elements 0 to P-1).
For H > 32, `einsum_row[32]` would be too small!
To be safe, let me use maximum reasonable sizes or dynamic shared memory.
Actually, PyTorch nn.Linear weights can be any size. For typical values, pair_dim could be up to 128 or 256, and hidden_dim up to 64 or 128.
For safety, let me change the kernels to use dynamic shared memory:
```cpp
extern __shared__ float smem[];
```
Then calculate offsets:
```cpp
float* row = smem;
float* warp_sums = row + P; // or use a separate small shared array
```
For warp_sums, since it's only 2 floats, I can declare it as a fixed-size array or use the dynamic memory too.
Actually, in CUDA, you can mix fixed and dynamic shared memory:
```cpp
__shared__ float warp_sums[2];
extern __shared__ float smem[];
```
But the actual layout depends on the compiler. To be safe, I'll put everything in dynamic shared memory.
For preprocess_kernel:
```cpp
extern __shared__ float smem[];
float* row = smem; // size P
// warp_sums can be a fixed array since it's small:
__shared__ float warp_sums[2];
```
For postprocess_kernel:
```cpp
extern __shared__ float smem[];
float* einsum_row = smem; // size H
float* pair_row = einsum_row + H; // size P
__shared__ float warp_sums[2];
```
At kernel launch, I need to pass the shared memory size:
```cpp
size_t smem_size = P * sizeof(float); // for preprocess
preprocess_kernel<<<grid, block, smem_size>>>(...);
size_t smem_size = (H + P) * sizeof(float); // for postprocess
postprocess_kernel<<<grid, block, smem_size>>>(...);
```
Wait, but I also fixed warp_sums[2] with fixed size. Let me verify: fixed-size `__shared__` arrays and `extern __shared__` can coexist. The fixed-size arrays are placed before or after the dynamic array depending on the compiler. On NVIDIA GPUs, fixed-size arrays typically come first in shared memory, followed by the dynamic array.
Actually, looking at the CUDA programming guide: "Variables declared with __shared__ but without the extern qualifier have static shared memory allocation. Variables declared with extern __shared__ have dynamic shared memory allocation. Static and dynamic shared memory can be used in the same kernel, but the total amount of shared memory cannot exceed the maximum allowed per block."
The memory layout: static allocations are placed first, then dynamic allocation follows.
So I can use:
```cpp
__shared__ float warp_sums[2];
extern __shared__ float smem[];
```
And `smem` starts after `warp_sums[2]`. This should work.
Let me update the code.
Actually, wait. I realized there's a potential issue with the layer norm. When `blockDim.x` > P, some threads have x = 0, which contributes correctly to the mean. When `blockDim.x` < P, the warp reduction only sums over the threads in the warp. I need to ensure `blockDim.x >= P` for the warp reduction to cover all elements.
In my code, I launch with blockDim.x = 64 threads, and P = 64. That's 2 warps of 32 threads each. The warp reduction within each warp sums 32 elements. Then I combine the two warp sums. This works for P=64.
But if P=32 and blockDim.x=64, threads 32-63 have x=0. Warp 1 sum = 0. Total = warp0_sum + 0 = warp0_sum. But warp0 only sums 32 elements (threads 0-31), which is exactly P elements. The mean = total / P = warp0_sum / 32. This is correct!
Wait, no. warp0_sum includes threads 0-31 with x = row[0..31]. That's P=32 elements. warp1_sum = 0 (since x=0 for threads 32-63). Total = warp0_sum. Mean = warp0_sum / P. Correct!
If P=128 and blockDim.x=64, we'd need 4 warps and more complex reduction, but that's not our case.
So for P <= 64 and H <= 32, with blockDim.x=64, the code works correctly.
What if P > 64? Then row[64] would overflow. The test might not use P > 64, but to be safe, I should use dynamic shared memory sized to P.
Let me use dynamic shared memory for all arrays. I'll make the code robust.
```cpp
extern __shared__ float smem[];
__shared__ float warp_sums[2];
// In preprocess: smem layout: row[P]
// In postprocess: smem layout: einsum_row[H], pair_row[P]
```
And pass the appropriate shared memory sizes at launch.
Let me also think about what happens if H > 64. Then `tid < H` might not be handled correctly with blockDim.x=64. But for the test, H=32.
For generality, I should launch with blockDim.x = max(P, H) at least. Or I should use different block sizes for each kernel.
Actually, the simplest is to launch with `blockDim.x = max(P, H)` or just `blockDim.x = max(32, max(P, H))`. For P=64, H=32, that gives 64.
But my kernels assume blockDim.x = 64 for the two-warp reduction. If blockDim.x is different, the warp_sums array size and reduction code need to change.
To keep it simple and correct for the given test case, I'll just hardcode to 64 threads. The problem clearly specifies pair_dim=64 and hidden_dim=32. If the evaluation uses different sizes, the code might fail, but that's a risk.
Actually, looking at the test infrastructure, it uses:
```
pair_dim = 64
hidden_dim = 32
seq_len = 32
batch_size = 2
```
But it might also test with different sizes for robustness. Let me make the code slightly more robust.
One option: compute `block_size = max(32, max(P, H))`, use dynamic shared memory sized accordingly, and compute `num_warps = (block_size + 31) / 32`. Then warp_sums should be sized dynamically or we can use dynamic shared memory for it too.
But this makes the reduction code more complex. For the given problem, let me just use 64 threads with fixed warp_sums[2] and accept that it works for P, H <= 64.
Actually, I just realized I should verify one more thing: the weights for nn.Linear in PyTorch. When you access `self.proj_a.weight`, it has shape (out_features, in_features) = (H, P). And the data layout is row-major: weight[h, p] is contiguous.
But wait - does PyTorch store Linear weights as (out_features, in_features) or transposed?
In PyTorch, `nn.Linear(in_features, out_features)` stores weight as `(out_features, in_features)`. The operation is `y = x @ A^T + b`, where A is the weight matrix. So mathematically, `y[i] = sum_j x[j] * A[i, j]`.
In my kernel, I compute `sum_d row[d] * proj_a_w[h * P + d]`, which corresponds to `A[h, d]`. This matches PyTorch's Linear.
Good.
One more thing: LayerNorm in PyTorch uses `weight` and `bias` parameters. The computation is `(x - mean) / sqrt(var + eps) * weight + bias`. My kernel does exactly this.
The `eps` in PyTorch LayerNorm is 1e-5 by default. I used `1e-5f` in the kernel. Good.
Now, for the `pair_mask`: the original code does `mask = pair_mask.unsqueeze(-1)`, then `a = ... * mask`. So mask has shape (B, N, N, 1) and broadcasts to (B, N, N, H). In the kernel, I multiply by `pair_mask[(b * N + i) * N + j]`, which is the scalar mask value for this (b, i, j). This is correct.
OK I'm fairly confident in the design. Let me now write the final code and compile it.
Wait, there's one more subtlety: in the original model, the layer norm is `self.layer_norm_in(pair_repr)`. This normalizes across the `pair_dim` dimension for each (B, N, N) element.
In my preprocess kernel, for each (b, i, j), I load `pair_repr[b, i, j, :]` and apply layer norm. But in the original code, `x = self.layer_norm_in(pair_repr)` normalizes ALL elements of pair_repr, not just one at a time. That is, for each spatial position (i, j), it normalizes across the pair_dim channel.
Wait, `nn.LayerNorm(pair_dim)` on input of shape (B, N, N, pair_dim) normalizes over the last dimension. Yes, for each (b, i, j), it computes mean and std over the pair_dim values. This is exactly what my kernel does!
Similarly, `self.layer_norm_out` normalizes over hidden_dim for each (b, i, j).
So my kernel is correct.
One more check: in the original code, `a = self.proj_a(x) * torch.sigmoid(self.gate_a(x)) * mask`. The `x` here is the layer-normalized pair_repr. My kernel applies layer norm first, then uses the normalized values for both proj_a and gate_a. Correct.
And `out = torch.einsum("bkih,bkjh->bijh", a, b)`. My bmm computes the same thing. Let me verify once more with shapes:
- a shape: (B, N, N, H) in original, but I write (B, H, N, N)
- b shape: same
einsum "bkih,bkjh->bijh":
- a[b, k, i, h] * b[b, k, j, h] summed over k
- out[b, i, j, h]
In my code:
- a has shape (B, H, N, N), a[b, h, i, j] = original_a[b, i, j, h]
- Wait no! I said a_out[((b * H + h) * N + i) * N + j] = a_orig[b, i, j, h]. So a[b, h, i, j] = original_a[b, i, j, h].
Then a_r = a.reshape(B*H, N, N). So a_r[bh, i, j] = a[b, h, i, j] = original_a[b, i, j, h].
For einsum: out[b, i, j, h] = sum_k original_a[b, k, i, h] * original_b[b, k, j, h].
In terms of a_r: original_a[b, k, i, h] = a_r[b*H + h, k, i].
Wait, a_r has shape (B*H, N, N). The first dimension is batch*H, second is N, third is N.
a_r[bh, i, j] where bh = b*H + h. So a_r[b*H + h, k, i] = original_a[b, k, i, h].
Similarly, b_r[b*H + h, k, j] = original_b[b, k, j, h].
Then `torch.bmm(a_r, b_r.transpose(-2, -1))` computes:
c[bh, i, j] = sum_k a_r[bh, i, k] * b_r[bh, j, k]
Wait! That's `sum_k a_r[bh, i, k] * b_r[bh, j, k]`. But we need `sum_k original_a[b, k, i, h] * original_b[b, k, j, h]`.
Let me expand:
- a_r[bh, i, k] = original_a[b, i, k, h]
- b_r[bh, j, k] = original_b[b, j, k, h]
So c[bh, i, j] = sum_k original_a[b, i, k, h] * original_b[b, j, k, h].
But we want sum_k original_a[b, k, i, h] * original_b[b, k, j, h].
These don't match! The indices are swapped!
In my a_r: a_r[bh, i, j] maps to original_a[b, i, j, h].
But einsum needs a[b, k, i, h] and b[b, k, j, h].
So for the a_r matrix, row i corresponds to fixed i, and column k corresponds to varying k. So a_r[bh, i, k] = original_a[b, i, k, h].
But einsum uses a[b, k, i, h] where the first index after b is k (the summation index), and the second is i. So in matrix form, we want A[k, i] and B[k, j].
If a_r[bh, i, k] = original_a[b, i, k, h], then to get original_a[b, k, i, h], we'd need a_r[bh, k, i] = original_a[b, k, i, h].
So I need to store a as a_r[bh, k, i] = original_a[b, k, i, h]. That is, the first spatial dimension of a should be k (the summation index), and the second should be i.
In my original layout planning:
a_out[((b * H + h) * N + i) * N + j] = a_orig[b, i, j, h]
This makes a_r[bh, i, j] = a_orig[b, i, j, h]. But for einsum, I need a_r[bh, k, i] = a_orig[b, k, i, h].
So I should write:
```
a_out[((b * H + h) * N + k) * N + i] = a_proj * sigmoidf(a_gate) * mask_val;
```
Wait, in the einsum, a uses indices (b, k, i, h) and b uses (b, k, j, h). The first spatial index is k for both.
So in a, for fixed (b, h, k), the values for different i should be contiguous. In b, for fixed (b, h, k), the values for different j should be contiguous.
For bmm: a_r has shape (B*H, N, N). We want a_r[b*H+h, k, i] = a[b, k, i, h]. So the inner two dimensions of a should be (k, i).
Similarly for b: b_r[b*H+h, k, j] = b[b, k, j, h]. Inner two dimensions should be (k, j).
Then `bmm(a_r, b_r.transpose(-2, -1))`:
c[bh, i, j] = sum_k a_r[bh, i, k] * b_r.transpose(-2, -1)[bh, k, j]
= sum_k a_r[bh, i, k] * b_r[bh, j, k]
Wait, that's still wrong because a_r[bh, i, k] is a[b, i, k, h] not a[b, k, i, h].
To fix this, we need a_r[bh, k, i] = a[b, k, i, h]. Then:
c[bh, i, j] = sum_k a_r[bh, i, k] * b_r[bh, j, k]
NO! `bmm(A, B^T)` does `C[i,j] = sum_k A[i,k] * B[j,k]`. This is A @ B^T.
If A = a_r[bh, :, :] with A[k, i] = a[b, k, i, h], then A is shape (N, N) where row k corresponds to fixed k and column i varies.
Then `bmm(a_r, b_r.transpose(-2, -1))`:
For a_r[bh, k, i] layout:
a_r[bh, :, :] is a matrix where a_r[bh, k, i] = a[b, k, i, h].
We want C[i,j] = sum_k a[b, k, i, h] * b[b, k, j, h].
This is equivalent to: C = a_r^T @ b_r, where a_r[bh, k, i] = a[b, k, i, h] and b_r[bh, k, j] = b[b, k, j, h].
Wait: if a_r[bh, k, i] = a[b, k, i, h], and we view this as matrix A with rows=k, columns=i, then A^T has rows=i, columns=k. Then (A^T @ B)[i, j] = sum_k A^T[i, k] * B[k, j] = sum_k a[b, k, i, h] * b[b, k, j, h]. This matches!
So we need `torch.bmm(a_r.transpose(-2, -1), b_r)` !
Or equivalently: `torch.bmm(b_r.transpose(-2, -1), a_r).transpose(-2, -1)`.
Wait let me re-examine:
- a_r[bh, k, i] original_a[b, k, i, h]
- b_r[bh, k, j] original_b[b, k, j, h]
We want C[bh, i, j] = sum_k original_a[b, k, i, h] * original_b[b, k, j, h]
This is: for each bh, C = A^T @ B where A[k, i] = a_r[bh, k, i] and B[k, j] = b_r[bh, k, j].
So C[i, j] = sum_k A[k, i] * B[k, j] = sum_k a_r[bh, k, i] * b_r[bh, k, j].
With `torch.bmm`, if we set `A_mat = a_r` (shape N, N) and `B_mat = b_r` (shape N, N), then:
- `bmm(A_mat, B_mat)` gives sum_k A_mat[i, k] * B_mat[k, j]. Not what we want.
- `bmm(A_mat.transpose(-2, -1), B_mat)` gives sum_k A_mat[k, i] * B_mat[k, j]. This matches!
So the correct call is `torch.bmm(a_r.transpose(-2, -1), b_r)`.
Wait, but a_r.transpose(-2, -1) creates a view with transposed strides. PyTorch bmm should handle this.
Let me double-check with a small example:
```python
import torch
a_orig = torch.tensor([[[[1,2],[3,4]], [[5,6],[7,8]]]]) # B=1, N=2, K=2, H=2
# Actually let me use proper shapes
```
Let's say B=1, N=2, H=2.
```
a_orig[b=0, k=0, i=0, h=0] = 1
a_orig[b=0, k=0, i=0, h=1] = 2
a_orig[b=0, k=0, i=1, h=0] = 3
a_orig[b=0, k=0, i=1, h=1] = 4
a_orig[b=0, k=1, i=0, h=0] = 5
a_orig[b=0, k=1, i=0, h=1] = 6
a_orig[b=0, k=1, i=1, h=0] = 7
a_orig[b=0, k=1, i=1, h=1] = 8
```
I want to store a_r[bh, k, i] = a_orig[b, k, i, h].
For h=0, bh=0:
a_r[0, 0, 0] = a_orig[0, 0, 0, 0] = 1
a_r[0, 0, 1] = a_orig[0, 0, 1, 0] = 3
a_r[0, 1, 0] = a_orig[0, 1, 0, 0] = 5
a_r[0, 1, 1] = a_orig[0, 1, 1, 0] = 7
For h=1, bh=1:
a_r[1, 0, 0] = 2
a_r[1, 0, 1] = 4
a_r[1, 1, 0] = 6
a_r[1, 1, 1] = 8
Now `bmm(a_r.transpose(-2, -1), b_r)` for bh=0:
a_r[0].transpose = [[1, 5], [3, 7]]
b_r[0] = [[1, 3], [5, 7]] (assuming same values for b)
bmm result = [[1+25, 3+35], [3+35, 9+49]] = [[26, 38], [38, 58]]
Expected einsum:
out[0, 0, 0, 0] = a[0,0,0,0]*b[0,0,0,0] + a[0,1,0,0]*b[0,1,0,0] = 1*1 + 5*5 = 26
out[0, 0, 1, 0] = a[0,0,0,0]*b[0,0,1,0] + a[0,1,0,0]*b[0,1,1,0] = 1*3 + 5*7 = 38
out[0, 1, 0, 0] = a[0,0,1,0]*b[0,0,0,0] + a[0,1,1,0]*b[0,1,0,0] = 3*1 + 7*5 = 38
out[0, 1, 1, 0] = a[0,0,1,0]*b[0,0,1,0] + a[0,1,1,0]*b[0,1,1,0] = 3*3 + 7*7 = 58
Yes! The bmm output is [[26, 38], [38, 58]], which matches.
So the correct call is `torch.bmm(a_r.transpose(-2, -1), b_r)` where a_r[bh, k, i] and b_r[bh, k, j].
Now I need to update my preprocess kernel to write in the layout a_r[bh, k, i]:
```
a_out[((b * H + h) * N + k) * N + i] = ...
```
Wait, in the original einsum, a is indexed as a[b, k, i, h]. But in the original PyTorch model, a has shape (B, N, N, H). So a[b, k, i, h] is a[k, i] spatial position for batch b and hidden h.
In my kernel, for each (b, i, j), I compute both a[b, i, j, :] and b[b, i, j, :]. But for the einsum, I need a at all k values for fixed (b, i) and b at all k values for fixed (b, j).
So the preprocess kernel should compute a for all k, not just for one k. This changes the design!
The original model computes a = proj_a(x) * sigmoid(gate_a(x)) * mask, where x = layer_norm(pair_repr). This computes a for ALL (b, i, j) positions. Then the einsum uses a[b, k, i, h] and b[b, k, j, h].
In my preprocess kernel, if I have one block per (b, i, j), I compute a[b, i, j, :] and b[b, i, j, :]. The einsum then uses a[b, k, i, :] and b[b, k, j, :].
For the einsum element out[b, i, j, h], it needs a[b, k, i, h] for all k and b[b, k, j, h] for all k.
If a is stored as (B, N, N, H), then a[b, k, i, h] is accessed with indices (b, k, i, h). This is NOT contiguous in memory for fixed (b, h). The stride between a[b, k, i, h] and a[b, k+1, i, h] is N*H (since k changes in the second dimension).
For bmm, we need the matrices to have contiguous rows or appropriate strides. If a is (B, N, N, H), and we want a_r[bh, k, i] = a[b, k, i, h], then the memory layout is:
offset = b*N*N*H + k*N*H + i*H + h
= ((b*N + k)*N + i)*H + h
For fixed bh = b*H + h, the matrix a_r[bh, :, :] has elements:
a_r[bh, k, i] at offset b*N*N*H + k*N*H + i*H + h
The stride between a_r[bh, k, i] and a_r[bh, k, i+1] is H.
The stride between a_r[bh, k, i] and a_r[bh, k+1, i] is N*H.
For bmm, a_r should be (batch, m, k). The leading dimension (lda) for cuBLAS is the stride between elements in the same row but adjacent columns. For a_r[bh, k, i] where k is the row index and i is the column index:
- lda = stride between a_r[bh, k, i] and a_r[bh, k, i+1] = H
Wait, but H=32. For a (N, N) matrix, lda=H means each row has stride H=32, but the matrix dimension is N=32. So lda = N, which means the rows are contiguous! Actually, for N=32 and H=32, lda=H=32 and the matrix is (32, 32). The row length is 32, so lda=32 means each row is contiguous.
Wait, let me recalculate:
- a has shape (B, N, N, H) = (2, 32, 32, 32)
- a[b, k, i, h] offset = b*32*32*32 + k*32*32 + i*32 + h
- For fixed b and h: offset = base + k*1024 + i*32
- Different i: stride 32
- Different k: stride 1024
For a_r of shape (B*H, N, N) where a_r[bh, k, i] = a[b, k, i, h]:
- bh = b*H + h
- offset = b*32*32*32 + k*32*32 + i*32 + h
= bh*32*32 + k*32*32 + i*32? No!
bh*1024 = b*H*1024 + h*1024 = b*32*1024 + h*1024 = b*32768 + h*1024
But the actual offset for a[b, k, i, h] is b*32768 + k*1024 + i*32 + h.
For bh = b*32 + h:
bh*1024 = b*32*1024 + h*1024 = b*32768 + h*1024
But the actual offset is b*32768 + k*1024 + i*32 + h.
These don't match! So a_r[bh, k, i] is NOT at offset bh*N*N + k*N + i.
The reason is that the h dimension is interleaved inside the innermost dimension, not in the batch dimension.
So a simple reshape of a (B, N, N, H) to (B*H, N, N) does NOT give a_r[bh, k, i] = a[b, k, i, h]. Instead, it would give some jumbled mapping.
To get a_r[bh, k, i] = a[b, k, i, h], we need to permute/reshape:
a.permute(0, 3, 1, 2) gives shape (B, H, N, N) with offset b*32768 + h*1024 + k*32 + i.
Then reshape(B*H, N, N) gives offset bh*1024 + k*32 + i, which is bh*N*N + k*N + i.
So `a.permute(0, 3, 1, 2).reshape(B*H, N, N)` correctly maps to a_r[bh, k, i] = a[b, k, i, h].
But as discussed earlier, `permute(0, 3, 1, 2)` on a (B, N, N, H) tensor gives non-contiguous memory. The `.reshape()` might create a copy.
If `a` is originally written in (B, H, N, N) layout from the kernel, then no copy is needed. Let me verify the layout again:
If my kernel writes: `a_out[((b * H + h) * N + k) * N + i]`
Then `a` has shape (B, H, N, N) and is contiguous.
`a.reshape(B*H, N, N)` gives a_r with a_r[bh, k, i] at offset bh*N*N + k*N + i = ((b*H+h)*N + k)*N + i.
And this equals the kernel write offset: `((b * H + h) * N + k) * N + i`. Yes!
So with the kernel writing to (B, H, N, N), `a.reshape(B*H, N, N)` gives a_r[bh, k, i] = a_orig[b, k, i, h]. Good.
Then `bmm(a_r.transpose(-2, -1), b_r)`:
a_r.transpose(-2, -1) has shape (B*H, N, N) with a_r^T[bh, i, k] = a_r[bh, k, i] = a_orig[b, k, i, h].
b_r has shape (B*H, N, N) with b_r[bh, k, j] = b_orig[b, k, j, h].
Then bmm(a_r^T, b_r):
c[bh, i, j] = sum_k a_r^T[bh, i, k] * b_r[bh, k, j]
= sum_k a_orig[b, k, i, h] * b_orig[b, k, j, h]
This is exactly the einsum!
Great, so the correct bmm call is `torch.bmm(a_r.transpose(-2, -1), b_r)`.
And the output c has shape (B*H, N, N). After reshape(B, H, N, N), c[b, h, i, j] = einsum_result[b, i, j, h].
Then in the postprocess kernel, I read `c[b, h, i, j]` using the offset `((b * H + h) * N + i) * N + j`. This is correct.
Now let me fix the preprocess kernel write location:
```
a_out[((b * H + h) * N + k) * N + i] = ...
```
Wait, in the original model, a[b, k, i, h] is the activation at spatial position (k, i) with hidden h. In the preprocess kernel, we iterate over (b, i, j) and compute a[b, i, j, :] and b[b, i, j, :]. So for each (b, i, j), the first two spatial indices are (i, j).
To match a[b, k, i, h], the kernel at block (b, k, i) should compute a[b, k, i, :]. But our grid is (B, N, N) where blockIdx.y = i and blockIdx.z = j.
Wait, in my original design, the grid was (B, N, N) with y=i and z=j, corresponding to computing a[b, i, j, :] and b[b, i, j, :].
For the einsum, a[b, k, i, h] uses the first spatial index as k (summation) and second as i (fixed). But our kernel computes a[b, i, j, :] where the first spatial index is i and second is j.
Since the variable names are arbitrary, as long as we're consistent:
- Our a tensor at spatial position (i, j) corresponds to a_orig[b, i, j, :] in the einsum.
- For einsum "bkih,bkjh->bijh", we need a at (k, i) and b at (k, j).
- If we relabel: our a[b, i, j, :] is what the einsum calls a[b, k=i, i=j, :]. This doesn't match.
Wait, I think I'm confusing myself. Let me be more explicit.
In the original model:
- `a = proj_a(x) * sigmoid(gate_a(x)) * mask` has shape (B, N, N, H)
- `b` has shape (B, N, N, H)
- `torch.einsum("bkih,bkjh->bijh", a, b)` means:
- a[b, k, i, h] is the value from a at batch=b, first_dim=k, second_dim=i, hidden=h
- b[b, k, j, h] is the value from b at batch=b, first_dim=k, second_dim=j, hidden=h
- out[b, i, j, h] = sum_k a[b, k, i, h] * b[b, k, j, h]
In the a tensor computed by `self.proj_a(x)`, the dimensions are:
- a[b, n1, n2, h] where n1 is the first spatial dim and n2 is the second spatial dim.
For the einsum, we're using a[b, k, i, h] where k is the FIRST spatial dimension and i is the SECOND spatial dimension.
In the original model, x has shape (B, N, N, P). x[b, n1, n2, :] is the pair representation at spatial position (n1, n2).
a[b, n1, n2, h] = proj_a(x[b, n1, n2, :]) * ...
For the einsum: a[b, k, i, h] = a[b, n1=k, n2=i, h]
And b[b, k, j, h] = b[b, n1=k, n2=j, h]
So a[b, n1, n2, h] is used with n1 as the summation index and n2 as the fixed index for the first operand.
For b[b, n1, n2, h], it's used with n1 as the summation index and n2 as the fixed index for the second operand.
So the einsum can be rewritten as:
`out[b, i, j, h] = sum_k a[b, k, i, h] * b[b, k, j, h]`
Where a[b, k, i, h] = original_a with first spatial dim = k, second = i.
If our preprocess kernel computes `a_block(b, n1, n2) = a[b, n1, n2, :]`, and we write it to a_out with layout a_out[b, h, n1, n2], then:
a_r = a_out.reshape(B*H, N, N) gives a_r[bh, n1, n2] = a[b, n1, n2, h].
For the einsum, we need a matrix where rows are k and columns are i/position:
A[bh, k, i] = a[b, k, i, h] = a_r[bh, k, i]
So a_r already has the right layout for A!
And B[bh, k, j] = b[b, k, j, h] = b_r[bh, k, j]
Then out[bh, i, j] = sum_k A[bh, k, i] * B[bh, k, j]
This is: C = A^T @ B in matrix form.
Because A^T[i, k] = A[k, i], so (A^T @ B)[i, j] = sum_k A^T[i, k] * B[k, j] = sum_k A[k, i] * B[k, j].
So `torch.bmm(a_r.transpose(-2, -1), b_r)` is correct.
And a_r[bh, k, i] corresponds to the original a[b, k, i, h] where k is the FIRST spatial dimension and i is the SECOND.
In the preprocess kernel, we compute a[b, i, j, :] where i is the first spatial dim and j is the second. So a_block(b, i, j, h) = a[b, i, j, h].
If we write this to a_out with a_out[b, h, i, j] = a[b, i, j, h], then:
a_r[bh, i, j] = a[b, i, j, h].
For the einsum, A[bh, k, i] = a[b, k, i, h] = a_r[bh, k, i].
Yes! a_r[bh, k, i] is exactly the value at original a[b, k, i, h] because k is the first index and i is the second.
So the layout is:
a_r[bh, first_spatial, second_spatial] = a[b, first_spatial, second_spatial, h].
This is exactly what our kernel produces!
OK so my analysis was correct, just a bit confused in the middle. Let me verify with the actual write:
preprocess kernel at block (b, i, j):
```
a_out[((b * H + h) * N + i) * N + j] = a[b, i, j, h]
```
So a_out[b, h, i, j] = a[b, i, j, h].
reshape(B*H, N, N):
a_r[bh, i, j] = a[b, i, j, h] where bh = b*H + h.
For einsum: A[bh, k, i] = a[b, k, i, h] = a_r[bh, k, i].
So A = a_r exactly, no transpose needed for the spatial dimensions.
Then out[bh, i, j] = sum_k A[bh, k, i] * B[bh, k, j]
This is `A^T @ B` where for each bh:
A^T[i, k] = A[k, i]
(A^T @ B)[i, j] = sum_k A^T[i, k] * B[k, j] = sum_k A[k, i] * B[k, j] ✓
So bmm(A.transpose(-2, -1), B) is correct.
I got it right eventually. Let me make sure the code says:
```python
c = torch.bmm(a_r.transpose(-2, -1), b_r)
```
NOT `torch.bmm(a_r, b_r.transpose(-2, -1))`.
OK, here's the final design confirmation:
1. preprocess_kernel computes a and b and writes them as (B, H, N, N) with a[b, h, n1, n2] = activation_at(b, n1, n2, h)
2. a_r = a.reshape(B*H, N, N), b_r = b.reshape(B*H, N, N)
3. c = bmm(a_r.transpose(-2, -1), b_r), so c[bh, i, j] = sum_k a_r[bh, k, i] * b_r[bh, k, j]
4. c.reshape(B, H, N, N) gives c[b, h, i, j] = einsum_result
5. postprocess reads c[b, h, i, j] and pair_repr[b, i, j, :] to produce output
Now, the postprocess kernel reads `einsum_out[((b * H + tid) * N + i) * N + j]` which is einsum_out[b, tid, i, j]. Since einsum result is stored as (B, H, N, N), this is correct.
And `pair_repr[((b * N + i) * N + j) * P + tid]` is the original pair representation at (b, i, j, tid).
For output: `output[((b * N + i) * N + j) * P + tid]` is (b, i, j, tid).
The output shape is (B, N, N, P), matching the original.
Great, everything is consistent now.
Let me write the final code!
Actually, wait. I should also check if there's an issue with the bmm output. `c = torch.bmm(a_r.transpose(-2, -1), b_r)` where a_r and b_r are both (B*H, N, N).
a_r.transpose(-2, -1) is (B*H, N, N) but with transposed strides.
b_r is (B*H, N, N).
PyTorch bmm with a transposed first argument should call cuBLAS with CUBLAS_OP_T for the first matrix. This should work without copying.
Actually, I need to double check: `bmm(A.transpose(-2, -1), B)` where A and B are batches of (N, N) matrices.
In cuBLAS terms:
- op(A) = A^T, where A is (N, N) stored in row-major? Or column-major?
Wait, PyTorch uses row-major tensors by default, but cuBLAS expects column-major. PyTorch handles this by transposing the operation.
For `torch.bmm(X, Y)`:
- X has shape (batch, m, k)
- Y has shape (batch, k, n)
- Output has shape (batch, m, n)
The actual cuBLAS call effectively computes C = A @ B where A and B are treated as column-major.
For row-major input X of shape (batch, m, k), PyTorch passes it as the transpose to cuBLAS, so cuBLAS computes `Y^T @ X^T`, then transposes the result.
This is getting complicated. Let me just test it or trust that PyTorch handles it correctly.
Actually, I realize there might be a simpler approach. Since N=32, H=32, I could write a custom CUDA kernel for the einsum too. This avoids the bmm complexity entirely.
For each (b, i, j, h) block of 32 threads, each thread reads a[b, k, i, h] and b[b, k, j, h] for its k, multiplies, and does warp reduce. Then writes the result.
This is very simple and avoids the layout issues with bmm. Let me add this as an option.
Custom einsum kernel:
```cpp
__global__ void einsum_kernel(
const float* __restrict__ a,
const float* __restrict__ b,
float* __restrict__ out,
int B, int N, int H
) {
int b = blockIdx.x;
int i = blockIdx.y;
int j = blockIdx.z;
int h = blockIdx.w;
int tid = threadIdx.x;
float val = 0.0f;
if (tid < N) {
int k = tid;
val = a[((b * H + h) * N + k) * N + i] * b[((b * H + h) * N + k) * N + j];
}
// Warp reduce sum
#pragma unroll
for (int offset = 16; offset > 0; offset /= 2) {
val += __shfl_down_sync(0xffffffff, val, offset);
}
if (tid == 0) {
out[((b * H + h) * N + i) * N + j] = val;
}
}
```
Wait, blockIdx.w doesn't exist in CUDA. The maximum grid dimension is 3D. So I'd need to combine dimensions.
I can use:
- grid = (B, N*N, H) where blockIdx.y = i*N + j
- Or grid = (B, N, N) with H threads per block
With H=32 threads per block and grid = (B, N, N):
Each block computes all H outputs for one (b, i, j).
Each thread h reads a[b, k, i, h] and b[b, k, j, h] for all k... that's a lot of reads.
Better: grid = (B*H, N, N) with N threads per block.
Each block computes one (b, h, i, j) output.
Threads 0..N-1 each read one k and multiply.
Warp reduce for sum.
But N=32 threads per block, one warp. Grid = B*H*N*N = 2*32*32*32 = 65536 blocks. Each with 32 threads. Total memory reads: 65536 * 32 * 2 * 4 bytes = ~16MB. That's more than the input tensor itself because each a and b element is read multiple times.
This is less efficient than bmm which would read each element once (with good caching).
So using bmm is still better for the einsum.
But to avoid the transpose layout complexity, I could write a and b in a layout that's naturally compatible with bmm.
If I write a as (B, H, N, N) and call bmm(a_r, b_r.transpose(-2, -1)):
a_r[bh, k, i] corresponds to a_orig[b, k, i, h]
b_r[bh, k, j] corresponds to b_orig[b, k, j, h]
b_r.transpose(-2, -1)[bh, j, k] = b_orig[b, k, j, h]
Then bmm(a_r, b_r.transpose(-2, -1)):
c[bh, i, j] = sum_k a_r[bh, i, k] * b_r.transpose(-2, -1)[bh, k, j]
Wait, a_r[bh, i, k] = a_orig[b, i, k, h], not a_orig[b, k, i, h].
So this doesn't work either.
If I write a as (B, H, N, N) where a[b, h, i, j] = a_orig[b, i, j, h]:
a_r[bh, i, j] = a_orig[b, i, j, h]
For einsum, we need A[bh, k, i] = a_orig[b, k, i, h] = a_r[bh, k, i]
Then bmm(A.transpose(-2, -1), B) works.
So the key is: a_r[bh, k, i] must equal a_orig[b, k, i, h].
And a_r[bh, k, i] = a_out[b, h, k, i] (since a_r is a.reshape(B*H, N, N)).
So a_out[b, h, k, i] = a_orig[b, k, i, h].
In the preprocess kernel, for block (b, n1, n2), we have n1=k and n2=i.
The output should be:
a_out[((b * H + h) * N + n1) * N + n2] = a_orig[b, n1, n2, h]
So if n1 is the FIRST spatial index and n2 is the SECOND:
a_out[b, h, n1, n2] = a_orig[b, n1, n2, h].
And a_r[bh, n1, n2] = a_orig[b, n1, n2, h].
For einsum: A[bh, k, i] = a_orig[b, k, i, h] = a_r[bh, k, i].
So A = a_r. Good.
Then bmm(A.transpose(-2, -1), B):
A.transpose(-2, -1)[bh, i, k] = A[bh, k, i] = a_orig[b, k, i, h].
B[bh, k, j] = b_orig[b, k, j, h].
bmm(A^T, B)[bh, i, j] = sum_k A^T[bh, i, k] * B[bh, k, j]
= sum_k a_orig[b, k, i, h] * b_orig[b, k, j, h].
Yes! This matches the einsum.
And a_r = a.reshape(B*H, N, N) is valid because a has shape (B, H, N, N) and is contiguous.
And b_r = b.reshape(B*H, N, N) is similarly valid.
And `torch.bmm(a_r.transpose(-2, -1), b_r)` should work with PyTorch optimally passing the transpose flag to cuBLAS.
Let me verify by checking if PyTorch handles the transpose in bmm without copying. In PyTorch's bmm implementation, when the input has a non-contiguous layout, it might copy. But for transpose(-2, -1), PyTorch's cuBLAS backend typically passes CUBLAS_OP_T rather than making a physical copy.
I believe this should be efficient. Let me proceed with this approach.
OK I'm finally ready to write the complete code.
One more thing: I need to make sure the .contiguous() calls in the Python wrapper are appropriate. Since pair_repr and pair_mask are already .contiguous() at the start, and the weights from nn.Module are typically contiguous, they should be fine.
For `a` and `b` created with `torch::empty`, they are contiguous by default.
For the `c` from bmm, it might be contiguous.
Let me just write the code. If there are issues, we'll see them in the correctness check.
Actually, I realize I should also consider whether the weights need `.t()` or `.contiguous()`. nn.Linear weights are stored as (out_features, in_features). In my kernel, I access `proj_a_w[h * P + d]` which assumes row-major layout with H rows and P columns. nn.Linear weight is exactly this. So no transpose is needed.
Let me write the complete code now.