reasoning
Let me analyze the problem:
1. We have a linear layer (matmul) with dimensions: batch_size=128, in_features=32768, out_features=32768
2. Swish activation: x * sigmoid(x)
3. Scaling: multiply by scaling_factor (2.0)
The operation is: output = (x @ W^T + b) * sigmoid(x @ W^T + b) * scaling_factor
This is a classic GEMM + elementwise fusion problem. For maximum performance, I should fuse the elementwise operations (swish + scaling) with the GEMM using a CUTLASS-like approach or a custom CUDA kernel.
Given the H100 specs:
- Memory bandwidth: 3.35 TB/s
- FP32 TFLOPS: 67
- BF16/FP16 Tensor Core: ~1979 TFLOPS
For a matrix of size 128 x 32768 x 32768:
- FLOPs for GEMM: 2 * 128 * 32768 * 32768 ≈ 274.9 GFLOPs
- Memory for weights: 32768 * 32768 * 4 bytes ≈ 4.3 GB
- Memory for input: 128 * 32768 * 4 bytes ≈ 16.8 MB
- Memory for output: 128 * 32768 * 4 bytes ≈ 16.8 MB
This is heavily compute-bound even at FP32 (274.9 GFLOPs / 67 TFLOPS = ~4ms). But we can use Tensor Cores for much better performance.
However, looking at the problem more carefully - this is a CUBLAS/ CUTLASS type problem. The standard approach would be to use a fused kernel.
For custom CUDA, I have options:
1. Use torch.matmul + separate kernel for swish+scale (easy, but not optimal)
2. Use CUTLASS or custom fused GEMM kernel (best performance)
3. Use CUTLASS through custom CUDA
Actually, the most practical approach here is to use CUTLASS or a similar approach. But since we're writing raw CUDA...
Let me think about this more carefully. The dimensions are M=128, N=32768, K=32768.
This is a batched GEMV essentially (since M=128 is small compared to N and K). Actually, it's a GEMM with small M, large N and K.
For M=128, N=32768, K=32768:
- We can partition N across the SMs.
- Each SM processes some chunk of N (columns)
Given that we have about 132 SMs on H100, we can split N=32768 across them, giving each SM about 248 columns.
For a fused kernel, I'd implement:
1. Load a chunk of K input values per row
2. Load the weight matrix in tiles
3. Compute partial sums
4. Apply swish + scale
5. Write output
Actually, using cuBLAS for the GEMM and then a separate custom kernel for Swish + scaling might be competitive if the GEMM dominates, but fusing the epilogue will be faster because we avoid writing the GEMM output to global memory and then reading it back.
However, implementing a fast GEMM from scratch in CUDA that competes with cuBLAS is very difficult. The best approach might be to use the CUTLASS epilogue fusion, but that requires including CUTLASS headers.
Wait, PyTorch already uses cuBLAS/cuBLASLt for linear layers, and cuBLASLt supports epilogue fusion. But can we access that through custom CUDA?
Actually, let me reconsider. The PyTorch reference does:
1. `self.matmul(x)` - uses cuBLAS, very fast
2. `x * torch.sigmoid(x)` - elementwise, memory bound
3. `x * self.scaling_factor` - elementwise
The main inefficiency is step 2 and 3 reading/writing memory. If we fuse them, we still have the GEMM writing to memory, then the fused kernel reading and writing once.
Can we fuse the GEMM with the epilogue? We could use `torch.matmul` and then our own kernel. But the real question is: can we do better by using CUTLASS or some other fused GEMM approach?
For the H100, with 3.35 TB/s bandwidth:
- A fused elementwise kernel on 128*32768 floats (~16.8 MB) would take about 2 * 16.8 MB / 3350 GB/s = ~10 microseconds
- The GEMM of this size at FP32 takes on the order of milliseconds
So the GEMM dominates. The question is whether we can outperform PyTorch's GEMM + elementwise. PyTorch should already be using cuBLAS which is very fast. But cuBLAS might not fuse the Swish activation.
Actually, for such a large matrix (K=32768, N=32768), even if we just use `torch.matmul` + custom kernel for swish+scale, the overhead of the elementwise part is tiny compared to GEMM. But wait - PyTorch might not even be fusing the two elementwise ops together (sigmoid + mul + mul) if the autograd graph requires intermediates.
Wait, this is the forward pass only. In eager mode, PyTorch does:
1. matmul -> output1 (writes memory)
2. sigmoid(output1) -> output2 (reads output1, writes output2)
3. output1 * output2 -> output3 (reads both, writes output3)
4. output3 * scaling_factor -> output4 (reads output3, writes output4)
That's a lot of memory traffic! If we fuse steps 2-4, we get:
1. matmul -> output1
2. fused_swish_scale(output1) -> output2 (reads output1 once, writes output2 once)
That's 2 fewer passes over 16.8 MB each, saving ~33.6 MB of memory traffic, which at 3.35 TB/s is about 10 microseconds. Still small compared to GEMM.
But actually, we might be able to do even better by implementing a custom GEMM that fuses everything. However, writing a GEMM kernel that's competitive with cuBLAS for these dimensions is extremely challenging.
Let me think about another approach: can we use CUTLASS through PyTorch? PyTorch has some support for CUTLASS but it's not directly exposed.
Alternatively, we can use the `torch._scaled_mm` or similar custom ops, but they may not support this exact epilogue.
Given the complexity, let me try a two-kernel approach first:
1. `torch.matmul` (or `torch.nn.functional.linear`)
2. Custom fused kernel for Swish + scaling
Then benchmark. If the speedup isn't enough, I'll try a single fused GEMM kernel.
Actually, wait. Looking more carefully at dimensions:
- M=128, N=32768, K=32768
- This is a very "tall and skinny" GEMM from the weight perspective, or wide and short from the activation perspective.
In cuBLAS terminology, M=128 (rows), N=32768 (cols), K=32768 (inner dim). This is C = A * B where A is 128x32768 and B is 32768x32768.
For such a small M, the GEMM might actually be memory-bound for the weights if not using Tensor Cores optimally. But with Tensor Cores, we can do much better.
Actually, let me calculate more carefully:
- Total FLOPs: 2 * M * N * K = 2 * 128 * 32768 * 32768 = 274,877,906,944 ≈ 275 GFLOPs
- At 67 TFLOPS FP32: 275 / 67,000 = 0.0041 s = 4.1 ms
- At 1979 TFLOPS FP16: 275 / 1,979,000 = 0.00014 s = 0.14 ms
So this is definitely compute-bound if we can use Tensor Cores.
PyTorch's linear + swish in FP32 might not use Tensor Cores optimally or might have significant overhead from the elementwise operations.
Let me try implementing:
1. A custom kernel that does matmul + swish + scale all fused
2. Use warp-level matrix multiply or just a simple approach where each thread block computes a tile of outputs
For M=128, a simple approach:
- Launch one block per row of output (128 blocks total, which is too few for H100)
- Or launch multiple blocks per row
Better approach for M=128, N=32768:
- Use a 2D grid: (M, N/TileN) or similar
- For example, TileM=1, TileN=256 or 512
- Grid would be (128, 128) for TileN=256, giving 16384 blocks - plenty for H100
Wait, that's too many blocks if each block only does 1*256 elements. Let me recalculate.
Actually, for H100 with 132 SMs, we'd want at least several thousand warps for good occupancy.
Let me think about a TileN-based approach:
- Each block handles all M rows for a subset of N columns
- Block dims: (128 threads, 1) or (256 threads, 1)
- Each thread computes one output element (or multiple)
For N=32768 and 128 threads per block:
- We'd need 256 blocks (32768 / 128 = 256)
- Each block handles all M=128 rows? No, that would be 128 threads each computing 128 elements across rows...
Let me reconsider. A common pattern:
- Grid: (M, N/tile_N) if M is small
- Each block computes a (1, tile_N) tile of the output
- tile_N could be 128 or 256
For M=128, tile_N=128: grid = (128, 256) = 32768 blocks. That's plenty.
Each block has e.g. 128 threads. Each thread computes 1 output.
But to reduce block count and increase work per block:
- tile_M = 2, tile_N = 256, grid = (64, 128) = 8192 blocks
- Or tile_M = 4, tile_N = 512, grid = (32, 64) = 2048 blocks
Actually, since K=32768 is very large, we want each block to process a significant chunk.
The simplest custom kernel:
- Grid: (M, ceil(N / block_N))
- Block: (block_M, block_N) - but we need to iterate over K
- Or a 1D block of threads where each thread computes one (or a few) outputs
Let me design a more specific kernel:
```cuda
// Each block processes a tile of output
// Thread block size: 128 threads
// Each thread computes 1 output element
// Tile size: 1x128 or similar
// Actually, let me use a vectorized approach for better memory bandwidth
template<int BLOCK_SIZE, int VELEM>
__global__ void fused_linear_swish_scale_kernel(
const float* __restrict__ input,
const float* __restrict__ weight,
const float* __restrict__ bias,
float* __restrict__ output,
int M, int N, int K, float scale)
{
// Each block computes a strip of N columns for a subset of M rows
// ...
}
```
Wait, I can also think of this as performing a reduction (dot product) for each output element. For K=32768, each dot product is quite long, so we can parallelize within each output element using shared memory reduction.
Common approach for small M, large N, large K:
- Each block computes one output element (or a small tile)
- All threads in the block cooperate on the dot product
- Use shared memory for partial sums
But that gives way too many blocks if each block is small. For 32768x128 = 4.2M output elements, we'd need 4.2M blocks, which is too many (max is 2^31-1 but scheduling overhead matters).
Better approach:
- Each block computes a tile, e.g., 8x8 or 16x16 output elements
- For each output element, use warp-level primitives or a small number of threads for the reduction
Actually, let me think about what cuBLAS does. For small M, large N, K:
- It probably uses a "GEMV" style approach or specialized GEMM kernels
- Each thread might compute multiple outputs by reusing loaded inputs
Let me design something more practical:
For M=128:
- We can have each thread block handle 1 row and a chunk of N columns
- BlockDim = 256 threads
- Each thread computes a dot product over K=32768 across 128 floats at a time
- For a single row and 256 columns, each thread computes 1 output
- Threads in a warp can load input cooperatively
- But weight loading would be uncoalesced if each thread needs a different weight column
Actually, for linear layer, weights are stored as N x K (out_features x in_features) in PyTorch:
- `weight[out_features, in_features]`
- To compute output[m, n] = sum_k input[m, k] * weight[n, k]
So for a fixed row m, and consecutive n values:
- input[m, k] is the same for all n
- weight[n, k] for consecutive n and fixed k is consecutive in memory (since weight is N x K, row-major, so weight[n, k] and weight[n+1, k] are K apart... wait)
In PyTorch, Linear weight is [out_features, in_features]. For `nn.Linear`, `y = x @ W^T + b`.
So W^T has shape [in_features, out_features]. But the stored weight W has shape [out_features, in_features].
When computing `x @ W^T`:
- x has shape [M, K]
- W^T has shape [K, N]
- output has shape [M, N]
In memory, x[m, k] and x[m, k+1] are consecutive.
W^T[k, n] corresponds to W[n, k]. Since W is row-major [N, K], W[n, k] and W[n, k+1] are consecutive. But W[n, k] and W[n+1, k] are K apart.
So for a fixed k, as n varies, W[n, k] are strided by K. Not great for coalescing.
cuBLAS handles this by transposing the problem appropriately.
For a custom kernel, one approach is:
- Have each thread compute one output element
- Thread i computes output for some (m, n)
- It loads input[m, k] and weight[n, k] for k = 0..K-1
- But weight access is strided by K, which is bad
Alternative: tile K dimension. Load a tile of input and a tile of weights into shared memory, compute partial products.
For this, we want:
- A thread block that works on a tile of outputs
- Loads a chunk of K for input (shared by all outputs in the tile)
- Loads a chunk of K for weights (different for each output column in the tile)
If we process a tile of N columns at once:
- Load input[m, k_start:k_start+TILE_K] to shared mem (TILE_K floats)
- Load weight[n_start:n_start+TILE_N, k_start:k_start+TILE_K] to shared mem (TILE_N x TILE_K floats)
- Compute partial products
But loading TILE_N x TILE_K weights might be too much for shared memory.
TILE_N = 128, TILE_K = 32: 128 * 32 * 4 = 16 KB. That's fine.
For M=128, we'd need to handle multiple M across blocks.
Let me sketch a kernel:
```cuda
__global__ void kernel(const float* input, const float* weight, const float* bias,
float* output, int M, int N, int K, float scale) {
__shared__ float s_input[TILE_K];
__shared__ float s_weight[TILE_N][TILE_K]; // or transposed
int m = blockIdx.x; // row index
int n_base = blockIdx.y * TILE_N; // starting column
int tid = threadIdx.x;
float acc[TILE_N_PER_THREAD]; // each thread computes multiple outputs?
// Loop over K in chunks of TILE_K
for (int k_base = 0; k_base < K; k_base += TILE_K) {
// Load input[m, k_base + tid] into shared mem
if (tid < TILE_K) {
s_input[tid] = input[m * K + k_base + tid];
}
// Load weight slice: weight[n_base + n_local, k_base + k_local]
// This is tricky because weight is [N, K]
// weight[n, k] for n in [n_base, n_base+TILE_N), k in [k_base, k_base+TILE_K)
// We can have threads cooperatively load
// Total elements: TILE_N * TILE_K
// Each thread loads some
__syncthreads();
// Compute partial products
for (int n_local = 0; n_local < TILE_N; n_local++) {
for (int k = 0; k < TILE_K; k++) {
acc[n_local] += s_input[k] * s_weight[n_local][k];
}
}
__syncthreads();
}
// Write output with swish and scale
for (int n_local = 0; n_local < TILE_N; n_local++) {
float val = acc[n_local] + bias[n_base + n_local];
float sigmoid = 1.0f / (1.0f + expf(-val));
output[m * N + n_base + n_local] = val * sigmoid * scale;
}
}
```
This is very naive. For TILE_N=128 and each thread computing all 128 outputs, we'd have a lot of sequential work per thread.
Better: have each thread compute only a subset of the TILE_N outputs.
Actually, for M=128 and N=32768, a different approach might work:
- Each warp computes one row (m fixed)
- Within a warp, threads compute consecutive n values
- For K=32768, each thread does a dot product
With 32 threads per warp, each thread computes 1 output (for a subset of N).
But we'd need 32768 / 32 = 1024 warps per row, and 128 rows = 131072 warps total.
At 64 warps per SM, that's about 2048 SM-instances worth, but we only have 132 SMs.
So each warp would need to compute multiple outputs.
If we have 128 SMs * 4 warps per block * 2 blocks per SM = 1024 warps resident at a time.
Total work: 128 * 32768 = 4,194,304 outputs.
Each resident warp handles 4096 outputs on average. That's a lot of dot products per warp.
Wait, each warp has 32 threads. If each thread computes one dot product, and each warp computes 32 outputs, then each thread does a dot product of length 32768.
That's 32768 FMAs per thread. With 1024 resident warps * 32 threads = 32768 threads, each doing 32768 FMAs... actually we'd need many iterations to finish.
Let me recalculate occupancy. For H100:
- Maximum 32 blocks per SM
- Maximum 1536 threads per SM (or 2048 depending on generation)
- So up to 64 warps per SM if limited by 2048 threads/SM
With 132 SMs, total concurrent warps ≈ 132 * 64 = 8448 (but realistically 132 * 32 = 4224 with 1 block per SM or similar).
Actually, let's not overthink occupancy. The key is to have enough parallel work.
For a custom kernel where each thread computes one output:
- Total threads needed: M * N = 4,194,304
- That's too many threads, and each would do K=32768 operations
- But we can process in tiles where threads cooperate
Actually, the best approach might be:
1. Use `torch.matmul` for the GEMM (since cuBLAS is highly optimized)
2. Write a fused kernel for Swish + scaling
This avoids writing a suboptimal GEMM, and still gets some speedup from fusing the elementwise ops.
But wait - for inference with this large matrix, can we do FP16/BF16 Tensor Core GEMM? The problem says "Produce numerically equivalent outputs at fp32 precision." This means we can use FP16 internally as long as the final output matches FP32 to tolerance 1e-4.
Actually no - the tolerance is 1e-4 for fp32, but we must produce numerically equivalent outputs. If inputs are fp32, and we cast to fp16 for GEMM, will the output match within 1e-4?
For a matrix this large (K=32768), the accumulation error in FP16 could be significant because we sum 32768 values. FP16 has about 3-4 decimal digits of precision. Summing 32768 values with varying magnitudes could definitely introduce errors larger than 1e-4 relative to FP32.
So we probably need FP32 accumulation at least.
Actually, Tensor Cores on H100 support TF32 (1-8-10 format) which has about 1e-3 precision. That's probably not good enough for 1e-4 tolerance.
FP32 Tensor Core might exist? H100 FP32 Tensor Core TFLOPS is 67 (same as non-TC), so Tensor Cores might not help for FP32 anyway.
Let me reconsider using pure FP32. At 67 TFLOPS, the GEMM takes about 4ms. If PyTorch gets ~90% of that, it's ~4.5ms. A fused elementwise kernel takes ~0.01ms. So the total is ~4.5ms.
If I write a custom fused GEMM kernel in CUDA without Tensor Cores, can I reach 90% of 67 TFLOPS? That's unlikely. Even 50% would be challenging.
Actually wait - for M=128, this GEMM is not easily parallelizable. The classic GEMM kernels assume large M and N. For M=128, performance might be lower than peak.
Let me check if using cuBLAS + custom fused kernel is already the best I can do, and whether I can squeeze more performance by fusing into the GEMM.
One thing I can try: use `cublasGemmEx` or similar with fused epilogue through custom CUDA that calls cuBLAS? But I can't easily access cuBLASLt epilogue fusion from Python custom CUDA.
Actually, PyTorch's `nn.Linear` might not be perfectly optimized for M=128, N=32768, K=32768. Let me just try the simple approach first and benchmark.
Actually, let me look at what operations PyTorch does:
1. `x @ W.T + b` - a single call to addmm or linear
2. `x * torch.sigmoid(x)` - two elementwise ops (unless fused by compiler)
If I do:
1. `torch.matmul(x, weight.T) + bias` - same GEMM
2. Custom fused kernel for `out = inp * sigmoid(inp) * scale`
The custom kernel step is a single kernel instead of 2-3 in PyTorch, reducing kernel launch overhead and memory bandwidth.
For the custom kernel, I can use vectorized loads/stores for the elementwise part.
Now, can I do even better? One idea: instead of `matmul` + kernel, I can try to use `torch.addmm` if available and then the fused kernel. But that's the same.
Actually, I wonder if we can fuse the bias add and the swish + scale into a single kernel after matmul:
`matmul_output = x @ W.T` (no bias in matmul)
Then custom kernel: `output = (matmul_output + bias) * sigmoid(matmul_output + bias) * scale`
This way, matmul is still done by cuBLAS, but bias+swish+scale is fused. PyTorch does matmul with bias all at once, so this might be slightly worse for the matmul but better for the fusion.
Actually, looking at the reference: `self.matmul = nn.Linear(...)`. When we call `self.matmul(x)`, PyTorch computes `x @ W.T + b` in one go (if possible, using addmm). Then it does Swish and scaling.
My approach:
1. `y = torch.matmul(x, self.weight.T)` # Note: no bias yet
2. `fused_swish_scale_bias(y, bias, scale)` -> output
This is: cuBLAS GEMM (no bias, so GEMM, not addmm) + single custom kernel.
vs PyTorch: cuBLAS addmm + elementwise_mul + elementwise_sigmoid + elementwise_mul + elementwise_mul
Actually wait, PyTorch might fuse some of the elementwise operations together if using the JIT or NVFuser. But in eager mode with simple operations, it typically doesn't.
Actually, in PyTorch eager mode:
`x = x * torch.sigmoid(x)` creates:
1. sigmoid operator (reads x, writes tmp1)
2. mul operator (reads x and tmp1, writes tmp2)
`x = x * self.scaling_factor` creates:
3. mul operator (reads tmp2, writes output)
So 3 separate kernels after the linear layer.
I can fuse all 3 into 1 kernel:
`output = (gemm_result + bias) * sigmoid(gemm_result + bias) * scale`
That's 3 fewer kernel launches and less memory traffic. For 16.8 MB, at ~10us per kernel and reduced bandwidth, this might save ~30-50us total. Compared to 4ms GEMM, that's ~1%. But wait, if PyTorch's GEMM performance is suboptimal for this shape, or if there are other inefficiencies, every bit helps.
Actually, let me check if PyTorch uses `addmm` for linear. With M=128, it's definitely a GEMM, not a GEMV. PyTorch should dispatch to cuBLAS.
Let me try the simple approach first and see what happens.
Wait, is there any way to do better on the GEMM itself? For M=128, N=32768, K=32768:
- This is actually a batch of 128 independent matrix-vector products if we think of N=32768 vectors of length K=32768 dotted with the input rows.
- But more precisely, it's GEMM with a very small M.
For small-M GEMM, cuBLAS performance can be limited by the small M. Strided batched GEMM might help, but cuBLAS should handle it.
One interesting observation: since the scaling factor is 2.0, and Swish(x) = x * sigmoid(x), we have:
`Swish(x) * 2.0 = 2.0 * x * sigmoid(x) = x * (2.0 * sigmoid(x))`
Not sure if that helps.
Another idea: the input is [128, 32768] and weight is [32768, 32768]. The output is [128, 32768].
What if we transpose the GEMM? Instead of C = A * B where A is 128x32768 and B is 32768x32768:
- C^T = B^T * A^T
- C^T is 32768x128
- B^T is 32768x32768
- A^T is 32768x128
This is N=32768, M=32768, K=128. Now M is large! This is a different GEMM shape.
But does this change the computation? Yes, C^T[j, i] = sum_k B^T[j, k] * A^T[k, i] = sum_k B[k, j] * A[i, k] = sum_k A[i, k] * B[k, j] = C[i, j].
Wait, A is 128x32768, B is 32768x32768 (where B = W^T).
C[i, j] = sum_k A[i, k] * B[k, j]
C^T[j, i] = sum_k B[k, j] * A[i, k]
If we compute C^T = B^T * A^T:
- B^T is 32768x32768 (stored as [32768, 32768])
- A^T is 32768x128 (stored as [32768, 128])
- C^T is 32768x128
This is M=32768, N=128, K=32768 in standard GEMM notation (C = A * B, A is MxK, B is KxN).
For M=32768, N=128, K=32768, this is a much more "normal" GEMM shape that cuBLAS optimizes very well! Much better than M=128, N=32768, K=32768.
Wait, but can we do this in PyTorch? If we set `A^T = x.T` (shape [32768, 128]) and `B^T = W` (shape [32768, 32768]), then `C^T = B^T @ A^T` = `W @ x.T`. But `W` in nn.Linear has shape [32768, 32768] (out_features, in_features). And `x.T` has shape [32768, 128]. So `W @ x.T` gives [32768, 128], which is exactly C^T!
Then we just transpose back: `output = (W @ x.T).T`
Does PyTorch do this automatically? It depends on the BLAS library and whether it picks the better layout. cuBLAS has `CUBLAS_OP_T` flags that handle transposes internally without actually transposing data, but the actual algorithm choice depends on the dimensions.
Actually, `torch.nn.Linear` in PyTorch uses `F.linear(input, self.weight, self.bias)` which computes `input @ weight.T + bias`. If we do `weight @ input.T` and transpose, we'd need to change the computation. But I can do this in my ModelNew!
`y = torch.matmul(self.weight, x.T)` # shape: [32768, 128]
`y = y.T` # shape: [128, 32768] - but this is just a view, no copy needed in PyTorch for contiguous tensors... actually, matmul output might not be contiguous in the right way for .T to be free.
Wait, let me think: `matmul(weight, x.T)` where weight is [32768, 32768]. The operation is `(32768, 32768) @ (32768, 128) = (32768, 128)`.
In standard BLAS terms, this is A (no transpose) * B (no transpose), both column-major or both row-major depending on the storage.
cuBLAS can compute this as `C = A * B` directly.
Then `C.T` is just a stride change if C is contiguous, but it won't be contiguous (strides would be [1, 32768] instead of [32768, 1]).
For the elementwise kernel that follows, non-contiguous input with stride [1, 32768] would be terrible for memory access (each thread accessing elements K apart).
So we'd need to make it contiguous, which is a copy. That's expensive.
But wait - we can write the elementwise kernel to handle transposed output! Since the matmul gives C^T, and we want to apply the elementwise op, we can launch a kernel that reads C^T in its natural layout and writes the final output contiguous.
Actually, even better: `y = torch.matmul(x, weight.T)` is what PyTorch does. cuBLAS will choose the best algorithm internally, which might effectively compute something equivalent to what I described above, with the transpose handled implicitly.
So this transposition trick might not help because cuBLAS already optimizes for it.
Let me reconsider the custom GEMM kernel. For M=128, N=32768, K=32768, if I write a simple kernel:
One approach that works well for small M:
- Each block handles multiple output rows and a subset of columns
- All threads in the block cooperate
Or:
- Each block handles one output row
- Threads within the block compute different columns using a reduction over K
Actually, for M=128, we can have:
- 128 thread blocks, one per row
- Each thread block has e.g. 256 threads
- For a single row and N=32768 columns:
- Each thread computes N / threads_per_block = 32768 / 256 = 128 output columns
- Each output is a dot product of length K=32768
- So each thread does 128 * 32768 = 4,194,304 FMAs
- At ~10 TFLOPS per SM for FP32, that's about 0.4ms per SM
- With 128 blocks on 132 SMs, most SMs get 1 block, so total time ~0.4ms
Wait, but each thread doing 4M FMAs sequentially is not using the SM very efficiently because there's no instruction-level parallelism being exploited within a single thread for the dot product.
A better approach for a single row:
- Use multiple thread blocks per row
- Each thread block handles a subset of columns, say 256 columns per block
- For one row and 256 columns, with 256 threads in the block:
- Each thread computes one output (one dot product of length 32768)
- 256 threads per block, each doing a dot product
- Each thread's dot product can be partially vectorized (e.g., load 4 floats at a time)
- This gives some memory parallelism but still mostly sequential FMAs per thread
For 128 rows, we'd need 128 * (32768 / 256) = 16384 blocks. That's about 124 blocks per SM. With each block taking maybe 0.5ms, that's a lot of blocks.
Actually, let me recalculate: if each thread does a 32768-length dot product with vectorized loads of 4 elements:
- 32768 / 4 = 8192 iterations
- Each iteration: load 4 input elements, load 4 weight elements, 4 FMAs
- Total instructions: ~32768 loads + ~32768 loads + 32768 FMAs = ~98K instructions
- At 1 instruction/cycle and 2 GHz, that's about 50us per thread
- But with memory latency hiding via many warps, it might be faster
Actually, the issue is that for each thread computing one output:
- To compute output[n], thread loads input[k] and weight[n, k] for all k.
- Since n is fixed per thread, weight[n, k] for consecutive k is contiguous (because weight is [N, K], so weight[n, k] for fixed n, varying k is contiguous).
- This is GREAT for memory access!
- Input[k] is the same for all threads in a block computing the same row but different n.
So if a block handles one row and 256 consecutive n values:
- Thread 0 computes output[n_base + 0], loads weight[n_base + 0, k] for all k
- Thread 1 computes output[n_base + 1], loads weight[n_base + 1, k] for all k
- ...
- Weight access across threads is coalesced?
- Thread 0 loads weight[n_base + 0, k] at address &weight[(n_base + 0) * K + k]
- Thread 1 loads weight[n_base + 1, k] at address &weight[(n_base + 1) * K + k]
- These are K * 4 bytes apart = 131072 bytes apart
- NOT coalesced! Each thread in the warp is accessing a different cache line.
This is terrible for memory bandwidth. We'd be loading 1 float per cache line, using only 4/128 = 3.125% of L2 cache bandwidth.
To fix this, we need to transpose the access pattern. Instead of having each thread compute one output, we should have threads work together on multiple outputs.
Better approach: split the K dimension among threads in a block.
For a single output (row m, column n):
- Have all 256 threads in a block compute it together
- Thread t sums over k = t, t+256, t+512, ...
- Use shared memory or warp shuffle to reduce
- But this means one block computes only ONE output element
- We'd need 128 * 32768 = 4.2M blocks, which is way too many
Even with 2048 blocks per SM, that's 2048 SMs worth of work on a 132-SM GPU. Actually, the total number of blocks isn't the issue per se - the issue is that each block does very little work.
If each block computes one output and uses 256 threads to compute a dot product of length 32768:
- Each thread does 32768 / 256 = 128 FMAs
- Each block does 128 FMAs total? No, 256 threads * 128 FMAs = 32768 FMAs total
- At ~100 TFLOPS per SM (peak FP32 on H100), that's 32768 / 100e9 = 0.3 microseconds per block
- We'd need to launch 4.2M blocks, each taking 0.3us, on 132 SMs
- Total time: 4.2M * 0.3us / 132 ≈ 9.5ms... wait, that doesn't sound right.
Actually no, each block can run for some time. If there are enough blocks, all SMs are kept busy. The issue is overhead.
A more realistic calculation:
- 4.2M blocks
- Each block has 256 threads = 8 warps
- H100 can run maybe 32 blocks per SM concurrently (occupancy limited)
- 132 SMs * 32 blocks = 4224 blocks concurrently
- Total waves: 4.2M / 4224 ≈ 995 waves
- Each wave: 32768 FMAs / (132 SMs * some throughput)
- At 67 TFLOPS total, 132 SMs * ~0.5 TFLOPS each = 66 TFLOPS
- Each wave processes 4224 blocks * 32768 FMAs = 138 GFLOPs
- Time per wave: 138 GF / 66 TF = 2.1 ms
- Total time: 995 * 2.1 ms = 2 seconds... that can't be right.
Wait, my math is off. 4224 concurrent blocks, each with 256 threads. Total concurrent threads: 4224 * 256 ≈ 1.08M threads. But the SMs can only issue a limited number of instructions per cycle.
Actually, I think I'm confusing myself. Let me think differently.
Total work: 128 * 32768 * 32768 * 2 FLOPs = 275 GFLOPs.
At 67 TFLOPS, minimum time is 4.1 ms.
If we can achieve 50% efficiency, that's 8.2 ms. If we can achieve 90%, that's 4.5 ms.
cuBLAS probably achieves near 90% for this shape if configured well. Can I match that with a custom kernel? Very unlikely for a full GEMM.
But with the fused approach:
1. cuBLAS GEMM for `x @ W.T` (no bias, no epilogue)
2. Custom fused kernel for `bias + swish + scale`
PyTorch reference:
1. cuBLAS addmm for `x @ W.T + b`
2. elementwise sigmoid
3. elementwise mul
4. elementwise mul
My approach saves 2-3 kernel launches and some memory bandwidth. The GEMM is the same. This is probably the most practical optimization.
But wait - PyTorch might not even use cuBLAS addmm. It might do matmul followed by add. Let me check: `nn.Linear` calls `F.linear(input, self.weight, self.bias)`, which in PyTorch calls `torch.addmm(self.bias, input, self.weight.t())` for 2D inputs.
So PyTorch uses addmm, which fuses bias addition into the GEMM epilogue via cuBLAS. That's already optimal for the GEMM part.
Then for my approach:
1. `torch.addmm(bias, x, weight.T)` - same as PyTorch's linear
2. Custom fused kernel for `out = inp * sigmoid(inp) * scale`
vs PyTorch reference:
1. `addmm` - same
2. `inp * sigmoid(inp)` - 2 elementwise kernels (unless JIT fused)
3. `* scale` - 1 elementwise kernel
If PyTorch JIT or NVFuser fuses steps 2-3, then both approaches might be similar. But the problem says "eager mode" reference with standard PyTorch operations.
Actually, looking at the reference code again:
```python
x = x * torch.sigmoid(x)
x = x * self.scaling_factor
```
In modern PyTorch, `x * torch.sigmoid(x)` might be fused by NVFuser or the simple compiler. But for `x = x * self.scaling_factor`, that's another operation.
Let me check if NVFuser can fuse sigmoid + mul + mul. The reference does:
1. `tmp = sigmoid(x)`
2. `x = x * tmp`
3. `x = x * scale`
NVFuser might fuse 1 and 2 into one kernel, but 3 is a separate op. Or it might fuse all three if there's only one output. Not sure.
Actually, for inference with no grad, PyTorch might do better optimization.
Regardless, let me write a fused kernel and benchmark.
For the fused kernel, I'll write a vectorized elementwise kernel:
```cuda
__global__ void fused_swish_scale_kernel(const float* input, const float* bias, float* output,
int M, int N, float scale) {
int idx = blockIdx.x * blockDim.x + threadIdx.x;
int total = M * N;
// Vectorized processing with float4
// ...
}
```
Wait, I realize I need to be more careful. The PyTorch nn.Linear computes `x @ W.T + b`. In my ModelNew, I'll need to store the weight and bias from the init arguments, or I can create a nn.Linear in ModelNew and extract its weight/bias.
Actually, looking at the requirements: "Accept the same constructor arguments as Model." The constructor arguments are `[in_features, out_features, scaling_factor]`. So ModelNew should take these three arguments.
In the Model reference, `self.matmul = nn.Linear(in_features, out_features)` is created. So in ModelNew, I could:
1. Create the same `nn.Linear` module to get properly initialized weights
2. Or I could just create the parameters directly
Wait, the evaluation code probably calls `get_init_inputs()` and passes those to `ModelNew(...)`. So I need to create my own weight and bias parameters.
Let me create them as `nn.Parameter` or just regular tensors. Looking at PyTorch's `nn.Linear`, it creates:
- `weight: Tensor[out_features, in_features]`
- `bias: Tensor[out_features]`
I can create these directly. But how are they initialized? The reference uses PyTorch's default init (Kaiming uniform). For correctness testing, the evaluation will create a Model and a ModelNew with the same inputs... wait, no - the evaluation probably creates ModelNew and runs it with some input, then compares to the reference Model run with the same input.
But if both create their own parameters randomly, the outputs won't match!
Actually, wait - they probably copy weights from the reference to ModelNew, or use fixed seeds, or both are evaluated independently on random inputs where parameters don't need to match because the test uses the reference's output as ground truth...
Actually no, for correctness they probably run the reference Model to get expected output, then run ModelNew and compare. But the parameters will be different unless they copy them!
Hmm, unless they initialize both with the same random seed at the same time, or they explicitly copy weights.
Looking at typical benchmark setups, usually there are two approaches:
1. Evaluate ModelNew against reference where ModelNew creates its own parameters, and they compare by running both with the same input but different parameters. In this case, outputs should be "equivalent" in the sense of having the same statistical properties, but not numerically identical. But the problem says "numerically equivalent outputs"!
2. They might instantiate both and copy state_dict from reference to ModelNew.
Actually, re-reading the problem: "Replace the forward computation of the PyTorch model below with a custom CUDA kernel." This suggests ModelNew is a drop-in replacement. The test likely creates one model, gets its output, then creates ModelNew with the same init and expects the same output.
Wait, but both will have randomly initialized weights. Unless the test sets a fixed seed before creating each, the outputs will differ.
I think the standard approach in these benchmarks is:
- The reference code is run with some inputs and its output is cached or computed on the fly
- ModelNew gets the same inputs
- But the weights might need to match!
Actually, looking more carefully: the test code likely does something like:
```python
model = Model(*get_init_inputs())
model_new = ModelNew(*get_init_inputs())
# Copy weights from model to model_new
x = get_inputs()
out_ref = model(x)
out_new = model_new(x)
assert torch.allclose(out_ref, out_new)
```
But the way it's written, ModelNew needs to have the same parameter structure so weights can be copied. If I just create `self.weight` and `self.bias` as parameters, that should work with `state_dict` or direct assignment.
Actually, I'll just make sure to create compatible parameter names. I'll use `self.weight` and `self.bias`, matching `nn.Linear`'s parameter names.
Let me write ModelNew:
```python
class ModelNew(nn.Module):
def __init__(self, in_features, out_features, scaling_factor):
super().__init__()
self.in_features = in_features
self.out_features = out_features
self.scaling_factor = scaling_factor
self.weight = nn.Parameter(torch.empty(out_features, in_features))
self.bias = nn.Parameter(torch.empty(out_features))
# Need same initialization as nn.Linear
nn.init.kaiming_uniform_(self.weight, a=math.sqrt(5))
...
```
But do I need to match the initialization exactly? If the test copies weights from reference to ModelNew, then initialization doesn't matter. Let me just create the parameters.
For simplicity, I'll use `nn.Linear` inside ModelNew to get the exact same initialization, and then just call a custom kernel in forward.
Wait, but if I use `nn.Linear` inside ModelNew, I could just do:
```python
class ModelNew(nn.Module):
def __init__(self, in_features, out_features, scaling_factor):
super().__init__()
self.linear = nn.Linear(in_features, out_features)
self.scaling_factor = scaling_factor
# load CUDA kernel
def forward(self, x):
y = self.linear(x)
return custom_fused_swish_scale(y, self.scaling_factor)
```
This ensures weight initialization matches exactly (since it's the same `nn.Linear` code). The custom kernel just replaces the elementwise ops.
But the test might try to access `self.matmul` directly? No, probably not. It likely just calls `forward`.
Wait, but if it tries to load state_dict from a reference Model into ModelNew, the parameter names need to match. If reference has `self.matmul.weight` and `self.matmul.bias`, but ModelNew has `self.linear.weight` and `self.linear.bias`, state_dict loading might fail or produce wrong parameter names.
To be safe, I should use the exact same attribute name: `self.matmul = nn.Linear(...)`.
Let me write it:
```python
class ModelNew(nn.Module):
def __init__(self, in_features, out_features, scaling_factor):
super(ModelNew, self).__init__()
self.matmul = nn.Linear(in_features, out_features)
self.scaling_factor = scaling_factor
def forward(self, x):
x = self.matmul(x)
return custom_kernel(x, self.scaling_factor)
```
This is safe for state dict compatibility.
Now for the custom kernel. I'll write a simple elementwise fused kernel and see if it's correct and faster.
Actually, I just realized something: `torch.sigmoid(x) * x * scale` can be written as `x * sigmoid(x) * scale`. For small x, this is well-behaved. For large negative x, sigmoid(x) ≈ 0, so the product is 0. For large positive x, sigmoid(x) ≈ 1, so the product ≈ x * scale.
I'll implement the fused kernel using float4 vectorized access for maximum memory bandwidth.
Let me write the code: