We need to write a custom CUDA kernel to replace the forward computation of an HMM log-likelihood (log-space forward algorithm). The reference model is a simple PyTorch module with parameters log_init, log_trans, log_emit. It does: - Normalize logs (subtract logsumexp). - B, T = observations.shape - log_B[:, observations[:, 0]].t() -> (B, S) - log_alpha = log_pi.unsqueeze(0) + emit_0 - for t in 1..T-1: log_alpha = log_alpha.unsqueeze(-1) + log_A.unsqueeze(0) # (B, S, S) log_alpha = torch.logsumexp(log_alpha, dim=1) # (B, S) emit_t = log_B[:, observations[:, t]].t() log_alpha = log_alpha + emit_t - log_lik = torch.logsumexp(log_alpha, dim=-1) We need a custom CUDA kernel to compute this faster. The dimensions: S=16, T=128, B=64, O=64. This is small. A per-sequence or per-batch element kernel doing the forward recursion in shared memory should be fast. Since S=16 is small, we can process each batch element with a warp or thread block. For each t, we need: new_alpha[j] = logsumexp_i(old_alpha[i] + A[i, j]) + B[j, obs_t] We have B batch elements, T time steps. We can have each block handle one batch element (or multiple). Since S=16, each thread block can have e.g., 16 threads (one per state). But logsumexp over i requires reduction across i. Actually new_alpha[j] = logsumexp_i(old_alpha[i] + A[i, j]) is a matrix-vector operation: A is SxS, old_alpha is length S. We can compute for each j, sum_i exp(old_alpha[i] + A[i][j]), but in log-space we do logsumexp. Since S=16, we can have each thread compute one j, and within each thread compute a loop over i=0..S-1 of old_alpha[i] + A[i][j], track max then compute logsumexp. That avoids inter-thread communication. That is efficient for small S. So a block of 16 threads, each handles one state j. Or maybe 32 threads with some idle. Since the batch size is 64, we can launch 64 blocks of 16 threads. Or maybe 64 blocks of 32 threads for better occupancy. But each warp of 32 threads will have 16 active threads; that is fine. In fact, occupancy might be limited by register usage or shared memory. If each block uses only a few registers and no shared memory, we can get high occupancy. But we need to load parameters and observation indices. However, to maximize throughput, we could also process multiple time steps per thread to amortize overhead, or fuse the T loop inside the kernel. Since T=128, we can write a kernel where each block processes one batch element (b) and iterates over t=0..T-1. At each t, each thread accumulates logsumexp over i for its j. Pseudo-code: __global__ void hmm_forward( const int* observations, // (B, T) const float* log_init, // (S,) const float* log_trans, // (S, S) const float* log_emit, // (S, O) float* log_lik, // (B,) int B, int T, int S, int O) { int b = blockIdx.x; if (b >= B) return; // load observation indices for this batch into shared memory or registers // T=128, so we can store obs indices in shared memory: 128 ints = 512 bytes. __shared__ int obs_shared[128]; // But if T varies, can be dynamic. Here T is fixed param passed to kernel? Actually in reference it's dynamic. But we can have T up to maybe 128. We can allocate dynamic shared memory or make it a template parameter. // For simplicity, we can read observations[b*T + t] each step from global memory. Since T=128, it's fine. // We can keep alpha in registers: each thread holds one alpha value (for its j). // Since blockDim.x = S, threadIdx.x = j. int j = threadIdx.x; float alpha; // t=0 int obs0 = observations[b * T + 0]; // log_pi = log_init[j] - logsumexp_k(log_init[k])? Actually normalized in python. // Wait, reference does _normalize_log on parameters before using them. // So log_init already normalized. We can pre-normalize on CPU or in kernel. Since parameters are passed as tensors, we can compute normalized versions in model.forward before calling kernel. But speedup requirement includes everything in forward. Could normalize with PyTorch then launch kernel. That's fine. // Actually, to avoid extra overhead, we can do normalization in Python with PyTorch as reference does, then call kernel with normalized parameters. // Let's examine: log_pi = self._normalize_log(self.log_init) etc. The reference does this every forward. So our ModelNew should do the same, or incorporate normalization in kernel. Doing it in PyTorch is fine because it's O(S^2 + S*O) negligible. alpha = log_init[j] + log_emit[j * O + obs0]; for (int t = 1; t < T; ++t) { int ob = observations[b * T + t]; // compute new_alpha_j = logsumexp_i(alpha_i + A[i*S + j]) float max_val = -1e20f; #pragma unroll for (int i = 0; i < S; ++i) { float val = alpha_i? But each thread only has its own old alpha? Wait, we need alpha_i for all i. // We can store alpha values in shared memory or do a shuffle. // Since S=16, we can have each thread keep its alpha in a register, and broadcast to other threads via __shfl_sync. } } Using __shfl_sync: Since S=16 <= warp size 32, all threads of a warp are in the same warp if blockDim.x <= 32. If we set blockDim.x = 32, then we have two warps? Actually if blockDim.x = 32 and S=16, the first 16 threads are active, the rest are idle. But __shfl_sync works within a warp, so we need to ensure all participating threads are in same warp. If blockDim.x = 32, threads 0..15 are in warp 0, threads 16..31 are in warp 1. If we only use threads 0..15, they are all in warp 0, so shfl is fine. But then half the threads in the block are idle. However, occupancy / throughput might still be okay. Alternatively, use blockDim.x = 16, which is half a warp. Then the warp has 16 active threads out of 32, but the other 16 are inactive. This reduces theoretical max occupancy because each block uses one warp, but only 16 threads active. Yet, the GPU can schedule warps, and the inactive threads within the warp don't consume execution resources beyond registers. Wait, if blockDim.x=16, the warp still has 32 threads, but only 16 are active; the hardware executes the warp with 16 active lanes. That's fine. Shfl requires mask of active threads. So using blockDim.x=S=16 is natural. But shfl only works within a warp. Since blockDim.x=16, there is only one warp per block. All threads are in the same warp. So we can do: float old_alpha = __shfl_sync(0xFFFFFFFF, alpha, i); // but mask should include all active lanes. Actually if blockDim.x=16, the warp has 16 active lanes (0..15). To be safe, use full mask 0xFFFFFFFF; the hardware will ignore inactive lanes. Or use __ballot to compute mask. But note: if blockDim.x=16, the warp is partially occupied. The __shfl_sync with mask 0xFFFFFFFF is okay as long as all threads in the mask are convergent and active. Since threads 16..31 are inactive (not launched), the mask still includes them but they are not executing. The shfl sync will just wait for the active threads. Hmm, actually CUDA says for __shfl_sync, the mask must include all threads that are source or destination. In a block of 16 threads, only threads 0..15 exist. The warp still has 32 logical threads, but 16..31 are not part of the block? Wait, in CUDA, a block with 16 threads occupies one warp. The warp has 32 lanes; lanes 0..15 map to the threads, lanes 16..31 are inactive. The __shfl_sync mask should have bits set for the participating threads. Typically, you use __activemask() or simply 0xFFFFFFFF for full warp, but if some lanes are inactive (because block doesn't cover them), does __shfl_sync hang? I think __shfl_sync with mask including inactive lanes is fine because inactive lanes are not waiting. But I recall issues: the mask should be a subset of active threads. Actually, the mask argument is used to specify which threads are participating in the collective operation. Threads not in the mask are not required to execute the instruction; but if they do, they can pass any value. The hardware may require all active threads in the warp to execute the instruction or none. Since threads 16..31 are not active, they are not executing it, which is okay. To be safe, we can put alpha values into shared memory of size S. That is simple and safe. So per block: __shared__ float sm_alpha[S]; // old alpha values Each thread writes its alpha to sm_alpha[j]. __syncthreads(); Then each thread reads sm_alpha[i] in a loop. But __syncthreads() inside a loop for each t may be a bit heavy but acceptable for T=128 and S=16. However, we can avoid syncthreads by having each thread compute the logsumexp using register values and shfl. But let's keep it simple with shared memory first. Since S=16, syncthreads cost is low. Alternatively, we can have each thread compute all j values, so no need for communication. Each thread computes the entire new alpha vector of length S. Then each block of e.g., 64 threads processes one batch element, with each thread handling one j. That's same as before. But if each thread handles multiple js or if we have more threads than S, we get underutilization. Alternatively, we could process multiple batch elements per block. For instance, each warp of 32 threads handles 2 batch elements (16 threads each). But that complicates. Given small dimensions, the simplest approach is fine. But to get speedup over PyTorch, PyTorch likely creates many intermediate tensors and uses generic logsumexp kernels, which for small S and B might be overhead-dominated. A custom kernel that avoids memory allocations and fuses everything should be much faster. Let's design the kernel: - Launch B blocks (or more if needed). Each block has S threads (or 32). We'll try S threads per block. - Each block reads observations for its batch index into shared memory or reads on the fly. - Each thread maintains its alpha_j in a register. - For t=1..T-1: - write alpha to shared mem (size S). - __syncthreads(); - read all alpha_i, compute max over i of alpha_i + A[i][j]. - compute logsumexp = max + log(sum_i exp(alpha_i + A[i][j] - max)). - read observation obs_t. - alpha_j = logsumexp + B[j][obs_t]. - After loop, write alpha_j to shared mem. - __syncthreads(); - compute logsumexp over j of alpha_j. - thread 0 stores result in log_lik[b]. We need to handle normalization of parameters. As in reference, we can normalize in Python using PyTorch before passing to kernel. The cost is tiny. Potential optimization: Precompute log_emit for the specific observation sequence? Since observations vary per batch and per t, we cannot precompute all. But inside the loop, each thread needs B[j][obs_t]. obs_t is same for all threads in a block (since it's the observation for batch b at time t). So we could load the emission vector for that observation into shared memory: emit_j = log_emit[j*O + obs_t]. Since obs_t is the same for all threads in the block, each thread can just read global memory: log_emit[j * O + obs_t]. Since O=64 and S=16, the access pattern is strided: thread j reads index j*O + obs_t. That's a strided memory access with stride O=64 floats = 256 bytes. Depending on cache, this might be okay, but coalescing is poor. Alternatively, we can load the entire column (for obs_t) into shared memory: one thread (or multiple) loads log_emit[i * O + obs_t] for i=0..S-1. Then each thread reads from sm. Because O*S is small, we can also store log_emit transposed (O, S) on the CPU side to get coalesced access. Or in kernel, we can have thread j load log_emit[j*O + obs_t]. Since j is threadIdx.x, addresses are j*O + obs_t. O=64, so addresses are spaced by 64*4=256 bytes. In a warp of 32 threads, that is 256 bytes apart. Global memory accesses will be separate cache lines (each 128 bytes). So 16 cache lines for 16 threads. That's not great but not terrible given small data. Better: transpose log_emit to shape (O, S) so that thread j reads index obs_t * S + j. That's contiguous: coalesced. Similarly, log_trans is (S, S). Thread j reads A[i][j] across i. If A is stored row-major (S, S), then for fixed i, A[i*S + j] is contiguous across j. Good. If we need to read A[i][j] for a given j across i, then accesses are strided by S. But each thread loops over i, reading A[i*S + j] for its own j. That's strided by S floats = 64 bytes. Again, each access per i is a separate cache line if S=16 (64 bytes, which is half a cache line). Actually, if S=16, each row is 64 bytes. So A[i*S + j] for a fixed j and varying i is separated by 64 bytes. That's exactly one cache line. So for a single thread accessing down a column, it will hit the same cache line as other threads? Let's see: thread j reads A[0*S+j] (address base + j*4), thread j+1 reads A[0*S+j+1] (base + (j+1)*4). These are contiguous, so the warp loads the first row coalesced. Then each thread loops i=1: reads A[1*S+j], which is base + S*4 + j*4. Again contiguous across j. So within each i iteration, the warp reads a contiguous row of A. That's coalesced! Good. So log_trans in (S, S) row-major is fine. For log_emit, if we store as (S, O) row-major, thread j reads log_emit[j*O + obs_t]. This is not coalesced across j (stride O). But if we transpose to (O, S), then thread j reads log_emit[obs_t*S + j], which is coalesced. So we can pass log_emit_t = log_emit.t().contiguous() from Python. Then in kernel, log_emit is shape (O, S). However, note that reference uses log_B[:, observations[:, t]].t() which yields (B, S) where B is batch. This is essentially gathering columns and transposing. So passing log_emit_T (O, S) and indexing by obs_t is natural. Similarly, log_init is just a vector of length S. For log_trans, reference uses log_A (S, S). The operation log_alpha.unsqueeze(-1) + log_A.unsqueeze(0) is (B, S, S) and then logsumexp over dim=1. The custom kernel computes for each j: logsumexp_i(old_alpha[i] + A[i][j]). If A is row-major (i, j), then thread j loops i and accesses A[i*S + j]. As we reasoned, each iteration i, the warp reads a contiguous row. Great. So the plan: In ModelNew.forward: - B, T = observations.shape - S = self.num_states - O = self.num_obs - Normalize parameters with _normalize_log (Python/PyTorch). - log_trans = normalized log_trans (S, S) contiguous. - log_emit_T = normalized log_emit.t().contiguous() (O, S). - log_init = normalized log_init (S,). - Allocate output log_lik = torch.empty(B, device=observations.device, dtype=torch.float32) - Launch kernel: blocks = B threads = S (or maybe next power of two, e.g., 16) func(observations.contiguous(), log_init, log_trans, log_emit_T, log_lik, B, T, S, O) Kernel code: extern "C" __global__ void hmm_forward_kernel( const int* __restrict__ obs, // (B, T) const float* __restrict__ log_init, // (S,) const float* __restrict__ log_trans,// (S, S) const float* __restrict__ log_emit, // (O, S) float* __restrict__ log_lik, int B, int T, int S, int O) { int b = blockIdx.x; if (b >= B) return; int j = threadIdx.x; if (j >= S) return; // if threads > S // load first observation int ob = obs[b * T + 0]; float alpha = log_init[j] + log_emit[ob * S + j]; // shared memory for old alpha (size S) extern __shared__ float smem[]; // dynamic shared mem? Or static. // We'll allocate dynamic shared mem of size S * sizeof(float) per block. // But we can also use static: __shared__ float sm_alpha[16]; but S is variable. However, maximum S might be known? The problem says num_states=16 but it's a parameter. We can allocate dynamic shared mem based on S. // Let's use dynamic shared memory: load_inline passes shared_mem=... we can set shared_mem=0 and allocate per block? Actually we can set shared_mem argument in load_inline? Wait, load_inline's cuda_sources just contains the kernel; the launch from Python can specify shared_mem_bytes. We'll pass shared_mem=0 if we use static size [64] or similar. Since S is small (<64), we can declare __shared__ float sm_alpha[64]; and only use first S. // But we need to compile with a fixed size. Since maximum S may be unknown but in reference it's 16. In general, we could allocate dynamic shared mem in the launch. In Python, when calling the CUDA extension function, we can specify shared_mem=... Actually the generated pybind wrapper for a raw kernel usually accepts blocks, threads, args..., and optional shared memory size. But it's easier to use a static max size. // Let's declare __shared__ float sm_alpha[64]; // enough for S up to 64. Then we index sm_alpha[j]. // To be safe for any S up to maybe 128, we can use dynamic shared memory: extern __shared__ float smem[]; and in Python when launching kernel, we pass shared_mem=S*sizeof(float). The wrapper generated by load_inline for a function declared as __global__ might accept an optional integer for shared memory? Actually, for raw kernels, the typical binding is: func(args, blocks, threads, shared_mem=0, stream=None). The load_inline auto-generated wrapper from pybind11 might not expose shared_mem directly if we don't define it, but I'm not 100% sure. In PyTorch's cpp_extension.load_inline, if we define a function with signature that includes the kernel launch config? Actually, we define a host wrapper function in C++ that launches the kernel. That gives us full control. Then we list that host wrapper in functions=[...]. Good. // Let's write a host wrapper in C++ that launches the kernel with dynamic shared memory. } Yes, we should write a host wrapper function in the C++ source (the "cpp_sources" or we can put it all in cuda_sources). Since we are using load_inline, we can provide both cpp_sources and cuda_sources, or just cuda_sources. The easiest is to write a .cu file via cuda_sources that contains both the kernel and a host wrapper function. Then functions=['hmm_forward_cuda']. The host wrapper: void hmm_forward_cuda( torch::Tensor observations, torch::Tensor log_init, torch::Tensor log_trans, torch::Tensor log_emit, // (O, S) torch::Tensor log_lik, int B, int T, int S, int O) { dim3 blocks(B); dim3 threads(S); // or maybe 16 size_t shared_mem_size = S * sizeof(float); hmm_forward_kernel<<<blocks, threads, shared_mem_size>>>( observations.data_ptr<int>(), log_init.data_ptr<float>(), log_trans.data_ptr<float>(), log_emit.data_ptr<float>(), log_lik.data_ptr<float>(), B, T, S, O); } But we need to be careful: if S < 32, using threads=S causes partial warp occupancy, but that's okay. To ensure all threads in a block are in a single warp (if S <= 32), we could round up threads to 32, but then we need to guard j >= S. With S=16, using 16 threads per block means one warp with 16 active lanes. That's okay. The kernel will have divergence only at the if (j >= S) which is uniform false for the first 16 lanes, but the other 16 lanes are inactive anyway. So it's fine. But wait: if we allocate shared memory size = S * sizeof(float), and block has S threads, that's fine. But if we round threads up to 32 for occupancy, shared memory still S*size, but we need to handle threads j>=S returning. Let's just use threads = S for simplicity. We also need to think about occupancy: B=64 blocks. If each block has 1 warp, that's 64 warps. An SM can hold many warps. On an A100, each SM can have up to 2048 threads = 64 warps. With 108 SMs, total capacity is huge. So occupancy is not an issue. Now, inside the kernel, we need to compute logsumexp efficiently. Specifically: sm_alpha[j] = alpha; __syncthreads(); float max_val = -1e20f; #pragma unroll for (int i = 0; i < S; ++i) { float val = sm_alpha[i] + log_trans[i * S + j]; if (val > max_val) max_val = val; } float sum = 0.0f; #pragma unroll for (int i = 0; i < S; ++i) { float val = sm_alpha[i] + log_trans[i * S + j]; sum += expf(val - max_val); } float logsumexp = max_val + logf(sum); int ob_t = obs[b * T + t]; alpha = logsumexp + log_emit[ob_t * S + j]; We need to be careful with numeric stability: use a large negative max_val. The log_init etc are normalized, but alpha can become very negative. Using -1e20f is fine, but maybe use -INFINITY macro. In CUDA, we can do max_val = -CUDART_INF_F; but we need to include <math_constants.h>? Actually we can just use -1e30f or -CUDART_INF_F. Let's use -1e30f to avoid includes. Since log probabilities are finite, that's safe. Unroll loops: Since S may be a run-time variable but small (16), the compiler may not unroll automatically. We can use #pragma unroll, but if S is not a compile-time constant, it may not unroll. However, S is passed as an argument, not a template parameter. We could make it a template parameter, but we'd need to instantiate for various S. Simpler: just pass S as argument and rely on compiler to unroll if it can see it's constant? Actually the compiler will see S as a kernel parameter (loaded from constant memory or register), but it's not a compile-time constant. However, the loop bound S is known at kernel launch, but the compiler compiles the kernel for generic S. Loop unrolling is less likely. But the loop is small (16 iterations), so overhead is negligible. But we can optimize by making S a template parameter and instantiating for common sizes (e.g., 16). Then loops will be fully unrolled. Since the reference uses num_states=16, we know the exact size. But the class accepts num_states as argument, so it could vary. In general, to handle arbitrary S, we can still write a generic loop. However, for maximum speed, we can write a templated kernel and instantiate for S=16. Or we can branch on S==16 at runtime. But unnecessary complexity. Let's write a generic kernel with loops. That should be fast enough. Wait: PyTorch's overhead might be larger than kernel time. A generic kernel with loops of 16 is tiny. So even without unrolling, the kernel time will be dominated by memory loads. Memory loads: Each block reads: - log_init[j] once. - For each t: - obs[b*T + t] once per block (all threads could load it, but we can have thread 0 load and broadcast via shared memory, or rely on L1 cache. Since all threads in block load same address, it will be a broadcast load, very fast. Actually, each thread does int ob_t = obs[b*T + t]; This is a uniform address across the warp, so the hardware broadcasts it. That's efficient. - log_emit[ob_t * S + j] once per thread per t. This is coalesced across j. So each warp loads one contiguous segment of S floats. Good. - log_trans[i * S + j] for each i loop. For each i, the warp loads row i contiguously. Coalesced. - sm_alpha[i] from shared memory, no bank conflicts if S <= 32? Actually, each thread reads sm_alpha[i] in a loop; i varies per iteration, same for all threads. So at iteration i, all threads read the same shared memory location. Broadcast, no bank conflict. Therefore, memory bandwidth is well utilized. Arithmetic intensity is low (few ops per load), so kernel is likely memory-bound. Since total memory per block is small and fits in L1/L2 cache, it should be very fast. We need to compute the final logsumexp across j. We can do this with a shared memory reduction: After the loop, store alpha into sm_alpha[j]. __syncthreads(). Then do a tree reduction in shared memory. Or, since S=16, have thread 0 compute a loop: if (j == 0) { float max_val = -1e30f; for (int i = 0; i < S; ++i) { if (sm_alpha[i] > max_val) max_val = sm_alpha[i]; } float sum = 0.0f; for (int i = 0; i < S; ++i) { sum += expf(sm_alpha[i] - max_val); } log_lik[b] = max_val + logf(sum); } That's fine. Alternatively, we can use warp shuffle to compute the logsumexp across j. But shared memory is fine. One more thing: The reference uses torch.logsumexp, which is numerically stable. Our manual implementation must also be stable. We are doing max + log(sum exp(x - max)). That's stable. Now, about the data types: Observations are long tensor (int64). In kernel, we can cast to int. Let's make observations tensor of type int (torch.int32) for convenience: observations = observations.to(torch.int32). Or we can use long and read as int64_t. Using int32 is fine since num_obs=64. Parameter dtypes: log_init, etc are float32 (since default nn.Parameter is float32). The observations tensor is long, but we convert to int. Let's write the code. ``` cuda_sources = ''' #include <torch/extension.h> #include <cuda_runtime.h> #include <math_constants.h> __global__ void hmm_forward_kernel( const int* __restrict__ obs, const float* __restrict__ log_init, const float* __restrict__ log_trans, const float* __restrict__ log_emit, float* __restrict__ log_lik, int B, int T, int S, int O) { int b = blockIdx.x; if (b >= B) return; int j = threadIdx.x; if (j >= S) return; extern __shared__ float smem[]; // initial step int ob0 = obs[b * T]; float alpha = log_init[j] + log_emit[ob0 * S + j]; for (int t = 1; t < T; ++t) { smem[j] = alpha; __syncthreads(); float max_val = -1e30f; for (int i = 0; i < S; ++i) { float val = smem[i] + log_trans[i * S + j]; if (val > max_val) max_val = val; } float sum = 0.0f; for (int i = 0; i < S; ++i) { float val = smem[i] + log_trans[i * S + j]; sum += expf(val - max_val); } int ob = obs[b * T + t]; alpha = max_val + logf(sum) + log_emit[ob * S + j]; __syncthreads(); // ensure smem not overwritten before others read? Actually after this all threads have read smem in the max and sum loops (all threads reach here after reading). Then we can safely overwrite smem in the next iteration. However, we need __syncthreads before writing smem at top of loop? The __syncthreads after the sum loop ensures all threads have finished reading smem before any thread proceeds to next iteration where it will write smem. But wait, threads might diverge? No divergence. So we can put __syncthreads() at the end of loop or at the beginning before write. Let's put __syncthreads() right before smem[j] = alpha in next iteration? Actually after computing alpha, all threads are synchronized (implied by warp?), but across warps we need explicit syncthreads. If block has only 1 warp, it's not needed, but to be safe, we put __syncthreads() at the end of loop. } smem[j] = alpha; __syncthreads(); if (j == 0) { float max_val = -1e30f; for (int i = 0; i < S; ++i) { if (smem[i] > max_val) max_val = smem[i]; } float sum = 0.0f; for (int i = 0; i < S; ++i) { sum += expf(smem[i] - max_val); } log_lik[b] = max_val + logf(sum); } } void hmm_forward_cuda( torch::Tensor observations, torch::Tensor log_init, torch::Tensor log_trans, torch::Tensor log_emit, torch::Tensor log_lik, int B, int T, int S, int O) { dim3 blocks(B); dim3 threads(S); size_t shared_mem_size = S * sizeof(float); hmm_forward_kernel<<<blocks, threads, shared_mem_size>>>( observations.data_ptr<int>(), log_init.data_ptr<float>(), log_trans.data_ptr<float>(), log_emit.data_ptr<float>(), log_lik.data_ptr<float>(), B, T, S, O); } ''' ``` Then Python ModelNew: ```python class ModelNew(nn.Module): def __init__(self, num_states, num_obs): super().__init__() self.num_states = num_states self.num_obs = num_obs self.log_init = nn.Parameter(torch.zeros(num_states)) self.log_trans = nn.Parameter(torch.zeros(num_states, num_states)) self.log_emit = nn.Parameter(torch.zeros(num_states, num_obs)) # compile inline here? Actually module-level load_inline. def _normalize_log(self, x, dim=-1): return x - torch.logsumexp(x, dim=dim, keepdim=True) def forward(self, observations: torch.Tensor) -> torch.Tensor: B, T = observations.shape S = self.num_states O = self.num_obs device = observations.device log_pi = self._normalize_log(self.log_init) log_A = self._normalize_log(self.log_trans, dim=-1) log_B = self._normalize_log(self.log_emit, dim=-1) # ensure contiguous and correct types log_pi = log_pi.contiguous().float() log_A = log_A.contiguous().float() # transpose log_B for coalesced access: (O, S) log_B_T = log_B.t().contiguous().float() obs_int = observations.contiguous().to(torch.int32) log_lik = torch.empty(B, device=device, dtype=torch.float32) hmm_cuda.hmm_forward_cuda(obs_int, log_pi, log_A, log_B_T, log_lik, B, T, S, O) return log_lik ``` But we need to define hmm_cuda module via load_inline. The module-level code will be executed when the file is exec'd. Example: ```python import torch import torch.nn as nn from torch.utils.cpp_extension import load_inline # compile extension hmm_cuda = load_inline( name='hmm_cuda', cuda_sources=cuda_source, functions=['hmm_forward_cuda'], extra_cuda_cflags=['-O3'], verbose=False, ) ``` Note: We should use `load_inline` with `cuda_sources` and `functions`. I need to be careful about the C++ host function signature. It should use `torch::Tensor`. The auto-generated pybind11 wrapper will handle it. Potential issues: - `torch.utils.cpp_extension.load_inline` expects the source to be proper C++/CUDA. Using `#include <torch/extension.h>` is fine. - Need to ensure the data_ptr types match. `observations.data_ptr<int>()` for int32 tensor. - Observations might be on CPU or GPU. The model expects GPU? The reference doesn't specify, but get_inputs returns a CPU tensor? Actually `torch.randint` returns a CPU tensor by default. The test harness may move it to the same device as model parameters. So our kernel will receive the tensor on the correct device. Let's think about numerical equivalence. The manual logsumexp should match PyTorch's logsumexp within tolerance. Using float32, expf/logf are standard. PyTorch may use a different algorithm but the result should be close. The tolerance is atol=rtol=1e-4 for fp32. Computations are deterministic (no randomness). One subtlety: In the reference, `log_alpha = log_alpha.unsqueeze(-1) + log_A.unsqueeze(0)` is (B, S, S), then `torch.logsumexp(log_alpha, dim=1)`. In our kernel, we compute for each j: max_i (old_alpha[i] + A[i][j]), then log(sum_i exp(...)). This is mathematically identical. Another subtlety: The reference does `emit_0 = log_B[:, observations[:, 0]].t()` which is (B, S). Since log_B is normalized, log_B[:, obs] picks column obs for each state, shape (S, B), then .t() -> (B, S). In our kernel, we use log_emit (which is log_B.t()) and index by [ob * S + j]. That's exactly log_B[j, ob]. So `log_init[j] + log_emit[ob*S + j]` = `log_init[j] + log_B[j, ob]`. Good. Make sure to use `float` type everywhere. Potential optimization: Since S is known to be 16 at runtime (but passed as variable), we can still create a templated version for S=16 and instantiate it. But the overhead of a small loop is negligible. However, we might want to fully unroll the loops to reduce instruction overhead. Let's see: For each t, we do 2*S loads from smem, 2*S loads from log_trans, 1 expf. With S=16, that's 32 loads, 16 expfs. The memory should dominate. But we can reduce shared memory usage and syncthreads by using warp shuffles. Let's consider a variant without shared memory or syncthreads. Since S=16 and blockDim.x=S=16, all threads are in one warp. We can do: float alpha = ...; for t in 1..T: // Each thread needs alpha_i for all i. // Since all threads are in same warp, we can use __shfl_sync to broadcast alpha of thread i to all threads. float max_val = -1e30f; #pragma unroll for (int i = 0; i < S; ++i) { float a_i = __shfl_sync(0xFFFFFFFF, alpha, i); float val = a_i + log_trans[i * S + j]; if (val > max_val) max_val = val; } float sum = 0.0f; #pragma unroll for (int i = 0; i < S; ++i) { float a_i = __shfl_sync(0xFFFFFFFF, alpha, i); float val = a_i + log_trans[i * S + j]; sum += expf(val - max_val); } int ob = ...; alpha = max_val + logf(sum) + log_emit[ob * S + j]; This avoids shared memory and syncthreads entirely, and since there's only one warp, no sync needed. However, __shfl_sync is a warp-level instruction; it takes variable i (lane index). The cost is similar to a register move. But the loop is over S=16, which is fine. But note: If we use blockDim.x = 16, the warp has 16 active threads and 16 inactive threads. The __shfl_sync mask must include the active threads. We can use `0xFFFFFFFF` safely? Let's verify: In a block of 16 threads, there is one warp with lanes 0..15 active, 16..31 inactive. `__shfl_sync(0xFFFFFFFF, val, i)` requires all threads in the mask to execute it. Threads 16..31 are not part of the block, so they do not execute the instruction. Is that okay? According to CUDA documentation, `__shfl_sync` is a warp-level instruction. All non-exited threads named in the mask must execute the same __shfl_sync with the same mask. Since threads 16..31 are not active and not named? Wait, they are part of the warp, but if blockDim.x=16, they are not created? Actually, in CUDA, a warp always has 32 threads. If a block has 16 threads, the warp still has 32 lanes, but lanes 16..31 are inactive. The __shfl_sync requires all threads in the mask to be active? Or at least all threads in the mask must eventually execute the instruction. The inactive threads do not execute it. This can lead to a deadlock? I think there is a subtlety: The mask should be the set of threads participating. If the block has only 16 threads, you should use `__activemask()` or `(1u << blockDim.x) - 1`? Let's recall: `__shfl_sync` is safe as long as all threads in the mask are active. But if the mask includes inactive threads, behavior is undefined. Actually, the docs say: "The new *_sync shfl intrinsics take in a mask indicating the threads participating in the call. A bit, representing the lane ID, must be set for each participating thread to ensure they are properly converged." In a block of 16 threads, threads 0..15 are active, 16..31 are inactive. If we pass mask 0xFFFFFFFF, bits 16..31 are set, but those threads are inactive, so it violates the requirement. The correct mask is `((1u << blockDim.x) - 1)` but limited to 32 bits. For blockDim.x=16, mask = 0xFFFF. But `blockDim.x` is not a compile-time constant. However, we can compute `unsigned mask = __activemask();` at the point of the shuffle. `__activemask()` returns the mask of active threads in the warp. In a block of 16 threads, it should return 0xFFFF. Then we can do `__shfl_sync(mask, alpha, i)`. But the docs also say `__activemask()` should be used with care. Alternatively, we can just compute the mask as `(1u << S) - 1` if S <= 32 and passed as a constant. But at compile time we don't know S. We can compute `unsigned mask = (blockDim.x == 32) ? 0xFFFFFFFFu : ((1u << blockDim.x) - 1u);` But left shifting by 32 is UB if blockDim.x==32. We can handle it. However, to avoid all this complexity, using shared memory with __syncthreads is much simpler and less error-prone. The overhead of __syncthreads for a single warp is minimal (it compiles to a memory barrier, but since it's one warp, the compiler may optimize it away? Actually, __syncthreads in a block of one warp still issues a barrier instruction, but it's very fast). So shared memory approach is fine. Wait: there is another issue with shared memory: bank conflicts. Since S=16, each thread accesses smem[i] in a loop where i is the same for all threads. That's a broadcast, no bank conflict. Writing smem[j] = alpha is contiguous; no bank conflict because each thread writes its own bank (assuming 32 banks, each float maps to a bank). For j=0..15, banks 0..15, safe. So shared memory is efficient. Thus, we'll stick with shared memory. One more optimization: Since T is small (128) and S is small (16), total work per block is tiny. If we launch 64 blocks, each block does 128 * 16 * 16 operations, maybe a few thousand FLOPs. The kernel launch overhead might dominate. We could process multiple batch elements per block, e.g., have each block process 2 or 4 batch elements sequentially. This reduces the number of blocks and increases work per block. However, 64 blocks is already enough to saturate an A100? Probably not; an A100 has 108 SMs, so 64 blocks means some SMs idle. But each block's work is small, so the kernel might finish very quickly. Actually, with 64 blocks, each block is tiny. The GPU schedule: each SM can run multiple blocks concurrently. With 108 SMs and 64 blocks, some SMs get 1 block, some 0. The total execution time is limited by the slowest SM. Since blocks are small (1 warp), an SM can hold many blocks at once (up to 32 warps per SM on A100? Actually up to 64 warps per SM). So one block per SM is fine. But the latency of memory loads may be hidden by other warps on the same SM. Here each block has only 1 warp, so an SM with one block has 1 warp. If memory latency is high, there are no other warps to switch to. That could hurt performance. To improve, we could process multiple batch elements per block. For example, each block has 32 threads: process 2 batch elements (16 threads each). Each warp handles 2 batch elements. Then each SM can run more warps. But we need to partition the 32 threads carefully. Alternatively, we can process 4 batch elements per block with 64 threads (2 warps). Or we can have each thread process all S states for one batch element (so block of e.g., 64 threads handles 64 batch elements? No, 64 threads each doing all S states means redundant work but no communication needed). Wait, that could be better: each thread computes the entire alpha vector of length S for one batch element. Then each block has e.g., 64 threads, each handling one batch element. This way we have 64 warps across 64 threads? No, 64 threads = 2 warps per block. Then we can launch B/64 blocks? If B=64, that's 1 block. 1 block with 2 warps is underutilized. Better: each block has 128 threads (4 warps) handling 128 batch elements. But B is only 64. So we could launch 1 block of 64 threads, each thread handling one batch element independently. Then we have 2 warps in a single block. The SMs can run this block. But only 2 warps, still not enough to hide latency. Alternatively, we can have each thread handle multiple time steps? No, time steps are sequential. Another approach: Instead of 1 thread per state, we can process multiple states with a single thread to increase arithmetic intensity? Actually, if each thread handles one batch element and loops over all states, we have B threads total. That's only 64 threads, 2 warps. That's worse. Better: each block processes one batch element with a warp of 32 threads (but only 16 active). Then 64 blocks = 64 warps. A100 has 108 SMs, so 64 warps can be distributed, but each SM can run up to 64 warps (A100). So some SMs will have 1 warp, others 0. With 64 warps total, that's far below the GPU's capacity (108 SMs * 64 warps = 6912 warps). But does that matter if the kernel is memory-bound and each warp's memory latency is hidden by other warps? On an SM with 1 warp, if it issues a load, the warp stalls, and since there are no other warps, the SM is idle until the load returns. However, if the loads hit L1 cache, latency is low (a few cycles). The working set per block is small (S floats ~64 bytes). It will likely stay in L1. Also, log_trans is S*S=256 floats = 1KB. It might also stay in L1. But there is another issue: global memory throughput. Since we have 64 blocks, each loading from global memory, the total memory traffic is small. The kernel may be limited by launch overhead rather than memory bandwidth. The reference PyTorch implementation does many kernel launches (one per time step for logsumexp and additions). Our single kernel launch avoids all that overhead. So even if our kernel is not perfectly optimized, the speedup over PyTorch's many kernels should be huge. Wait, PyTorch might fuse some operations? In the loop: ``` log_alpha = log_alpha.unsqueeze(-1) + log_A.unsqueeze(0) # (B, S, S) log_alpha = torch.logsumexp(log_alpha, dim=1) # (B, S) emit_t = log_B[:, observations[:, t]].t() log_alpha = log_alpha + emit_t ``` `unsqueeze` + `+` is broadcasted addition. PyTorch likely launches a kernel for the addition. Then `logsumexp` is a reduction kernel. Then `log_B[:, obs]` is an index select, which may launch a kernel. Then another addition kernel. So per time step, 4 kernel launches. For T=128, that's ~512 kernel launches. Our fused single kernel will be much faster just because of launch overhead. Thus, a straightforward kernel is likely sufficient for a large speedup. But to be safe, let's also consider using vectorized loads or transposed log_trans. Let's try to keep the kernel simple but correct. One possible issue: `load_inline` might not compile `-O3` by default? Actually extra_cuda_cflags=['-O3', '--use_fast_math'] can help. `--use_fast_math` uses approximate intrinsics for expf and logf, which are faster but slightly less accurate. Tolerance is 1e-4 for fp32. Fast math should be fine for probabilities? Let's evaluate: log probabilities are moderate (normalized), so exp of differences might be in a reasonable range. Fast math exp/log relative error is maybe ~1e-5, which is safe for 1e-4 tolerance. But to be safe, we can avoid `--use_fast_math` or use it for speed. Since speedup is important, we can add it. But we need to pass correctness first. Let's try without fast math first to ensure correctness, then add it if needed. We should also test with `compile_kernel` and `run_correctness`. Let's write the kernel code now and run these tools. I will write the full Python module with ModelNew and the inline loading. One thing: the `get_gpu_specs` might be useful to see what GPU we are on. Let's call it.

Let me analyze this problem: 1. **Problem**: Hidden Markov Model forward algorithm in log-space 2. **Dimensions**: - `num_states = 16` - `num_obs = 64` - `seq_len = 128` - `batch_size = 64` 3. **Reference implementation**: - Normalizes log_init, log_trans, log_emit - For each batch and time step, does: - `log_alpha = log_alpha.unsqueeze(-1) + log_A.unsqueeze(0)` → (B, S, S) - `log_alpha = torch.logsumexp(log_alpha, dim=1)` → (B, S) - Add emission probability 4. **GPU**: H100 with 3.35 TB/s bandwidth, 67 TFLOPS FP32, 228KB shared memory per SM The bottleneck is the forward recursion. With S=16, B=64, T=128, each step involves: - Computing `(B, S, S)` matrix (64 * 16 * 16 = 16384 elements) - logsumexp over one dimension - Adding emission The reference does this sequentially over T. But since T=128 is small and each step depends on the previous, we can't parallelize across T easily. However, with S=16 being very small, we can: 1. Process multiple time steps in a single kernel, perhaps unrolling the loop 2. Use shared memory for log_A and log_B lookups 3. Process multiple batches in parallel Actually, looking more carefully, S=16 is tiny. This means: - The (B, S, S) tensor is small enough that computation is cheap - But T=128 sequential steps means 128 kernel launches in PyTorch, which has overhead - We can fuse the entire T-loop into a single CUDA kernel! Let me think about a fused kernel approach: For each batch element b: for t = 1 to T-1: for j in 0 to S-1: log_alpha[b][j] = logsumexp_i(log_alpha[b][i] + log_A[i][j]) + log_B[j][obs[b][t]] With S=16, B=64, T=128, we can have each thread block handle one batch element, and threads within the block compute the S states in parallel. For logsumexp with S=16, we can do this efficiently with shared memory or registers. Let me design the kernel: ```cuda __global__ void hmm_forward_kernel( const float* log_pi, // (S,) const float* log_A, // (S, S) const float* log_B, // (S, O) const long* observations, // (B, T) float* log_lik, // (B,) int B, int T, int S, int O ) { int b = blockIdx.x; int tid = threadIdx.x; if (b >= B) return; // Shared memory __shared__ float alpha_prev[16]; __shared__ float alpha_curr[16]; __shared__ float log_A_shared[16 * 16]; // Load log_A into shared memory for (int i = tid; i < S * S; i += blockDim.x) { log_A_shared[i] = log_A[i]; } __syncthreads(); // Initial step: alpha[j] = log_pi[j] + log_B[j, obs[0]] if (tid < S) { int obs0 = observations[b * T]; alpha_prev[tid] = log_pi[tid] + log_B[tid * O + obs0]; } __syncthreads(); // Forward recursion for (int t = 1; t < T; t++) { int obs_t = observations[b * T + t]; if (tid < S) { // Compute logsumexp_i(alpha_prev[i] + log_A[i, tid]) float max_val = -INFINITY; for (int i = 0; i < S; i++) { float val = alpha_prev[i] + log_A_shared[i * S + tid]; if (val > max_val) max_val = val; } float sum = 0.0f; for (int i = 0; i < S; i++) { sum += expf(alpha_prev[i] + log_A_shared[i * S + tid] - max_val); } alpha_curr[tid] = max_val + logf(sum) + log_B[tid * O + obs_t]; } __syncthreads(); // Swap float* temp = alpha_prev; alpha_prev = alpha_curr; alpha_curr = temp; __syncthreads(); } // Final logsumexp if (tid < S) { // Find max and compute logsumexp float max_val = alpha_prev[0]; for (int i = 1; i < S; i++) { if (alpha_prev[i] > max_val) max_val = alpha_prev[i]; } // Need reduction... use shared memory } } ``` Wait, I need to be more careful about the shared memory swap (can't swap pointers to shared memory in CUDA easily). Let me use indexing instead. Actually, since S=16 is so small, I could just have each thread handle one state and use warp-level reductions or shared memory for finding the max and sum. With warp size 32 and S=16, a single warp could handle all S states for a batch element. But then we need to process multiple batch elements per warp or have idle threads. Let me think about occupancy: B=64, so 64 blocks. Each block could have 16 threads (one per state). With 132 SMs on H100, we'd only use 64 SMs, but each SM can run multiple blocks. With 16 threads per block and S=16: - Each thread computes one state's new alpha value - For logsumexp, each thread iterates over all S=16 states to find max then sum Since there are only 16 threads per block and they all access alpha_prev, this is fine in shared memory. For the final logsumexp across states, we need a reduction. With 16 threads: 1. Thread 0 finds max by reading all S values 2. Or we can do a parallel reduction in shared memory Actually, since each thread computes one state's value, and then for the final reduction each thread can compute its contribution. Let me refine: ```cuda __global__ void hmm_forward_kernel( const float* log_pi, const float* log_A, const float* log_B, const long* observations, float* output, int B, int T, int S, int O ) { int b = blockIdx.x; if (b >= B) return; int tid = threadIdx.x; extern __shared__ float smem[]; float* alpha0 = smem; float* alpha1 = smem + S; float* log_A_s = smem + 2 * S; // Load transition matrix for (int i = tid; i < S * S; i += blockDim.x) { log_A_s[i] = log_A[i]; } __syncthreads(); // Initialize long obs0 = observations[b * T]; if (tid < S) { alpha0[tid] = log_pi[tid] + log_B[tid * O + obs0]; } __syncthreads(); float* prev = alpha0; float* curr = alpha1; for (int t = 1; t < T; t++) { long obs_t = observations[b * T + t]; if (tid < S) { float max_val = -1e30f; #pragma unroll for (int i = 0; i < S; i++) { float val = prev[i] + log_A_s[i * S + tid]; if (val > max_val) max_val = val; } float sum = 0.0f; #pragma unroll for (int i = 0; i < S; i++) { sum += expf(prev[i] + log_A_s[i * S + tid] - max_val); } curr[tid] = max_val + logf(sum) + log_B[tid * O + obs_t]; } __syncthreads(); float* tmp = prev; prev = curr; curr = tmp; __syncthreads(); } // Final logsumexp if (tid < S) { // First get max // We can use a simple loop since S is small // Actually let's use shared memory reduction } } ``` For the final logsumexp with S=16 and blockDim.x=16, I have each thread read all values. Or I can do a warp shuffle reduction. Actually, since we have exactly 16 threads and they all have the `prev` array in shared memory: ```cuda // Final reduction if (tid == 0) { float max_val = prev[0]; for (int i = 1; i < S; i++) { if (prev[i] > max_val) max_val = prev[i]; } float sum = 0.0f; for (int i = 0; i < S; i++) { sum += expf(prev[i] - max_val); } output[b] = max_val + logf(sum); } ``` But wait - there's an issue with numerical stability. The reference does `_normalize_log` on log_init, log_trans, and log_emit. This means they subtract logsumexp from each. But my kernel just adds them directly. Since we're computing in log-space and doing logsumexp correctly, the normalization shouldn't affect the final result... Actually wait, it does matter because normalization changes the individual values. Let me check. The reference does: 1. `log_pi = log_init - logsumexp(log_init)` 2. `log_A[i,j] = log_trans[i,j] - logsumexp(log_trans[i,:])` 3. `log_B[j,o] = log_emit[j,o] - logsumexp(log_emit[j,:])` Then uses these normalized values. But in the kernel, if I use the raw parameters without normalizing, I get different results. So I need to normalize in the kernel, or do it on the Python side before calling the kernel. The cleanest approach is to have Python do the normalization and pass normalized arrays to the kernel. But the reference computes normalization in forward(), which means the gradients flow through it too. For correctness with autograd, we need to be careful. Actually, the parameters are nn.Parameter, so normalization happens inside forward() and gradients propagate through it. If we want exact numerical equivalence, we need to replicate this. But since we're replacing the entire forward() with a custom kernel, and backward will be handled by PyTorch's autograd... wait, for a custom CUDA kernel called via cpp_extension, PyTorch won't automatically compute gradients unless we use torch.ops or implement backward. Hmm, but the evaluation probably just checks forward() output, not backward. Let me re-read the instructions: "Replace the forward computation... with a custom CUDA kernel". And "Produce numerically equivalent outputs at fp32 precision." It doesn't mention backward. So I can: 1. In Python, normalize the parameters before passing to CUDA 2. Or in CUDA, normalize them The simplest approach is to do normalization in Python before kernel launch, since it's O(S^2 + S*O + S) which is tiny. Actually, looking more carefully at the reference: `log_pi = self._normalize_log(self.log_init)` etc. are computed in forward(). If we compute them separately and pass to the kernel, we should get the same result for forward(). Let me just do normalization in Python. Wait, but `self.log_init` etc. are Parameters with grad. For correctness checking, they probably just compare outputs. Let me compute normalization in Python and pass the normalized values to the CUDA kernel. But there's a subtlety: the `_normalize_log` function does `x - torch.logsumexp(x, dim=dim, keepdim=True)`. For log_emit with shape (S, O), dim=-1 means normalize across O for each state. OK, let me structure the code: ```python class ModelNew(nn.Module): def __init__(self, num_states, num_obs): super().__init__() self.num_states = num_states self.num_obs = num_obs self.log_init = nn.Parameter(torch.zeros(num_states)) self.log_trans = nn.Parameter(torch.zeros(num_states, num_states)) self.log_emit = nn.Parameter(torch.zeros(num_states, num_obs)) def forward(self, observations): log_pi = self.log_init - torch.logsumexp(self.log_init, dim=-1, keepdim=True) log_A = self.log_trans - torch.logsumexp(self.log_trans, dim=-1, keepdim=True) log_B = self.log_emit - torch.logsumexp(self.log_emit, dim=-1, keepdim=True) B, T = observations.shape output = torch.empty(B, device=observations.device, dtype=torch.float32) # Call CUDA kernel cuda_ext.hmm_forward( log_pi, log_A, log_B, observations, output, B, T, self.num_states, self.num_obs ) return output ``` Now for the CUDA kernel. Let me think about optimization: 1. **Kernel fusion**: Fusing T=128 steps into one kernel eliminates 127 kernel launch overheads and intermediate memory reads/writes. 2. **Memory access**: - `log_A` is (16, 16) = 256 floats = 1KB - fits in shared memory - `log_B` is (16, 64) = 1024 floats = 4KB - fits in shared memory - `observations` is (64, 128) = 8192 longs = 32KB - We only need to read each observation once per batch element 3. **Shared memory usage**: - With log_A in shared memory: 1KB - With log_B in shared memory: 4KB - Two alpha buffers: 2 * 16 * 4 = 128 bytes - Total per block: ~5KB, well within limits Since each batch element accesses log_B in a scattered way (depending on observations), putting log_B in shared memory might help, but we need to access `log_B[:, obs_t]` for each t. Since obs_t varies per t and per batch, preloading all of log_B is probably best. Actually, we can load the relevant column of log_B on the fly from global memory. With 16 states and coalesced access (if threads read consecutive states for the same obs), it should be OK. Let me also consider processing multiple batch elements per block. With B=64 and 16 threads per block for 1 batch element, we have 64 blocks. On an H100 with many SMs, this might not fully saturate the GPU. But the computation per batch element is 128 * 16 * 16 = 32768 operations, which is pretty small. Actually, let me try having each block process multiple batch elements, or each warp process one batch element. Option A: 1 warp (32 threads) per batch element, process 2 batch elements per block (64 threads) Option B: Have each thread process one batch element - but then how to handle the inner loop? With S=16, if each thread processes one batch element: - The thread needs to maintain an array of 16 alpha values in registers - For each state j, compute logsumexp over i: alpha[i] + log_A[i,j] - This is 16 sequential logsumexp operations per thread Actually, this could work well! Each thread handles one batch element completely independently. We launch 64 threads total (one per batch element), or more if we want to occupancy. Wait, but 64 threads is very few. We should launch at least 256-512 threads to saturate the GPU. We could have each thread handle one batch element and launch 64 threads per block with multiple blocks... but we only have 64 batch elements total. Better: each thread handles one batch element, launch (B, ) threads with some threads/blocks configuration. Actually, let me think about this differently. With S=16, T=128: - Total work: 64 * 128 * 16 * 16 = 2,097,152 operations - This is quite small For small workloads, the overhead of kernel launch matters a lot. A single kernel launch is already a big win over 128 PyTorch operations. Let me go with: one thread per batch element, or 16 threads per batch element. I think 16 threads per batch element is cleaner for the shared-memory access pattern of log_A. Actually, let me benchmark both approaches mentally: - 16 threads per batch element × 64 blocks = good parallelism - Each block does 128 iterations - Shared memory for log_A and alpha buffers - Could also put log_B in shared memory If I add log_B to shared memory: 16 × 64 × 4 = 4KB Total shared memory: 1KB (log_A) + 4KB (log_B) + 0.128KB (alpha) = ~5KB This is very comfortable. With log_B in shared memory, access is just s_obs[threadIdx.x] for the current observation. Actually, we need `log_B[state, obs_t]`, so for each t, each thread accesses `log_B_s[state * O + obs_t]`. Wait, that's not right. log_B has shape (S, O), so log_B[state, obs] is at index state * O + obs. Actually, if obs_t is the same for all threads in a block (per batch element), and each thread tid represents one state, then thread tid accesses log_B[tid * O + obs_t]. These are strided by O=64, not coalesced. But with only 16 threads, coalescing doesn't matter much. Let me reconsider: we could transpose log_B to (O, S) to get coalesced access. Thread tid accesses transposed_log_B[obs_t * S + tid], which is coalesced! So I'll transpose log_B before passing to the kernel. Let me write the kernel more carefully now: ```cuda __global__ void hmm_forward_kernel( const float* log_pi, // (S,) const float* log_A, // (S, S) - row major, A[i,j] at i*S+j const float* log_Bt, // (O, S) - transposed, Bt[o,j] = log_B[j,o] const long* observations, // (B, T) float* output, // (B,) int B, int T, int S, int O ) { int b = blockIdx.x; int tid = threadIdx.x; if (b >= B) return; extern __shared__ float sdata[]; float* alpha_prev = sdata; float* alpha_curr = sdata + S; float* log_A_s = sdata + 2 * S; // Load log_A into shared memory for (int i = tid; i < S * S; i += blockDim.x) { log_A_s[i] = log_A[i]; } __syncthreads(); // Initialize alpha long obs0 = observations[b * T]; if (tid < S) { alpha_prev[tid] = log_pi[tid] + log_Bt[obs0 * S + tid]; } __syncthreads(); for (int t = 1; t < T; t++) { long obs_t = observations[b * T + t]; if (tid < S) { // logsumexp_i(alpha_prev[i] + log_A[i, tid]) float max_val = -1e30f; #pragma unroll for (int i = 0; i < S; i++) { float val = alpha_prev[i] + log_A_s[i * S + tid]; max_val = fmaxf(max_val, val); } float sum = 0.0f; #pragma unroll for (int i = 0; i < S; i++) { sum += expf(alpha_prev[i] + log_A_s[i * S + tid] - max_val); } alpha_curr[tid] = max_val + logf(sum) + log_Bt[obs_t * S + tid]; } __syncthreads(); // Swap float* tmp = alpha_prev; alpha_prev = alpha_curr; alpha_curr = tmp; __syncthreads(); } // Final logsumexp if (tid == 0) { float max_val = alpha_prev[0]; for (int i = 1; i < S; i++) { max_val = fmaxf(max_val, alpha_prev[i]); } float sum = 0.0f; for (int i = 0; i < S; i++) { sum += expf(alpha_prev[i] - max_val); } output[b] = max_val + logf(sum); } } ``` Shared memory per block: 2 * 16 + 16 * 16 = 288 floats = 1152 bytes. Very small. Hmm, wait. If we have 64 blocks with 16 threads each, that's only 1024 threads total. On an H100, we have 132 SMs × 2048 threads/SM = 270,336 max threads. 1024 threads is very low. But since each thread does 128 * 16 * 16 ≈ 32K operations, the total work is about 32M operations which is quite small. The arithmetic intensity might be low. Can we increase occupancy? We could have each block process multiple batch elements sequentially. Or we could use more threads per block and do parallel reduction for logsumexp. Actually, with 16 threads doing independent work (each computing one state's new alpha), and S=16, the logsumexp for each state j requires reading all alpha_prev[i]. This means each thread reads the entire alpha_prev array. With 16 threads, that's 16 concurrent reads to the same shared memory bank. Since it's the same array and different threads read the same values simultaneously, bank conflicts shouldn't be too bad (broadcast). Wait, each thread reads alpha_prev[0], alpha_prev[1], ..., alpha_prev[15] sequentially. With 16 threads in lockstep (since we're in CUDA SIMT), thread 0 reads alpha_prev[0] while thread 1 reads alpha_prev[0] too. Actually no, each thread is doing a sequential loop: Thread 0: reads alpha_prev[0] + A[0*S+0] Thread 1: reads alpha_prev[0] + A[0*S+1] ... Actually, within the inner loop over i: - Thread tid reads alpha_prev[i] (same for all threads when i is fixed in iteration i) - Thread tid reads A[i*S + tid] (different for each thread) So alpha_prev[i] is broadcast to all threads. A[i*S + tid] is coalesced (consecutive elements). This is a good memory access pattern! For log_Bt[obs_t * S + tid], this is also coalesced across threads in a block. Good. Now, one issue: 64 blocks is very few. Can we have more blocks by processing partial work? Not really, since each batch element is independent and we have exactly 64. What if we have each block process, say, 2 batch elements? Then we'd have 32 blocks, which is even worse. What if we have each warp process one batch element, and a block has multiple warps processing different batch elements? For example, 4 warps per block = 128 threads per block. Then we'd have ceil(64/4) = 16 blocks. Still very few. Hmm, actually with 64 blocks × 16 threads = 1024 threads total, and each thread doing ~32K ops, total FLOPs is ~33M. At 67 TFLOPS, this would take ~0.5 microseconds if we were compute bound. We're not compute bound. The issue is more about memory bandwidth and latency. The kernel should be memory-bound since we read 128 observations per batch element, plus some parameters. Memory traffic per batch element: - Initial read of log_pi: 16 floats = 64 bytes (amortized over T) - log_A: read once to shared memory: 256 floats = 1KB (amortized) - observations: 128 longs = 512 bytes - log_B: 128 * 16 = 2048 floats = 8KB (from shared memory or global) - Total: ~9.5KB per batch element - For 64 batch elements: ~608KB At 3.35TB/s, 608KB takes ~0.18 microseconds. Plus overhead. The PyTorch reference would have much more overhead from 128+ kernel launches. Wait, I think the main win is avoiding 128 kernel launches. Let me also look for micro-optimizations. One optimization: instead of having thread 0 do the final reduction, we can have all threads participate or use warp primitives. But with only 16 threads, it doesn't matter much. Another optimization: we can unroll the loop over T using pragma unroll if T is a template parameter or constant. But T is a runtime parameter (128). We could use loop unrolling manually or with `#pragma unroll 4` or similar. Let me check if the compiler can unroll a runtime-bounded loop... it can unroll with a pragma but the bound is dynamic. Actually, `#pragma unroll` on a loop with dynamic bound might just unroll the body and keep the loop structure. Let me try it. Actually, for better optimization, I can make the kernel a template with S=16 as a compile-time constant. This allows better unrolling and register allocation. ```cuda template <int S> __global__ void hmm_forward_kernel(...) ``` Then `#pragma unroll` on the inner loop over S will actually unroll since S is a template parameter. Let me write this more carefully. One concern: using `expf` and `logf`. The reference uses `torch.logsumexp` which uses stable log-sum-exp. My kernel does the same: find max, subtract max, exp, sum, log. Wait, there might be numerical issues when all values are -inf. But since log_pi and log_A are normalized, they shouldn't be -inf. Let me handle edge cases? Actually, to match PyTorch exactly, I should be careful about precision. PyTorch's logsumexp is quite accurate. My hand-rolled version should match within fp32 tolerance (1e-4). Another thought: can we vectorize the inner loops? With S=16, we could use float4 loads for log_A and use vector math. But given the small size, it's probably not worth the complexity. Let me also consider putting log_B in shared memory. If we preload the relevant observations' columns of log_B, we'd need to know all observations in advance for a batch element. Since we read obs_t sequentially, we can preload the entire log_B (4KB) once per block. Actually, let me put log_B in shared memory too: ```cuda extern __shared__ float sdata[]; float* alpha_prev = sdata; float* alpha_curr = sdata + S; float* log_A_s = sdata + 2 * S; float* log_B_s = sdata + 2 * S + S * S; ``` Total: 2*16 + 256 + 1024 = 1312 floats = 5248 bytes per block. With 228KB shared memory per SM and 32 blocks max per SM, each block uses ~5KB, so we could have ~45 blocks per SM. But we only have 64 blocks total. Since there are 132 SMs, we have less than 1 block per SM. That's fine. Wait, actually 64 blocks / 132 SMs = 0.48 blocks per SM on average. But blocks are distributed across SMs, so some SMs run 1 block and others run 0. With 16 threads per block and each block taking ~128*16*16 = 32768 operations, each SM with 1 block does 32K operations, which takes very little time. The GPU will be mostly idle. But there's no helping it since the total work is small. The real win is fusing all operations into a single kernel launch, eliminating 128+ kernel launch overheads from PyTorch. Let me think about warp-level execution. Since blockDim.x = 16 and warp size = 32, the entire block fits in half a warp. Some threads in the warp are inactive. This might not be optimal for instruction throughput. What if I set blockDim.x = 32, with 2 batch elements per block? Each warp handles one batch element. - Block 0: warp 0 → batch 0, warp 1 → batch 1 - Block 1: warp 0 → batch 2, warp 1 → batch 3 - etc. With 64 batches, I'd have 32 blocks, each with 32 threads (1 warp per batch element). This means: - 32 blocks instead of 64 - Full warp utilization Actually, let me think about this. Each block has 2 warps, handling 2 batch elements. The warps can execute independently. Within each warp (32 threads), only 16 are active (tid < 16 check). The other 16 threads are masked off. Is this better or worse? With 16 threads per block: half a warp is active, but the entire block runs together. Block-level barriers synchronize 16 threads. With 32 threads per block (2 batch elements): each half-warp is a different batch element. The __syncthreads() synchronizes all 32 threads. I think 16 threads per block is simpler and likely fine. The inactive threads in the warp don't really hurt much because the active 16 threads still execute at full rate. The main issue is thread divergence: threads 16-31 are permanently inactive, but that's only within the block. Since each block occupies half a warp, SMs might only run 64 half-warps. Actually, on NVIDIA GPUs, scheduling is at the warp level. A block with 16 threads is still one warp (32 threads, 16 active). The warp scheduler issues instructions for the active threads. This is fine. But having only 64 warps total across the entire GPU... with 4 warp schedulers per SM on H100, we have 4 * 132 = 528 warp schedulers. 64 warps means each scheduler runs 0.12 warps on average. The GPU will be severely underutilized, but that's unavoidable for this small workload. Wait, I just realized: 64 blocks with 16 threads each = 64 warps. Each warp does ~64K instructions (32K FLOPs with multiple instr per op). At H100's clock (~2GHz) and 4 schedulers per SM, a single SM could process ~4 warps per clock cycle. So 64 warps would take ~16 clock cycles to dispatch across all SMs, and each warp takes maybe 64K / 32 ≈ 2000 cycles to execute. Total time might be ~2000 cycles = 1 microsecond. Very fast. Actually, the PyTorch reference might be much slower because: 1. Each time step involves several kernel launches 2. Memory is allocated and freed 3. Tensors are created and destroyed So a single fused kernel should be much faster. Let me also consider whether we can reduce the number of expf calls. In the inner loop, we have: 1. First loop: find max (2 * S reads, S fmaxf ops) 2. Second loop: compute sum (2 * S reads, S expf ops, S adds) 3. One logf We can't avoid this for correctness. One optimization: precompute alpha_prev[i] + log_A[i, j] for all i, j using direct products, but that's what the reference does with matmul. Since S=16 is so small, iterating is actually faster than setting up tensor operations. Let me write the complete kernel now. Wait, I need to double-check the normalization. The reference normalizes log_trans over dim=-1, meaning each row sums to 1 (in probability space). So log_A[i,j] = log_trans[i,j] - logsumexp(log_trans[i,:]). For log_emit, log_B[j,o] = log_emit[j,o] - logsumexp(log_emit[j,:]). In my kernel, I need to use these normalized values. I'll compute them in Python and pass them to the kernel. Actually, I just realized: the reference stores `log_pi`, `log_A`, `log_B` as local variables and then uses them. For autograd, PyTorch would trace these operations. If I compute them in Python and pass them to a custom CUDA kernel, autograd won't be able to compute gradients through the kernel (unless I write a custom backward). But the task says "Replace the forward computation", and the evaluation probably only checks forward outputs. And since the model isn't trained, backward doesn't matter. Let me write the actual code now. ```python import torch import torch.nn as nn cuda_src = r''' #include <torch/extension.h> #include <cuda_runtime.h> #include <cmath> template <int S> __global__ void hmm_forward_kernel( const float* __restrict__ log_pi, const float* __restrict__ log_A, const float* __restrict__ log_Bt, const long* __restrict__ observations, float* __restrict__ output, int B, int T, int O ) { int b = blockIdx.x; int tid = threadIdx.x; if (b >= B) return; extern __shared__ float sdata[]; float* alpha_prev = sdata; float* alpha_curr = sdata + S; float* log_A_s = sdata + 2 * S; // Load log_A into shared memory #pragma unroll for (int i = tid; i < S * S; i += blockDim.x) { log_A_s[i] = log_A[i]; } __syncthreads(); // Initialize alpha long obs0 = observations[b * T]; if (tid < S) { alpha_prev[tid] = log_pi[tid] + log_Bt[obs0 * S + tid]; } __syncthreads(); for (int t = 1; t < T; t++) { long obs_t = observations[b * T + t]; if (tid < S) { float max_val = -1e30f; #pragma unroll for (int i = 0; i < S; i++) { float val = alpha_prev[i] + log_A_s[i * S + tid]; max_val = fmaxf(max_val, val); } float sum = 0.0f; #pragma unroll for (int i = 0; i < S; i++) { sum += expf(alpha_prev[i] + log_A_s[i * S + tid] - max_val); } alpha_curr[tid] = max_val + logf(sum) + log_Bt[obs_t * S + tid]; } __syncthreads(); // Swap float* tmp = alpha_prev; alpha_prev = alpha_curr; alpha_curr = tmp; __syncthreads(); } // Final logsumexp if (tid == 0) { float max_val = alpha_prev[0]; #pragma unroll for (int i = 1; i < S; i++) { max_val = fmaxf(max_val, alpha_prev[i]); } float sum = 0.0f; #pragma unroll for (int i = 0; i < S; i++) { sum += expf(alpha_prev[i] - max_val); } output[b] = max_val + logf(sum); } } torch::Tensor hmm_forward_cuda( torch::Tensor log_pi, torch::Tensor log_A, torch::Tensor log_Bt, torch::Tensor observations, torch::Tensor output ) { int B = observations.size(0); int T = observations.size(1); int S = log_pi.size(0); int O = log_Bt.size(0); const int threads = S; // 16 threads const int shared_mem = (2 * S + S * S) * sizeof(float); hmm_forward_kernel<16><<<B, threads, shared_mem>>>( log_pi.data_ptr<float>(), log_A.data_ptr<float>(), log_Bt.data_ptr<float>(), observations.data_ptr<long>(), output.data_ptr<float>(), B, T, O ); return output; } ''' cpp_src = r''' torch::Tensor hmm_forward_cuda( torch::Tensor log_pi, torch::Tensor log_A, torch::Tensor log_Bt, torch::Tensor observations, torch::Tensor output ); ''' class ModelNew(nn.Module): def __init__(self, num_states, num_obs): super().__init__() self.num_states = num_states self.num_obs = num_obs self.log_init = nn.Parameter(torch.zeros(num_states)) self.log_trans = nn.Parameter(torch.zeros(num_states, num_states)) self.log_emit = nn.Parameter(torch.zeros(num_states, num_obs)) def forward(self, observations): log_pi = self.log_init - torch.logsumexp(self.log_init, dim=-1, keepdim=True) log_A = self.log_trans - torch.logsumexp(self.log_trans, dim=-1, keepdim=True) log_B = self.log_emit - torch.logsumexp(self.log_emit, dim=-1, keepdim=True) log_Bt = log_B.t().contiguous() B, T = observations.shape output = torch.empty(B, device=observations.device, dtype=torch.float32) hmm_cuda = torch.utils.cpp_extension.load_inline( name="hmm_forward", cpp_sources=cpp_src, cuda_sources=cuda_src, functions=["hmm_forward_cuda"], extra_cuda_cflags=["-O3", "--use_fast_math"], extra_cflags=["-O3"], verbose=False ) return hmm_cuda.hmm_forward_cuda(log_pi, log_A, log_Bt, observations, output) ``` Wait, I shouldn't call `load_inline` inside `forward()`! That would recompile the kernel every forward pass. It should be done at module load time or in `__init__`. Let me fix this: ```python import torch import torch.nn as nn cuda_src = r''' #include <torch/extension.h> #include <cuda_runtime.h> #include <cmath> template <int S> __global__ void hmm_forward_kernel( const float* __restrict__ log_pi, const float* __restrict__ log_A, const float* __restrict__ log_Bt, const long* __restrict__ observations, float* __restrict__ output, int B, int T, int O ) { int b = blockIdx.x; int tid = threadIdx.x; if (b >= B) return; extern __shared__ float sdata[]; float* alpha_prev = sdata; float* alpha_curr = sdata + S; float* log_A_s = sdata + 2 * S; #pragma unroll for (int i = tid; i < S * S; i += blockDim.x) { log_A_s[i] = log_A[i]; } __syncthreads(); long obs0 = observations[b * T]; if (tid < S) { alpha_prev[tid] = log_pi[tid] + log_Bt[obs0 * S + tid]; } __syncthreads(); for (int t = 1; t < T; t++) { long obs_t = observations[b * T + t]; if (tid < S) { float max_val = -1e30f; #pragma unroll for (int i = 0; i < S; i++) { float val = alpha_prev[i] + log_A_s[i * S + tid]; max_val = fmaxf(max_val, val); } float sum = 0.0f; #pragma unroll for (int i = 0; i < S; i++) { sum += expf(alpha_prev[i] + log_A_s[i * S + tid] - max_val); } alpha_curr[tid] = max_val + logf(sum) + log_Bt[obs_t * S + tid]; } __syncthreads(); float* tmp = alpha_prev; alpha_prev = alpha_curr; alpha_curr = tmp; __syncthreads(); } if (tid == 0) { float max_val = alpha_prev[0]; #pragma unroll for (int i = 1; i < S; i++) { max_val = fmaxf(max_val, alpha_prev[i]); } float sum = 0.0f; #pragma unroll for (int i = 0; i < S; i++) { sum += expf(alpha_prev[i] - max_val); } output[b] = max_val + logf(sum); } } torch::Tensor hmm_forward_cuda( torch::Tensor log_pi, torch::Tensor log_A, torch::Tensor log_Bt, torch::Tensor observations, int S ) { int B = observations.size(0); int T = observations.size(1); int O = log_Bt.size(0); torch::Tensor output = torch::empty({B}, log_pi.options()); const int threads = S; const int shared_mem = (2 * S + S * S) * sizeof(float); hmm_forward_kernel<16><<<B, threads, shared_mem>>>( log_pi.data_ptr<float>(), log_A.data_ptr<float>(), log_Bt.data_ptr<float>(), observations.data_ptr<long>(), output.data_ptr<float>(), B, T, O ); return output; } ''' cpp_src = r''' torch::Tensor hmm_forward_cuda( torch::Tensor log_pi, torch::Tensor log_A, torch::Tensor log_Bt, torch::Tensor observations, int S ); ''' # Load the extension at module load time hmm_cuda = torch.utils.cpp_extension.load_inline( name="hmm_forward", cpp_sources=cpp_src, cuda_sources=cuda_src, functions=["hmm_forward_cuda"], extra_cuda_cflags=["-O3", "--use_fast_math"], extra_cflags=["-O3"], verbose=False ) class ModelNew(nn.Module): def __init__(self, num_states, num_obs): super().__init__() self.num_states = num_states self.num_obs = num_obs self.log_init = nn.Parameter(torch.zeros(num_states)) self.log_trans = nn.Parameter(torch.zeros(num_states, num_states)) self.log_emit = nn.Parameter(torch.zeros(num_states, num_obs)) def forward(self, observations): log_pi = self.log_init - torch.logsumexp(self.log_init, dim=-1, keepdim=True) log_A = self.log_trans - torch.logsumexp(self.log_trans, dim=-1, keepdim=True) log_B = self.log_emit - torch.logsumexp(self.log_emit, dim=-1, keepdim=True) log_Bt = log_B.t().contiguous() return hmm_cuda.hmm_forward_cuda(log_pi, log_A, log_Bt, observations, self.num_states) ``` Wait, using `--use_fast_math` might cause numerical issues since `expf` and `logf` are slightly less accurate. The tolerance is 1e-4 for fp32. Fast math should still be within that, but let me be careful. Actually, fast math exp/log have relative error around 1e-5 to 1e-6, and we're doing log-sum-exp which should be stable. I'll keep `--use_fast_math` for speed and see if it passes. Actually, looking at the kernel again, there might be a subtle issue. In the reference, the `log_alpha` computation does: ```python log_alpha = log_alpha.unsqueeze(-1) + log_A.unsqueeze(0) # (B, S, S) log_alpha = torch.logsumexp(log_alpha, dim=1) # (B, S) ``` This means: new_log_alpha[b, j] = logsumexp_i(old_log_alpha[b, i] + log_A[i, j]) So `log_A[i, j]` corresponds to transition from state i to state j. In my kernel, `log_A_s[i * S + tid]` where tid=j. That matches! Good. For emission: `emit_t = log_B[:, observations[:, t]].t()` → shape (B, S), where emit_t[b, j] = log_B[j, obs_t[b]]. In my kernel: `log_Bt[obs_t * S + tid]` where log_Bt is the transpose of log_B. So log_Bt[o * S + j] = log_B[j, o]. This matches! For initialization: `log_alpha = log_pi.unsqueeze(0) + emit_0` In my kernel: `alpha_prev[tid] = log_pi[tid] + log_Bt[obs0 * S + tid]`. Matches! For final logsumexp: `torch.logsumexp(log_alpha, dim=-1)` In my kernel: max over alpha_prev, then log(sum(exp(alpha - max))). Matches! Great, the logic seems correct. One concern: the reference computes `log_lik = torch.logsumexp(log_alpha, dim=-1)` where log_alpha still contains the last emission. My kernel's alpha_prev after the loop also contains the last emission. Good. Now, one performance thought: can we avoid the Python-side normalization overhead? It's tiny (O(S^2 + SO + S) = 256 + 1024 + 16 = ~1296 elements), so it's negligible. But it does involve a few kernel launches. We could move normalization into the CUDA kernel to save even more. But it's fine. Actually, wait. The normalization uses `torch.logsumexp`. Could we compute it once and cache? Since the parameters might change (if we're training), caching is tricky. But for inference/evaluation, parameters are fixed. Let me not worry about it. Let me think about another optimization: what if `observations` doesn't need to be read from global memory for each t inside the loop? We could preload the entire observation sequence for each batch element into shared memory. For B blocks and T=128, each block would need 128 longs = 512 bytes. Total shared memory per block would be ~5KB + 512B = ~5.5KB. Still very comfortable. This saves 128 global memory reads of observations, replacing them with shared memory reads. Let me add this. ```cuda extern __shared__ float sdata[]; float* alpha_prev = sdata; float* alpha_curr = sdata + S; float* log_A_s = sdata + 2 * S; long* obs_seq = (long*)(sdata + 2 * S + S * S); // Preload observations for this batch element if (tid < T) { obs_seq[tid] = observations[b * T + tid]; } __syncthreads(); ``` Then use `obs_seq[t]` instead of `observations[b * T + t]`. Wait, but `sdata` is float*, and obs_seq is long*. We need to be careful with alignment. Let me compute the shared memory layout precisely. Actually, let me just use a separate long* or compute the address properly. Or I can make obs_seq a `long*` at a known offset. But `__shared__ float sdata[]` means `sdata` starts at a float-aligned address. sizeof(long) is 8 bytes on 64-bit systems. If 2*S + S*S is even, the offset is 8-byte aligned (since float is 4 bytes). Let me check: 2*16 + 256 = 288. 288 * 4 = 1152. 1152 % 8 = 0. Good. So I can cast: ```cuda long* obs_seq = reinterpret_cast<long*>(sdata + 2 * S + S * S); ``` Then preload: ```cuda for (int t = tid; t < T; t += blockDim.x) { obs_seq[t] = observations[b * T + t]; } __syncthreads(); ``` This is cleaner and might be faster. Let me add it. Updated shared memory size: (2 * S + S * S) * sizeof(float) + T * sizeof(long) = (32 + 256) * 4 + 128 * 8 = 288 * 4 + 1024 = 1152 + 1024 = 2176 bytes per block. Still very small. Actually, T might not be a compile-time constant. But we can handle it since we pass T as a parameter. One more optimization: since `log_Bt` is small (64 * 16 = 1024 floats = 4KB), we could preload it entirely into shared memory too. But since each obs_t is different per batch and per t, and we preload observations, each step only needs to read `log_Bt[obs_seq[t] * S + tid]`. With log_Bt in global memory, this is 16 coalesced floats = 64 bytes per time step per block. With 128 time steps, that's 8KB of global memory reads per block. Putting log_Bt in shared memory saves these reads. Let me preload log_Bt too: ```cuda float* log_Bt_s = sdata + 2 * S + S * S; long* obs_seq = reinterpret_cast<long*>(sdata + 2 * S + S * S + O * S); ``` Wait, this is getting complicated with the mixed float/long. Let me be very careful. Total floats before obs_seq: 2*S (alpha) + S*S (log_A) + O*S (log_Bt) = 2*16 + 256 + 1024 = 1312 floats 1312 * 4 = 5248 bytes 5248 % 8 = 0. Good. So: ```cuda float* alpha_prev = sdata; // 0 .. S-1 float* alpha_curr = sdata + S; // S .. 2*S-1 float* log_A_s = sdata + 2 * S; // 2*S .. 2*S + S*S - 1 float* log_Bt_s = sdata + 2 * S + S * S; // 2*S + S*S .. 2*S + S*S + O*S - 1 long* obs_seq = reinterpret_cast<long*>(sdata + 2 * S + S * S + O * S); ``` Total shared memory in bytes: (2*S + S*S + O*S) * sizeof(float) + T * sizeof(long) = (32 + 256 + 1024) * 4 + 128 * 8 = 1312 * 4 + 1024 = 5248 + 1024 = 6272 bytes. Still well under limits. Then in the kernel, access log_Bt_s[obs_seq[t] * S + tid] instead of global memory. This means almost all accesses during the T-loop are to shared memory! Only initial loads of log_pi, log_A, log_Bt, and observations are from global memory. The T-loop is entirely shared-memory bound. This should be very fast. Let me write the updated kernel: ```cuda template <int S> __global__ void hmm_forward_kernel( const float* __restrict__ log_pi, const float* __restrict__ log_A, const float* __restrict__ log_Bt, const long* __restrict__ observations, float* __restrict__ output, int B, int T, int O ) { int b = blockIdx.x; int tid = threadIdx.x; if (b >= B) return; extern __shared__ float sdata[]; float* alpha_prev = sdata; float* alpha_curr = sdata + S; float* log_A_s = sdata + 2 * S; float* log_Bt_s = sdata + 2 * S + S * S; long* obs_seq = reinterpret_cast<long*>(sdata + 2 * S + S * S + O * S); // Load log_A and log_Bt into shared memory int total_params = S * S + O * S; #pragma unroll for (int i = tid; i < total_params; i += blockDim.x) { if (i < S * S) { log_A_s[i] = log_A[i]; } else { log_Bt_s[i - S * S] = log_Bt[i - S * S]; } } // Load observations for this batch element for (int t = tid; t < T; t += blockDim.x) { obs_seq[t] = observations[b * T + t]; } __syncthreads(); // Initialize if (tid < S) { alpha_prev[tid] = log_pi[tid] + log_Bt_s[obs_seq[0] * S + tid]; } __syncthreads(); for (int t = 1; t < T; t++) { if (tid < S) { int obs_t = obs_seq[t]; float max_val = -1e30f; #pragma unroll for (int i = 0; i < S; i++) { float val = alpha_prev[i] + log_A_s[i * S + tid]; max_val = fmaxf(max_val, val); } float sum = 0.0f; #pragma unroll for (int i = 0; i < S; i++) { sum += expf(alpha_prev[i] + log_A_s[i * S + tid] - max_val); } alpha_curr[tid] = max_val + logf(sum) + log_Bt_s[obs_t * S + tid]; } __syncthreads(); float* tmp = alpha_prev; alpha_prev = alpha_curr; alpha_curr = tmp; __syncthreads(); } if (tid == 0) { float max_val = alpha_prev[0]; #pragma unroll for (int i = 1; i < S; i++) { max_val = fmaxf(max_val, alpha_prev[i]); } float sum = 0.0f; #pragma unroll for (int i = 0; i < S; i++) { sum += expf(alpha_prev[i] - max_val); } output[b] = max_val + logf(sum); } } ``` I should be careful about the `for` loop with `total_params`. The `if-else` branch within the loop might cause divergence since threads in the same warp might take different paths. But since we have only 16 threads, and total_params = 256 + 1024 = 1280, each thread does ceil(1280/16) = 80 iterations. The first 16 iterations all go to the `if` branch (loading log_A), and the rest go to `else`. This means the first warp (threads 0-15) has all threads taking the same branch in each iteration. Actually wait, all threads in the block execute the same branch in lockstep if the condition is the same, which it is (i < S*S doesn't depend on tid). So there's no divergence - all active threads take the same path. Good. Actually, I could split the loading into two separate loops for clarity: ```cuda for (int i = tid; i < S * S; i += blockDim.x) { log_A_s[i] = log_A[i]; } for (int i = tid; i < O * S; i += blockDim.x) { log_Bt_s[i] = log_Bt[i]; } ``` This is cleaner and has the same performance since the compiler will optimize similarly. Now, with all this shared memory, the T-loop is mostly doing arithmetic on data in shared memory. Let me think about whether we can further optimize. One thing: `__syncthreads()` after every time step is needed because all threads must finish writing `alpha_curr` before any thread reads from `alpha_prev` in the next iteration. But with only 16 threads and S=16, each thread writes one element. Since there's no bank conflict (each thread writes to a different shared memory bank), and threads in a warp execute in lockstep, the `__syncthreads()` is technically needed for correctness but might be a tiny overhead. Actually, wait. The 16 threads in my block are within one warp (32 threads, 16 active, 16 inactive). Since all active threads are in the same warp and execute in lockstep, do we still need `__syncthreads()`? Yes, because CUDA doesn't guarantee that a thread won't be scheduled ahead. Even within a warp, memory writes might not be visible to other threads in the same warp immediately. So `__syncthreads()` (or at least `__syncwarp()`) is needed. But `__syncthreads()` on a block with only 16 active threads and 16 inactive threads... I wonder how efficient this is. Actually, since threads 16-31 are inactive (masked off), `__syncthreads()` waits for all active threads in the block. I think this is fine. Another thought: I could use `__syncwarp()` instead since all active threads are in the same warp. But `__syncwarp()` is only for threads within a warp and requires all threads in the warp to call it (or be active). Since threads 16-31 are inactive, `__syncwarp()` might hang or have undefined behavior if those threads don't participate. Let me stick with `__syncthreads()` to be safe. Actually, I just realized: with blockDim.x=16, the block consists of threads 0-15, which is exactly half a warp. But CUDA organizes threads into warps of 32. Threads 0-15 are in warp 0, threads 16-31 don't exist. So all threads in the block are in the same warp. `__syncwarp(0xFFFF)` (mask for threads 0-15) should work. But `__syncthreads()` also works and might even be faster for such small blocks because it has less overhead than a full warp sync? I'm not sure. Let me just use `__syncthreads()`. Hmm, let me double check: can I use fewer __syncthreads()? The writes to alpha_curr happen, then we need them visible for the next iteration. Can I use `__threadfence_block()`? That's for global memory coherence. For shared memory within a warp, I might not need any sync if the compiler doesn't reorder. But CUDA compilers can reorder memory operations, so some memory fence is needed. Actually, within a single warp, all threads execute in lockstep, and shared memory has sequential consistency within a thread. When thread 0 writes `alpha_curr[0]`, thread 1 reads `alpha_curr[0]` in the next iteration (after swap). Since they're in the same warp and we have a `__syncthreads()`, it's safe. But do we need it? In practice, without `__syncthreads()`, the compiler might reorder the write to `alpha_curr` after the pointer swap, or the hardware might not have made the write visible. Let me keep `__syncthreads()` for correctness. The overhead should be minimal compared to the arithmetic in the loop. Now let me think about the observation preloading. We have `for (int t = tid; t < T; t += blockDim.x)`. With 16 threads and T=128, each thread loads 8 longs, total 128 loads. Then all threads use `obs_seq`. Good. One more optimization: prefetch. Could we prefetch the next observation while computing the current step? With S=16, the arithmetic is very cheap. But prefetching from shared memory doesn't make sense. Let me not overcomplicate. Let me also think about whether I should process multiple batch elements per block. Since each batch element takes a separate block, and we have 64 blocks total, and each block does 128 * (compute) work. If I process 2 batch elements per block, I'd have 32 blocks, each with 32 threads (2 warps). Each warp handles one batch element independently. This might improve occupancy and reduce block launch overhead. Let me try this: each block has `2 * S = 32` threads. Threads 0-15 handle batch element `2*blockIdx.x`, threads 16-31 handle batch element `2*blockIdx.x + 1`. We'd have `B/2 = 32` blocks. This approach: 1. Halves the number of blocks 2. Each block still needs shared memory for 2 sets of alpha buffers, or we can use registers. 3. Each warp is fully active (32 threads, though each warp is split into 2 independent batch elements) 4. Warps within a block execute independently if they don't need to sync Actually, if threads 0-15 and 16-31 work on different data and don't need to sync, we can remove `__syncthreads()` between them! But they both access shared memory for log_A_s and log_Bt_s, which is read-only after loading. And each half uses its own alpha buffers. Wait, but `__syncthreads()` is a block-level sync. If both halves are in the same block, they need to sync together. But since they don't depend on each other, the sync is harmless but unnecessary. Actually, if we make each batch element use its own block, we avoid this issue. With only 64 blocks, the overhead of extra blocks is minimal. Let me keep it simple. One more idea: use 1 thread per batch element and do everything in registers. There are 255 registers per thread on H100. Each thread needs: - alpha[16] floats = 16 registers - log_A is shared, but each thread needs to read it. If log_A is in shared memory, we need fewer registers. - obs_seq[128] longs = too many for registers (128 * 8 = 1024 bytes, ~256 registers). Can't fit. So registers-only isn't feasible due to the observation sequence. What if we read observations on the fly from global memory? ```cuda for (int t = 1; t < T; t++) { int obs_t = observations[b * T + t]; ... } ``` This avoids storing obs_seq in shared memory or registers. With 64 batch elements, the observations tensor is (64, 128), which is 32KB total. Each batch element reads 512 bytes sequentially. With 64 threads (one per batch), the reads are strided by T=128 longs = 1024 bytes. So thread 0 reads obs[0*T + t], thread 1 reads obs[1*T + t]. These are 1024 bytes apart, so not coalesced. But if we have 16 threads per batch element, thread tid within batch b reads obs[b*T + t], which is the same address for all threads in the block. This is a broadcast, which is actually efficient! So with 16 threads per batch, the observation access is broadcast. We don't need to preload into shared memory. But let me think: is global memory broadcast faster than shared memory? Global memory broadcast: when all threads in a warp read the same address, it gets broadcast. This is fast but still has higher latency than shared memory. However, with the T loop being the outer loop and each t reading a different address, we can't really cache this well in L1 unless the observations for all t fit in L1. For a single batch element, observations are 128 longs = 512 bytes. With 64 batch elements, total is 32KB. This fits in L1 cache! So repeated reads might hit cache. But each block runs once and reads sequentially through the observation array. Actually, let me just try the preloaded version in shared memory. It's cleaner and guaranteed fast. OK I think I've optimized enough. Let me write the complete code. One more thing: `logf(sum)` when `sum` is 0 can produce `-inf`. And `logf(sum)` when `sum < 0` is NaN. But `sum` is a sum of positive exponentials, so it's always >= 0. If all values are -inf (which shouldn't happen due to normalization), sum would be 0 and log(0) = -inf. This is fine and matches PyTorch behavior. Also, `max_val = -1e30f` is a large negative number. Could there be values smaller than this? Since the parameters are normalized log probabilities, and we're adding them up over 128 steps, the minimum reasonable value is about -128 * (something). log(1/16) ≈ -2.77, so over 128 steps it's about -354. And -1e30 is much smaller. So it's safe. But to be absolutely safe, I should initialize with `alpha_prev[0] + log_A_s[0 * S + tid]` in the first iteration. Or I can use: ```cuda float max_val = alpha_prev[0] + log_A_s[0 * S + tid]; for (int i = 1; i < S; i++) { ... } ``` This is more correct but requires an extra add for the first element. Since S is tiny, the overhead is negligible. And it's more numerically stable. Actually wait, is there any case where alpha_prev[0] could be -inf? In principle, yes, if we have zero probabilities. But with normalized parameters, no. Let me keep the explicit initialization to be safe and more accurate: ```cuda float max_val = alpha_prev[0] + log_A_s[tid]; for (int i = 1; i < S; i++) { float val = alpha_prev[i] + log_A_s[i * S + tid]; max_val = fmaxf(max_val, val); } ``` Wait, `log_A_s[0 * S + tid]` is at index `tid`, not `0 * S + tid`... Actually, `i * S + tid` with i=0 gives `tid`. So the first element is `log_A_s[tid]`. Correct. For the sum loop: ```cuda float sum = expf(alpha_prev[0] + log_A_s[tid] - max_val); for (int i = 1; i < S; i++) { sum += expf(alpha_prev[i] + log_A_s[i * S + tid] - max_val); } ``` This saves one subtraction and is slightly more accurate. Let me use this. Similarly for the final reduction: ```cuda float max_val = alpha_prev[0]; for (int i = 1; i < S; i++) { ... } float sum = expf(alpha_prev[0] - max_val); for (int i = 1; i < S; i++) { ... } ``` Good. Let me also consider: is there any benefit to using shared memory for log_A in this scenario? Since each thread within a block reads the entire log_A column by column (for its tid), and across iterations the same log_A values are reused, shared memory helps. But log_A is only 256 floats = 1KB. If all 64 blocks are on the GPU simultaneously, and each has its own 1KB copy, that's 64KB total. L1 cache per SM is much larger (128KB+ on H100). So log_A might already stay in L1 cache if multiple blocks run on the same SM sequentially. But shared memory is still faster than L1 because it's directly addressable without cache lookups. OK, let me finalize the code. I'll compile and test it. One concern: the cpp_extension.load_inline might fail if we call it multiple times with the same name. I should make sure the name is unique. "hmm_forward" should be fine since it only executes once at module load time. Also, I need to make sure the observations tensor is contiguous and on the correct device. Let me write the complete code now.

The problem is that `observations` is being passed as `long` type. Let me check - in PyTorch, `.data_ptr<long>()` might not work correctly. Actually, the issue is that `observations` might not be `int64` on the device where it's used. Let me look at the error --- all trials failed to match, and there's a runtime error: "runtime_error: tensors used as indices must be long, int, byte or bool tensors" This is odd. Let me check if the issue is with `long` in the CUDA code. In CUDA, `long` could be 64-bit or 32-bit depending on platform. Let me use `int64_t` instead for portability. That's likely the issue in the CUDA code. Also, the obs_seq is declared as `long*` and used. Let me use `int64_t` from `<cstdint>` to be safe.

The error says "tensors used as indices must be long, int, byte or bool tensors". This is happening in the torch code (`observations_i64 = observations.to(torch.int64)`) or maybe observations is already int. Actually, looking again, the error says the tensor used as indices must be a specific type. But my `observations.to(torch.int64)` creates a new tensor and passes it to CUDA. So the error might not be from my CUDA code. Wait - this is the SAME error as before. But I changed the type on the Python side. Hmm, maybe the error is actually coming from somewhere in the backward pass or in the reference code that's being run for comparison? No, the reference model should be called independently. Actually wait - maybe `observations` is not coming as the right dtype. The error "tensors used as indices must be long, int, byte or bool tensors" usually comes from indexing with a non-integer tensor. In the reference, `observations[:, 0]` is used as an index. If observations is some floating point type, that would cause issues. But the test setup says `torch.randint(0, num_obs, (batch_size, seq_len))` which should be long. Hmm, could the issue be that in torch, `log_Bt_s[obs_t * S + tid]` in the CUDA kernel is okay, but the error is somewhere else? Wait - the test environment might be using a model comparison that's somehow running my code through different path... Wait, actually I think the error might be in my `_normalize_log` function or some reference comparison. Let me look at the error again: "runtime_error: tensors used as indices must be long, int, byte or bool tensors". This happens when `observations` is passed and it's maybe not an integer type. But the test `get_inputs()` returns `torch.randint` which is `torch.int64`. Let me think... Oh! Maybe the test is checking with a different dtype? Let me just make sure my model handles any observations type by explicitly converting. Actually, the simplest explanation is that the error is happening inside the reference model comparison, which means when it tries to evaluate my model, something goes wrong internally. But the stack trace points to a runtime_error. Wait, I just realized - maybe the `observations` in get_inputs() is actually torch.int32 on some platforms. Let me ensure the conversion happens. Let me try another approach - maybe using torch.utils.cpp_extension.load_inline is causing some issue with the compiled module being cached or something. I should use a unique name. Actually let me re-read the error message: "runtime_error: tensors used as indices must be long, int, byte or bool tensors". In my CUDA wrapper function, I'm using `observations.data_ptr<int64_t>()`. The error message is a Torch error, not a CUDA assert. This means it's happening in Python land, not in my kernel. Where could this happen? The `_normalize_log` function uses `torch.logsumexp` which doesn't use indices. The only indexing happens in: - `log_B[:, observations[:, 0]].t()` - in reference - In my code, the CUDA kernel accesses the observation indices in shared memory. But if my CUDA kernel is launching and failing somehow... Actually no, a CUDA kernel failure would show differently. OH! I think I see it. The issue might be that `load_inline` has compiled a module named "hmm_forward" before, and it's being reused from cache but with the old function signature (that took int, not int64_t). When I changed to int64_t, the cache might still have the old version. I should use a unique name for the compiled module. Or clean the cache. Let me try using a different name. Actually, `load_inline` creates a shared library in a temporary directory. If we call it again with the same name, it may reuse the cached version. Let me use a unique hash in the name.

The compile passed but it seems like the .so cache isn't being regenerated. Let me try a completely different approach: I'll use a much simpler name scheme and try to clean the cache. Actually, the best approach might be to use torch.tensor.long() or explicitly check what the issue is. Wait, looking at this more carefully - the runtime_error says "tensors used as indices must be long, int, byte or bool tensors". This error typically happens during the `.to(torch.int64)` part... No wait, that's not about indexing. Actually, the problem might be: 1. The `.to(torch.int64)` creates a new tensor that's not contiguous properly 2. Or the issue is with how we're referring to module-level variables Let me try copying observation explicitly and use torch.int64 dtype check. Actually wait - let me also make sure the variable names in the CUDA code and Python are aligned. Actually hah! I just realized that I imported torch.utils.cpp_extension before torch in the very first attempt, but then the actual compilation might have cached the torch.utils module... At this point, the caching issue could be very real. Let me try the simplest thing: use torch.int32 for the observations tensor in the CUDA code instead of int64. This avoids any memory layout issues and is what we need. Also, let me try to use a completely unique module name each time (using full hash, plus timestamp if needed). Actually, here's another idea: what if we use int for the observations indexing in CUDA but pass observations as int32? Let me do that.

Let me run correctness check before submitting. I have 22 tool calls remaining so I should verify correctness before submitting.

The issue is that `observations` is coming in as `torch.long` (int64) but my kernel is expecting `int` (int32). I need to explicitly cast observations to int32 in Python before calling the CUDA kernel. Let me also fix the module name issue - I need to use a truly unique name. Let me also make sure I'm using `load_inline` correctly. Actually wait, looking back at the error message - "tensors used as indices must be long, int, byte or bool tensors" - this isn't a CUDA error. This is a PyTorch error that happens when someone tries to use a tensor as an index but it's not an integer type. But this error happens in the reference code, not in my code. Unless... the error is happening in my `_normalize_log` function? No, that's just subtraction and logsumexp. Wait, maybe the error is happening because the observations tensor has the wrong dtype in some runs. Let me just convert to int32 explicitly in my kernel call. Also, I should check if the dtype of observations passed to `hmm_forward_cuda` matters. Since my CUDA kernel does `observations.data_ptr<int>()`, I need the tensor to actually be int32. Let me put `observations = observations.to(torch.int32)` before calling the CUDA kernel. Actually, wait. The error "tensors used as indices must be long, int, byte or bool tensors" can only come from PyTorch indexing. Since my forward method doesn't do any indexing after the CUDA call, maybe the issue is that my `log_Bt` or something isn't right? No, the error should be from the reference or from somewhere else. Looking at the run_correctness output again - it says "runtime_error: tensors used as indices must be long, int, byte or bool tensors". This is during the execution of my ModelNew. But my ModelNew doesn't do any indexing. Unless it's from the `_normalize_log` step... no, that uses `torch.logsumexp` and subtraction. Oh wait! Maybe in some runs `observations` is coming in as a float tensor, not an integer tensor. Let me check by adding `.long()` or `.int()` conversion. Actually, looking at `get_inputs()`, observations are created with `torch.randint(0, num_obs, (batch_size, seq_len))` which returns `torch.long`. But maybe in some test cases they use a different dtype. Let me cast to int32 in Python. Let me also fix the module name properly and use `import time` to ensure uniqueness. I'll also check that the kernel works with int32 observations. Let me submit with these fixes: