reasoning
Let me analyze this HMM forward algorithm implementation carefully.
The reference implementation computes the log-likelihood of observation sequences under an HMM using the forward algorithm in log-space.
Key operations:
1. Normalize log probabilities (log_init, log_trans, log_emit) using logsumexp
2. Initialize log_alpha with log_pi + emit_0
3. For each time step:
- Compute transition: log_alpha[b,j] = logsumexp_i(log_alpha[b,i] + log_A[i,j])
- Add emission probability for current observation
4. Final log-likelihood is logsumexp of final log_alpha
The dimensions are:
- num_states = 16
- num_obs = 64
- seq_len = 128
- batch_size = 64
The main computation is:
1. log_alpha is (B, S) = (64, 16)
2. For each time step (128 steps):
- Expand log_alpha to (B, S, S) = (64, 16, 16)
- logsumexp over dimension 1 to get (B, S)
- Add emission (B, S)
Let me think about how to optimize this with a CUDA kernel.
The logsumexp operation is the key computational challenge. For each batch element and each target state j, we need:
log_alpha_new[b, j] = logsumexp_i(log_alpha[b, i] + log_A[i, j])
This can be computed efficiently using a parallel logsumexp reduction.
For the CUDA kernel, I'll:
1. Pre-normalize the log parameters on the host or in a separate kernel
2. Launch a kernel that processes each batch element in parallel
3. For each time step, compute the transition and emission updates
Let me design the kernel:
Option 1: Single kernel for entire forward pass
- Each thread block handles one batch element
- Shared memory for intermediate results
- Sequential processing of time steps
Option 2: One kernel per time step
- More kernel launch overhead
- But can parallelize across batch elements and states
I'll go with Option 1 since it reduces kernel launch overhead and the state space is small enough (16 states) to fit efficiently in shared memory.
The kernel needs to handle the observation emission lookup, compute logsumexp transitions across states at each timestep, and accumulate the final log-likelihood. For 16 states, a thread block can be configured with 16 threads per batch element to parallelize the state computations, using shared memory reduction for the logsumexp operations.
Since the batch size is 64 and states are 16, I'll assign one thread block per batch element with 16-32 threads per block. The kernel loads normalized parameters into shared memory, then for each timestep, looks up emission probabilities and computes the logsumexp transition in parallel. The logsumexp itself requires finding the maximum value across states and summing exponentials, which can be parallelized across the thread block. ] += log_A[i * S + j];
}
}
__syncthreads();
// Compute logsumexp across sources
float max_val = -INFINITY;
for (int i = 0; i < S; i++) {
max_val = max(max_val, log_alpha_shared[i]);
}
__syncthreads();
float sum = 0.0f;
for (int i = 0; i < S; i++) {
sum += exp(log_alpha_shared[i] - max_val);
}
float logsum = max_val + log(sum);
log_alpha[j] = logsum + emit;
}
}
```
Wait, I'm not handling this correctly. Let me reconsider the data dependencies. For each timestep and target state, I need to compute a logsumexp over all source states, where each source contributes its log probability plus the transition log probability. The shared memory approach lets me accumulate these contributions in parallel, then reduce to find the maximum and compute the sum for the logsumexp operation. The thread's target state j comes from its ID within the block. First, I load log_alpha and log_A values, computing the sum for each source state i and storing in shared memory. After synchronizing, I perform the logsumexp reduction by finding the maximum value across all sources, then summing the exponentials shifted by that maximum. The result is the logsumexp plus the emission probability, which becomes the new log_alpha for that target state.
But wait—there's a potential issue with how the thread block is organized. With blockDim.x = S and blockDim.y = B, each thread handles one (target_state, batch) pair. So threadIdx.x tells me which target state I'm responsible for, and threadIdx.y tells me which batch. I need to reconsider the kernel design. Each batch element needs its own computation, so mapping one block per batch makes sense. Within each block, I'll assign threads to target states, using 32 threads total where 16 handle the actual state calculations and the rest assist with reductions.
The shared memory layout should hold `alpha_acc[S][S]` for accumulating log-sum-exp across source states, plus `log_alpha[S]` for the current alpha values. Each thread computes contributions for its assigned target state j by iterating over source states i. After the first timestep, I need to write out the updated log_alpha values and prepare for the emission step. Wait, I need to reconsider the thread mapping. If blockDim.x = S and blockDim.y = B, then blockIdx.x becomes redundant. The better approach is using blockIdx.y for batch indexing, threadIdx.y for target states, and threadIdx.x for source states to parallelize the summation more effectively. However, with S being small (16), having threads compute serially per target state might actually be more efficient, so I should start with that simpler design and optimize later if needed.
For the initial implementation, each block handles one batch element with one thread per target state, computing the serial summation across source states. I need to implement a logsumexp reduction using shared memory, update log_alpha in registers, and iterate through all timesteps. The kernel accepts the normalized log probabilities (init, transition matrix, and emission distributions) along with the observation sequence, and for each thread I extract its state index and verify it's within bounds.
For the first timestep, I compute the initial log_alpha by combining the log initial probabilities with the emission probability for the first observation, then synchronize threads. For subsequent timesteps, I compute the forward pass by iterating through each source state to accumulate log_alpha[i] plus log_A[i,j] in shared memory, then perform the logsumexp reduction to update log_alpha[j] with the transition probability and current emission.
However, using shared memory this way risks overwriting log_alpha values that other threads still need to read. A better approach is to have each thread directly compute the logsumexp without intermediary shared memory storage.
```cpp
// For each source state i
float max_val = -INFINITY;
float sum = 0.0f;
for (int i = 0; i < S; i++) {
float val = log_alpha[i] + log_A[i * S + j];
max_val = max(max_val, val);
}
for (int i = 0; i < S; i++) {
float val = log_alpha[i] + log_A[i * S + j];
sum += exp(val - max_val);
}
float new_log_alpha = max_val + log(sum) + emit;
```
After computing the updated log_alpha values, I need to synchronize threads before proceeding to the next timestep. The final log-likelihood is obtained by performing a logsumexp reduction across all states using shared memory to find the maximum value and compute the sum. Normalizing the log probabilities ensures they represent proper probability distributions. The log_init, log_trans, and log_emit parameters each get normalized along their respective dimensions using logsumexp, then the forward algorithm proceeds with these normalized values to compute the sequence likelihood.
For the CUDA implementation, I need to handle the normalize_log operation for each parameter. The first observation's emission probabilities combine with the initial state distribution, then the forward recursion iterates through remaining timesteps, computing transition probabilities and incorporating emission likelihoods at each step. The transition computation expands the log_alpha tensor to broadcast with the transition matrix, applies logsumexp reduction along the state dimension, then adds the emission probabilities for the current timestep.
For the CUDA implementation, I need to handle logsumexp reductions and observation lookups. I'll pass pre-normalized parameters to the kernel along with the observation tensor, then structure the kernel code to compute these operations efficiently on the GPU. h.device());
auto out = hmm_forward_cuda(obs, log_pi, log_A, log_B, num_states);
return out;
}
```
I should implement logsumexp normalization on the CPU side using existing PyTorch operations since it's a one-time preprocessing step. Now I'll structure the complete implementation with proper CUDA kernel definitions and the model class.
```cpp
#include <torch/extension.h>
#include <cuda.h>
#include <cuda_runtime.h>
#include <cfloat>
template <typename scalar_t>
__global__ void hmm_forward_kernel(
const int64_t* __restrict__ observations,
const scalar_t* __restrict__ log_pi,
const scalar_t* __restrict__ log_A,
const scalar_t* __restrict__ log_B,
scalar_t* __restrict__ log_lik,
int B, int T, int S, int O) {
int b = blockIdx.x; // batch index
int j = threadIdx.x; // target state index
if (b >= B || j >= S) return;
extern __shared__ char smem[];
scalar_t* log_alpha = (scalar_t*)smem; // [S]
scalar_t* scratch = (scalar_t*)(smem + S * sizeof(scalar_t)); // [S]
// Initialize log_alpha with emission probability for first observation
int obs_0 = observations[b * T + 0];
log_alpha[j] = log_pi[j] + log_B[j * O + obs_0];
__syncthreads();
// Forward pass through remaining timesteps
for (int t = 1; t < T; t++) {
int obs_t = observations[b * T + t];
scalar_t emit = log_B[j * O + obs_t];
// Accumulate log_alpha[i] + log_A[i,j] for all source states i
scalar_t max_val = -INFINITY;
for (int i = 0; i < S; i++) {
scalar_t val = log_alpha[i] + log_A[i * S + j];
max_val = max(max_val, val);
}
scalar_t sum = 0;
for (int i = 0; i < S; i++) {
scalar_t val = log_alpha[i] + log_A[i * S + j];
sum += exp(val - max_val);
}
scalar_t new_alpha = max_val + log(sum) + emit;
scratch[j] = new_alpha;
__syncthreads();
// Update log_alpha for next iteration
if (j < S) {
log_alpha[j] = scratch[j];
}
__syncthreads();
}
```
The kernel computes forward probabilities by accumulating transitions across states using log-space arithmetic to maintain numerical stability. For each timestep, it applies logsumexp to combine previous alpha values with transition probabilities, then adds the emission probability for the current observation. __global__ void hmm_forward_kernel(
const int64_t* __restrict__ observations,
const float* __restrict__ log_pi,
const float* __restrict__ log_A,
const float* __restrict__ log_B,
float* __restrict__ log_lik,
int B, int T, int S, int O) {
int b = blockIdx.x; // batch index
int j = threadIdx.x; // target state index
if (b >= B || j >= S) return;
extern __shared__ float smem[];
float* log_alpha = smem; // [S]
float* scratch = smem + S; // [S]
// Initialize with first emission
int64_t obs_0 = observations[b * T];
log_alpha[j] = log_pi[j] + log_B[j * O + obs_0];
__syncthreads();
// Iterate through time steps
for (int t = 1; t < T; t++) {
int64_t obs_t = observations[b * T + t];
float emit = log_B[j * O + obs_t];
// Compute logsumexp over incoming transitions
float max_val = -INFINITY;
for (int i = 0; i < S; i++) {
float val = log_alpha[i] + log_A[i * S + j];
if (val > max_val) max_val = val;
}
float sum = 0.0f;
for (int i = 0; i < S; i++) {
float val = log_alpha[i] + log_A[i * S + j];
sum += expf(val - max_val);
}
float new_alpha = max_val + logf(sum) + emit;
scratch[j] = new_alpha;
__syncthreads();
// Swap buffers for next iteration
log_alpha[j] = scratch[j];
__syncthreads();
}
// Compute final log-likelihood via logsumexp reduction
float max_alpha = -INFINITY;
for (int i = 0; i < S; i++) {
if (log_alpha[i] > max_alpha) max_alpha = log_alpha[i];
}
scratch[j] = max_alpha;
__syncthreads();
// Sum exponentials across states
float total = 0.0f;
for (int i = 0; i < S; i++) {
total += expf(log_alpha[i] - scratch[i]);
}
if (j == 0) {
log_lik[b] = scratch[0] + logf(total);
}
}
```
Actually, there's a synchronization issue here. I'm writing max_alpha to scratch[j], but then reading from scratch[i] which other threads wrote to. This requires another __syncthreads() between the write and read to avoid race conditions. Also, using a single thread for the final reduction would be more efficient than having all threads participate redundantly. Let me restructure this to have thread 0 handle the final logsumexp computation after all threads have written their max values.
For the shared memory allocation, I'm using dynamic shared memory sized at 2*S*sizeof(float) passed as the third kernel argument. But I should verify this works correctly when blockDim.x is less than S, since only some threads would be writing to the shared buffer.
In my current setup with blockDim.x = S = 16 and blockDim.y = 1, each thread handles one state for its batch element. The observation indexing needs to account for batch and time dimensions properly, so obs_0 should be observations[b * T + 0] and subsequent observations would be observations[b * T + t]. transition probabilities to float upfront for the kernel, and the observation tensor should be converted to long type for indexing. The log-likelihood output needs to match the batch size, and I'm using 32 threads per block with the first 16 handling actual computation while the remainder stay idle. I could alternatively launch exactly 16 threads per block to keep all threads active. Looking back at the reference implementation, log_A has dimensions (S, S) where the first dimension represents source states and the second represents target states. The transition update computes log_alpha[b, i] + log_A[i, j] for each target state j, summing over all source states i. This indexing needs to be reflected in the kernel when loading log_A values.
For the emission lookup, log_B is organized as (S, O) and the indexing `log_B[:, observations[:, t]]` selects the emission probability for each state given the current observation. The `.t()` operation transposes this to (B, S), so each batch element gets a vector of emission probabilities across all states for that timestep. In the kernel, this means for a given observation value `obs_t`, I need to access `log_B[j * O + obs_t]` for each state j to get the corresponding emission probability. I need to set up the shared memory allocation carefully. Each block requires S * 2 * sizeof(float) bytes for the shared arrays. When launching the kernel with B blocks and S threads per block, I should ensure S is a power of 2 for optimal warp efficiency, though the actual computation still uses num_states threads.
The memory access pattern could become a bottleneck since each thread reads all S values of log_alpha sequentially during the transition step, creating potential bank conflicts. I should consider alternative layouts like having threadIdx.x handle source states instead, but the current approach should work for correctness first. For the logsumexp operation specifically, I'm computing the maximum value across states, then accumulating the exponentials of deviations from that maximum.
Once I have the sum of exponentials, I can compute the log-sum as max_val + log(sum). To update the state probabilities, I'll store the result in shared memory, synchronize threads to ensure all writes complete, then read back the updated values for the next iteration. After processing all timesteps, a final logsumexp reduction across states gives the log-likelihood for each batch element.
Rather than having all threads participate redundantly, I can have thread 0 handle the entire computation serially for efficiency. It iterates through all states to find the maximum log probability, then accumulates the sum of exponentials normalized by that maximum, finally storing the result. Since thread 0 is the only one writing to the output array, no synchronization is needed afterward.
For the Python wrapper, I need to handle parameter normalization first—computing log probabilities for the initial, transition, and emission matrices using logsumexp along the appropriate dimensions. Then I'll extract the batch and sequence dimensions from the observations tensor and prepare the output tensor for the log-likelihoods. I should wrap this as a proper PyTorch module with parameters, but first I want to verify the core kernel logic is sound. Looking at the transition accumulation loop, I'm iterating through all source states correctly, computing the log probability contributions for each one. I need to verify the indexing matches my PyTorch reference. The log_A matrix uses `log_A[i * S + j]`, which accesses element (i, j) in row-major format. My reference code computes `log_alpha[b, i] + log_A[i, j]` and takes logsumexp over dimension i, then indexes the result at position j for the target state—this aligns with what the kernel is doing. The indexing is correct. Now I need to compile and test the kernel.