reasoning
Let me analyze the reference model. It's a batched Metropolis-Hastings MCMC step with:
- Input x: (B, dim)
- Proposal: x + step_size * randn_like(x)
- log_prob: -0.5 * sum(((x - mu) / sigma)^2)
- log_alpha = log_prob(proposal) - log_prob(x)
- log_u = log(rand(B)).clamp(min=1e-10)
- accepted = log_u < log_alpha
- x_new = torch.where(accepted.unsqueeze(-1), proposal, x)
This is a memory-bandwidth bound operation because:
1. We need to read x, mu, sigma
2. Generate random numbers for proposal and log_u
3. Compute log_prob differences
4. Write x_new and accepted
For B=4096 and dim=64, total elements per batch = 262144.
The PyTorch reference does this with multiple separate kernel launches:
1. torch.randn_like(x) - generates random numbers
2. proposal = x + step_size * noise - element-wise
3. log_prob(proposal) - reads proposal, mu, sigma, computes, reduces
4. log_prob(x) - reads x, mu, sigma, computes, reduces
5. log_alpha = log_prob(proposal) - log_prob(x)
6. torch.rand(x.shape[0]) - generates random numbers
7. log_u = log(rand).clamp(min=1e-10)
8. accepted = log_u < log_alpha
9. x_new = torch.where(accepted.unsqueeze(-1), proposal, x)
We can fuse most of this into a single CUDA kernel to reduce memory bandwidth. The key operations are:
- Generate proposal noise (needs random numbers)
- Generate uniform random for acceptance (needs random numbers)
- Compute log_prob of x and proposal
- Compute acceptance
- Select x_new
However, generating random numbers inside a CUDA kernel is non-trivial. We can use the Philox random number generator from cuRAND or implement a simple one.
Actually, wait - we need to think about this carefully. The reference uses `torch.randn_like(x)` and `torch.rand(x.shape[0])`. If we fuse everything into one kernel, we need to generate random numbers with the same distribution. But we don't need to match the exact same random numbers - just the distribution, since it's a stochastic algorithm.
For maximum performance on H100, I should fuse everything into one kernel. Let me think about how to generate random numbers. We can use Philox4_32_10 which is deterministic and can be seeded.
Actually, for a fused kernel, each thread can handle one element of (B, dim). But wait, we also need per-batch random numbers for the acceptance (B numbers) and per-element random numbers for the proposal (B*dim numbers).
Let me structure the kernel:
- Each thread block handles multiple rows (batches)
- Threads within a block can cooperate on row reductions for log_prob
Actually, let me think about this more carefully. For each batch element b:
1. We need dim normal random numbers for proposal
2. We need 1 uniform random number for acceptance
For log_prob computation, we need to sum dim elements.
A good strategy might be:
- Use 1D thread blocks where each thread handles one (b, d) pair
- Use shared memory for row-wise sums (log_prob_x and log_prob_proposal per batch)
- Then have threads in the first warp compute acceptance and write results
But actually, with B=4096 and dim=64, we have 262K elements total. On H100 with huge bandwidth, we might not need complex reductions.
Let me think about a simpler approach first. What if we just write a straightforward fused kernel where each thread handles one element:
1. Thread (b, d) generates a normal random number
2. Computes proposal = x + step_size * noise
3. Writes proposal to output or shared memory
4. Computes contribution to log_prob_x and log_prob_proposal
5. Uses atomicAdd or shared memory reduction for the row sums
With dim=64, we could have each block handle one row with 64 threads, then use warp shuffle for reduction. 64 threads is small but might be fine.
Or we could have each block handle multiple rows. Let's say block size = 256 threads. Each block handles 256/dim = 4 rows. Then we need shared memory for 4 sums.
Actually, for simplicity and coalescing, let me use a 2D grid where:
- Each block has 32x8 = 256 threads (32 in dim direction, 8 in batch direction)
- This way we get good memory coalescing
But reduction across dim for each batch element might be tricky.
Let me try a simpler 1D thread assignment: each thread handles one (b,d) pair.
- Thread tid handles element at flat index tid
- b = tid / dim, d = tid % dim
- After computing local contributions, we reduce within each row
For reduction, we could use shared memory. With blockDim.x = 256 and dim = 64:
- If we let each block handle 4 complete rows (4*64=256 threads), then each thread computes one element
- We need shared memory for 4 partial sums
- Each warp handles 64/32 = 2 elements per thread for a given row
Wait, 64 threads would handle one row if blockDim.x = 64. But to fully utilize warps, we want blockDim.x to be a multiple of 32.
Let me try blockDim.x = 128, handling 2 rows (2 * 64 = 128 threads).
- Each thread handles one (b,d) element
- We use shared memory s_log_p[2] and s_log_p_prop[2]
- Each thread atomicAdds its contribution
- Then thread 0 of each row computes acceptance and fills in x_new
Actually, atomicAdd to shared memory for floats is supported and should be fast for only 64 threads contending.
Wait, a cleaner approach: blockDim.x = 32 (one warp per row), blockDim.y = rows_per_block.
- Each warp handles one row with 32 threads, each doing 2 elements (since dim=64)
- Warp shuffle reduction for sum
- This avoids shared memory for reduction entirely
Let me think more carefully:
- gridDim = (B / rows_per_block, 1)
- blockDim = (32, rows_per_block)
- Each warp handles one row
- Thread i in warp j handles elements d = i and d = i+32
That sounds good. With B=4096, if rows_per_block = 8, we have 4096/8 = 512 blocks.
For random number generation, we can use Philox. Each thread needs separate counter and key.
- We can base the seed on (blockIdx.x, threadIdx.x, threadIdx.y) and some global seed.
Actually, for simplicity and to match PyTorch's RNG behavior approximately, let's just use a simple but good RNG. We need:
- Normal distribution for proposal noise
- Uniform distribution for acceptance
We can use the Box-Muller transform or the Ziggurat algorithm for normal. But for simplicity, let's use the Central Limit Theorem approximation or just a simple Box-Muller with uniform inputs.
Actually, cuRAND isn't easily available in inline CUDA unless we include the headers. But we can easily implement Philox since it's just integer operations, then convert to uniform, then to normal via Box-Muller.
Wait, Box-Muller needs sin/cos which is expensive. For dim=64, we might want to use the polar method or just accept the cost.
Alternatively, since exact RNG sequence doesn't matter (it's stochastic), we can use a simple LCG or xorshift.
Actually, for maximum speed, let me check if we can use the built-in curand header by including <curand_kernel.h>. Yes, we can use it with `torch.utils.cpp_extension.load_inline`.
Let me write a fused kernel using curand:
```cuda
#include <curand_kernel.h>
__global__ void mcmc_kernel(const float* x, const float* mu, const float* sigma,
float step_size, int B, int dim, unsigned long long seed,
float* x_new, bool* accepted) {
// Each block handles multiple rows
int row = blockIdx.x * blockDim.y + threadIdx.y;
int col = threadIdx.x;
curandStatePhilox4_32_10_t state;
// Initialize RNG
curand_init(seed, row * dim + col, 0, &state);
if (row >= B) return;
float log_p_x = 0.0f;
float log_p_prop = 0.0f;
// Each thread in warp handles a subset of columns
// For dim=64 and warp_size=32, each thread handles 2 columns
for (int c = col; c < dim; c += 32) {
float x_val = x[row * dim + c];
float noise = curand_normal(&state) * step_size;
float prop_val = x_val + noise;
float diff_x = (x_val - mu[c]) / sigma[c];
log_p_x += -0.5f * diff_x * diff_x;
float diff_prop = (prop_val - mu[c]) / sigma[c];
log_p_prop += -0.5f * diff_prop * diff_prop;
// Store proposal temporarily - we can write to x_new and overwrite if rejected
x_new[row * dim + c] = prop_val;
}
// Warp reduce
#pragma unroll
for (int offset = 16; offset > 0; offset /= 2) {
log_p_x += __shfl_down_sync(0xffffffff, log_p_x, offset);
log_p_prop += __shfl_down_sync(0xffffffff, log_p_prop, offset);
}
// Thread 0 generates uniform random and decides acceptance
if (col == 0) {
float u = curand_uniform(&state);
u = fmaxf(u, 1e-10f); // clamp minimum
float log_u = logf(u);
float log_alpha = log_p_prop - log_p_x;
bool acc = log_u < log_alpha;
accepted[row] = acc;
if (!acc) {
// Need to copy x to x_new for this row
// Have other threads help? Or just do it ourselves
// For dim=64, thread 0 can copy all
for (int c = 0; c < dim; c++) {
x_new[row * dim + c] = x[row * dim + c];
}
}
}
}
```
Hmm, there's an issue. If a row is rejected, only thread 0 copies x back. But it's okay since we wrote proposal first, and if rejected, thread 0 overwrites. No race condition because only thread 0 writes in the rejection case.
Wait, there IS a race condition! All threads write proposal to x_new first. Then thread 0 might overwrite those with x values. In CUDA, memory operations within a warp are not guaranteed to serialize in any particular order without __syncwarp(). So thread 0 might read/write before other threads in the warp finish their writes. We need __syncwarp() before thread 0 does the copy.
Let me revise:
```cuda
// Write proposal first
for (int c = col; c < dim; c += 32) {
x_new[row * dim + c] = prop_val;
}
__syncwarp();
// Thread 0 decides and potentially overwrites
if (col == 0) {
// ... compute acceptance ...
if (!acc) {
for (int c = 0; c < dim; c++) {
x_new[row * dim + c] = x[row * dim + c];
}
}
}
```
But wait, `__syncwarp()` only synchronizes threads in the warp. Since all threads in this row are in the same warp (they have the same threadIdx.y, and threadIdx.x ranges 0-31), this should be fine. After __syncwarp(), all proposal writes are visible to thread 0.
Actually, let me reconsider. We could avoid the race entirely by having threads only write after the decision. Thread 0 computes acceptance and stores the bool. Then all threads check and write either proposal or x. But that requires broadcasting the bool to all threads.
```cuda
__shared__ bool s_acc[rows_per_block];
// ...
if (col == 0) {
s_acc[threadIdx.y] = acc;
}
__syncwarp();
bool acc_val = s_acc[threadIdx.y];
for (int c = col; c < dim; c += 32) {
x_new[row * dim + c] = acc_val ? prop_val : x[row * dim + c];
}
```
This is cleaner! No race conditions.
Wait, but shared memory for just one bool per row... with rows_per_block=8, that's tiny. Actually we can use warp broadcast:
```cuda
bool acc_val = __shfl_sync(0xffffffff, acc, 0);
for (int c = col; c < dim; c += 32) {
x_new[row * dim + c] = acc_val ? prop_val : x_val;
}
```
Even better! `__shfl_sync` broadcasts from thread 0 to all threads.
Let me refine the kernel:
```cuda
#include <curand_kernel.h>
__global__ void mcmc_kernel(const float* x, const float* mu, const float* sigma,
float step_size, int B, int dim, unsigned long long seed,
float* x_new, bool* accepted) {
int row = blockIdx.x * blockDim.y + threadIdx.y;
int col = threadIdx.x;
if (row >= B) return;
curandStatePhilox4_32_10_t state;
curand_init(seed, row * dim + col, 0, &state);
float log_p_x = 0.0f;
float log_p_prop = 0.0f;
for (int c = col; c < dim; c += 32) {
float x_val = x[row * dim + c];
float noise = curand_normal(&state) * step_size;
float prop_val = x_val + noise;
float diff_x = (x_val - mu[c]) / sigma[c];
log_p_x += -0.5f * diff_x * diff_x;
float diff_prop = (prop_val - mu[c]) / sigma[c];
log_p_prop += -0.5f * diff_prop * diff_prop;
// We'll write after computing acceptance
// Need to keep prop_val - we can recompute or store in registers
// For dim=64 with 2 elements per thread, we can store in local array
}
// Warp reduce
#pragma unroll
for (int offset = 16; offset > 0; offset /= 2) {
log_p_x += __shfl_down_sync(0xffffffff, log_p_x, offset);
log_p_prop += __shfl_down_sync(0xffffffff, log_p_prop, offset);
}
bool acc = false;
if (col == 0) {
float u = curand_uniform(&state);
u = fmaxf(u, 1e-10f);
float log_u = logf(u);
float log_alpha = log_p_prop - log_p_x;
acc = log_u < log_alpha;
accepted[row] = acc;
}
acc = __shfl_sync(0xffffffff, acc, 0);
// Now write the result
// But we need prop_val for each c again! We recomputed it.
// For dim=64 and 2 elements per thread, we can just regenerate the noise
// Actually we should store prop_val. Let's use a small array.
}
```
Hmm, we need to either store prop_val or recompute it. With only 2 elements per thread (dim=64, warp=32), a 2-element array is fine. But what if dim changes? The constructor can take any dim.
Actually, wait. In the problem statement, the reference implementation takes `dim` as a constructor argument and uses it in `forward`. So our kernel needs to handle arbitrary dim. But the benchmark only tests with dim=64.
For correctness with arbitrary dim, we can't use fixed-size arrays. But we can recompute the noise if we re-seed the RNG the same way. Or we can store prop_val in registers with dynamic allocation... actually we can use a loop that handles one element at a time, storing in a register.
Actually, since the problem says the kernel should accept the same constructor arguments and implement forward with the same signature, it should work for any dim. But the benchmark might only test with dim=64.
Let me think of a cleaner approach. We can do TWO passes:
1. First pass: compute proposal and log_probs, reduce
2. Thread 0 decides acceptance
3. Second pass: write x_new based on acceptance
For pass 1, we can store proposal in a temporary buffer or recompute. Using a temporary buffer is simplest and most general. But that uses extra memory.
Alternatively, we can recompute the noise. If we re-seed the RNG with the same seed for pass 2, we get the same noise. That's elegant!
Wait, but we advance the RNG state in pass 1 when we call curand_normal. So to get the same numbers, we'd need to save and restore the state, or use a separate state initialized the same way. Or, we can just save the state after pass 1 and use it for... no, that won't give the same normals unless we rewind.
Actually, the simplest approach: initialize TWO curand states - one for the proposal normals and one for the uniform acceptance. Or better: compute all normals first, then after the decision, reinitialize the state and recompute.
Reinitializing curand_init might be expensive. Let me think...
Actually, for dim up to maybe a few hundred, we can just recompute. With Philox, curand_init is not that expensive - it's mainly setting up the key and counter. But calling it twice per thread is wasteful.
Alternative: use a simple custom RNG. For example, a simple xorshift+ that we can "rewind" by saving the state. Or we can just generate all the normals and store them. But with B=4096 and dim=64, storing all proposal values is B*dim*4 = 1MB, which is not much. Actually wait, we could just write proposal to x_new first, then if rejected, copy x back. But we had the race condition issue.
With __syncwarp() it's safe:
1. All threads write proposal to x_new
2. __syncwarp()
3. Thread 0 computes acceptance
4. If rejected, thread 0 copies x over x_new
Wait, is there a race between step 1 and step 4? After __syncwarp(), all writes from step 1 are guaranteed to be visible to all threads in the warp. Then thread 0 starts copying x. But what if another thread (from the same warp) is still executing step 1? No - __syncwarp() ensures all threads have reached this point. So it's safe.
But actually, the order of memory operations matters. __syncwarp() is a memory barrier for threads in the warp, so all prior memory stores are visible. Yes, this should work.
Actually, let me double check: in the CUDA memory model, `__syncwarp()` provides a memory fence. It ensures that all memory accesses by the calling threads before the call are completed before any thread in the warp executes any instructions after the call. So yes, this is safe.
Let me write the kernel again:
```cuda
#include <curand_kernel.h>
__global__ void mcmc_kernel(const float* x, const float* mu, const float* sigma,
float step_size, int B, int dim, unsigned long long seed,
float* x_new, bool* accepted) {
int row = blockIdx.x * blockDim.y + threadIdx.y;
int col = threadIdx.x;
if (row >= B) return;
curandStatePhilox4_32_10_t state;
curand_init(seed, row * dim + col, 0, &state);
float log_p_x = 0.0f;
float log_p_prop = 0.0f;
for (int c = col; c < dim; c += 32) {
float x_val = x[row * dim + c];
float noise = curand_normal(&state) * step_size;
float prop_val = x_val + noise;
float diff_x = (x_val - mu[c]) / sigma[c];
log_p_x += -0.5f * diff_x * diff_x;
float diff_prop = (prop_val - mu[c]) / sigma[c];
log_p_prop += -0.5f * diff_prop * diff_prop;
x_new[row * dim + c] = prop_val;
}
__syncwarp();
if (col == 0) {
float u = curand_uniform(&state);
u = fmaxf(u, 1e-10f);
float log_u = logf(u);
float log_alpha = log_p_prop - log_p_x;
bool acc = log_u < log_alpha;
accepted[row] = acc;
if (!acc) {
for (int c = 0; c < dim; c++) {
x_new[row * dim + c] = x[row * dim + c];
}
}
}
}
```
This looks correct. But wait, there's a subtle issue: when thread 0 copies x over x_new in the rejection case, other threads in the warp see this. But since thread 0 is the only one writing after __syncwarp(), and other threads are either done or waiting at the next instruction, there's no race. Thread 0 does all the copying.
However, there's a potential performance issue: thread 0 does a lot of serial work (copying dim elements) while other threads in the warp are idle. For dim=64, that's 64 loads and 64 stores. Not too bad, but we could parallelize it.
Actually wait, I just realized: the other threads in the warp are NOT idle - they continue past the `if (col == 0)` block. If we add `__syncwarp()` after the if block to ensure thread 0 finishes before anyone reads x_new... but in this kernel, no one reads x_new after, so it's fine.
Actually, if we want to be more efficient, we can have all threads participate in the copy-back using predicated execution. But with the current structure, it's not possible because only thread 0 knows the result. We'd need to broadcast first:
```cuda
__shared__ bool s_acc[rows_per_block]; // or use __shfl_sync
if (col == 0) {
// ... compute acc ...
s_acc[threadIdx.y] = acc;
}
__syncwarp();
bool acc = s_acc[threadIdx.y];
for (int c = col; c < dim; c += 32) {
float x_val = x[row * dim + c];
float prop_val = x_new[row * dim + c]; // we already wrote this
x_new[row * dim + c] = acc ? prop_val : x_val;
}
```
Wait, this has a race too! Thread col might read x_new[col] before thread (col+32) writes x_new[col+32]? No, x_new[col] is only written by thread col. So reading it back is safe because we wrote it ourselves.
Actually, the first version's race condition was because thread 0 might write x_new[c] while thread c is still writing proposal. With __syncwarp() in between, it's fine. But then thread 0 does serial copy.
With the approach of "write proposal, sync, broadcast acc, rewrite with select", all threads can participate but we need to re-read x_new. x_new is in global memory, so reading it back has latency. But since each thread just wrote it, it might still be in cache (L1).
Actually, a cleaner approach: don't write proposal to x_new at first. Write it nowhere. After computing acceptance, broadcast it, then all threads compute proposal again and write either proposal or x. We just need to recompute the noise, which means reusing the RNG state... but we consumed the normal randoms already.
OR: we can just store prop_val in registers. Each thread handles dim/32 elements. For dim=64, that's 2 elements. We can store them in a register array. But for general dim, we'd need a dynamic size.
Actually, we can make a specialized fast path for dim <= some value (like 128), and a general path otherwise. But let's keep it simple.
For the register approach with runtime dim: we can loop twice. First loop computes and accumulates log probs. Second loop reinitializes RNG and recomputes.
Wait, double initialization of curand per thread might be acceptable. Let me check how expensive curand_init is. For Philox, it's moderately expensive but maybe okay for our problem size. Alternatively, we can use a simple hash-based RNG.
Actually, let me use a custom simple RNG. For normal distribution, we can use the Box-Muller transform on a simple uniform RNG.
For uniform in [0,1), we can use a 32-bit integer hash and divide by 2^32:
```cpp
inline float uint32_to_float(uint32_t x) {
return (x >> 8) * 0x1.0p-24f; // x / 2^24 in [0, 1)
}
```
Or we can use:
```cpp
inline float uint32_to_float_open(uint32_t x) {
return (x + 0.5f) * (1.0f / 4294967296.0f);
}
```
For a simple hash, we can use MurmurHash3 or xxHash or just a simple LCG/xorshift.
Actually, let's use a very simple and fast generator: SplitMix64 or xoshiro128+. But we need to seed it per thread.
Here's a simple approach using a hash function:
```cpp
uint32_t hash(uint32_t x) {
x = ((x >> 16) ^ x) * 0x45d9f3b;
x = ((x >> 16) ^ x) * 0x45d9f3b;
x = (x >> 16) ^ x;
return x;
}
```
No wait, we need a stream of random numbers, not just one hash. Better to use an LCG that we can step:
```cpp
uint32_t lcg(uint32_t *state) {
*state = 1664525u * (*state) + 1013904223u;
return *state;
}
```
For normal distribution, we can use the Box-Muller transform. But that requires pairs of uniforms per normal. For dim elements, we need dim normals, which means dim*2 uniforms. Not a big deal, but we need to handle the pairing.
Actually, for performance, since exact RNG doesn't matter, let me just use curand. It's well-tested and likely optimized.
But I want to avoid recomputing by reinitializing. So let me try the "write proposal, sync, thread 0 serial copy back" approach. For dim=64, 64 memory operations on one thread is maybe 100-200ns. With 512 blocks and 8 warps each, the GPU will have plenty of other warps to run. Let's see if this is a bottleneck.
Actually, let me think about this differently. With 4096 rows and each row taking ~the same time, we have perfect parallelism. The issue is within a warp: if thread 0 takes longer, the warp stalls. But for dim=64 with 2 iterations of the first loop, then warp shuffle (fast), then thread 0 does ~64 memory ops while other 31 threads are idle. If 64 memory ops take ~200 cycles at 3000MHz = ~67ns, and the whole kernel takes maybe 200ns, then this is a 33% overhead. Significant but maybe still faster than PyTorch.
A better approach: have all warps write proposal, sync, then use `__shfl_sync` to broadcast acc, and have all threads conditionally write. To avoid recomputing, we read back from x_new:
```cuda
// Phase 1: write proposals
for (int c = col; c < dim; c += 32) {
float x_val = x[row * dim + c];
float noise = curand_normal(&state) * step_size;
x_new[row * dim + c] = x_val + noise;
// ... compute log probs ...
}
__syncwarp();
// Phase 2: decide acceptance
bool acc;
if (col == 0) {
// ... compute acc ...
accepted[row] = acc;
}
acc = __shfl_sync(0xffffffff, acc, 0);
// Phase 3: fix up rejected rows
for (int c = col; c < dim; c += 32) {
if (!acc) {
x_new[row * dim + c] = x[row * dim + c];
}
}
```
Wait, this does have a subtle issue. In phase 3, thread c reads x_new[row*dim+c]... no, it just writes x[row*dim+c] to x_new[row*dim+c]. But we already wrote proposal in phase 1. If acc=true, we do nothing (proposal stays). If acc=false, we overwrite with x. This is correct as long as the phase 1 writes are visible before phase 3 begins, which __syncwarp() ensures.
Actually wait, do we need another __syncwarp() between phase 2 and phase 3? Thread 0 is the only one computing acc and writing accepted[]. But all threads need to know acc before phase 3. __shfl_sync handles that - it's synchronous. So no additional sync needed.
Let me also reconsider: __syncwarp() after phase 1 ensures all phase 1 writes are visible to all threads in the warp. After that, thread 0 proceeds to compute acc. Other threads also proceed past __syncwarp(), execute __shfl_sync(), get the acc value, and proceed to phase 3. But thread 0 might still be computing acc when other threads call __shfl_sync(). __shfl_sync(0xffffffff, acc, 0) will block until thread 0 reaches it. But thread 0 is computing log_u, log_alpha, etc. So there's some divergence, but after thread 0 finishes and reaches __shfl_sync, all threads continue.
Actually, __shfl_sync is a warp-level intrinsic where all threads in the mask must execute it. If thread 0 is the source, it must reach the instruction for the shuffle to complete. If other threads reach __shfl_sync before thread 0, they block until thread 0 also reaches it. This is fine.
Let me verify: in the code, thread 0 is inside `if (col == 0)`. Other threads are NOT inside this if. Will other threads still execute __shfl_sync? Yes, because __shfl_sync is outside the if block. So:
- Threads 1-31 execute __syncwarp(), hit the if (false), then execute __shfl_sync(), block waiting for thread 0.
- Thread 0 executes __syncwarp(), enters if (true), computes acc, then hits __shfl_sync(), broadcasts acc to all.
This is correct.
Now, one more concern: when dim is exactly 64, and col ranges 0-31, the first loop iterates with c=0,32. This gives 2 iterations per thread, fine.
For the log_prob computation, we need ((x - mu) / sigma)^2. Since mu and sigma are buffers. In the reference, sigma is ones(dim). But we should implement the general case.
Actually, I can optimize the log_prob a bit: when sigma is all ones (which it is in the reference), we can skip the division. But the constructor might be called with different sigma values in the future? Actually, looking at the reference, sigma is always `torch.ones(dim)`. We could check if it's all ones at runtime... but that's extra logic. Let me just keep it general.
Actually, wait. The `__init__` in Model creates `mu = zeros(dim)` and `sigma = ones(dim)`. In `ModelNew` we should do the same. But should we optimize for this specific case? Probably not necessary for speed - a division on H100 isn't that slow compared to memory.
Hmm, but one thing: in the reference, `_log_prob` computes `((x - self.mu) / self.sigma) ** 2`. When sigma is ones, this simplifies to `(x - mu)^2`. We can check if sigma is 1.0 to skip division. But this adds a branch. Maybe not worth it.
Let me also think about launching the kernel. We need to pass seed. We can use a fixed seed or the current time.
Now, let me also think about whether we should handle the case where step_size, mu, sigma are on CPU or GPU. In the reference, they are registered as buffers on the same device. In our ModelNew, we'll keep them as buffers too.
Actually, looking at the problem more carefully: the `forward` takes `x` as input, and `mu`, `sigma` are module buffers. We need to ensure they're on the same device as x.
Let me draft the full code now.
Wait, one more thing: the `accepted` return type in reference is a boolean tensor. In PyTorch, `torch.where(accepted.unsqueeze(-1), ...)` where accepted is bool. And the return is `x_new, accepted` where accepted is bool.
In CUDA, we'll write `bool* accepted`. When we create the output tensor, we should use `torch::kBool` or `torch::kUInt8`. Let me check: in PyTorch C++ API, `at::Tensor` with `at::kBool` exists. But `bool*` in CUDA might have alignment issues. Usually 1 byte per bool.
Actually, in PyTorch's `TensorAccessor<bool, 1>`, bool is 1 byte. But in the kernel, `bool*` should work. Or we can use `uint8_t*` and set to 0 or 1. Let me just use `bool*`.
Wait, in C++ `bool` is 1 byte, and in CUDA it's also typically 1 byte for `bool`. But sometimes nvcc treats it specially. To be safe, let me use `unsigned char*`.
Actually, in PyTorch CUDA kernels, `bool` is commonly used. Let me check... yes, `bool` is fine.
Now let me write the complete Python module.
One concern with Philox + curand_normal: it returns `float` with mean 0 and std 1. `curand_uniform` returns (0, 1]. The reference uses `torch.rand` which is [0, 1). Actually, `torch.rand` is uniform on [0, 1), and then `.clamp(min=1e-10)` makes it [1e-10, 1).
`curand_uniform` returns (0, 1]. That's slightly different - it can return values arbitrarily close to 0 but never exactly 0, and can return exactly 1. PyTorch's `torch.rand` can return exactly 0 but not exactly 1. After clamp(min=1e-10), both exclude 0 below 1e-10.
Actually, `curand_uniform` documentation says it returns (0.0, 1.0]. That's fine because we clamp anyway. Wait, the reference clamps to min=1e-10. Since curand_uniform never returns exactly 0, we're fine on the lower end. It could return 1.0, and log(1.0) = 0.0, which is also fine.
Hmm, but `torch.rand` gives [0, 1) and curand_uniform gives (0, 1]. The difference is negligible for a Monte Carlo method, but for exact numerical equivalence... the problem says outputs must match within tolerance, and the method is stochastic, so exact sequence shouldn't matter. However, the test harness might call the model multiple times and check statistical properties or something. More likely, they call it once with a fixed input and check that the output matches within tolerance.
If they use a fixed seed for PyTorch's RNG... our kernel would produce different random numbers and therefore different outputs. That would cause the correctness check to fail if they compare element-wise.
OH WAIT. The evaluation might not fix the seed, and since it's a stochastic operation, they might call it multiple times and check tolerance. But if they call it once and compare exact values, we'd fail because of different RNG.
Hmm. Actually, the problem says "numerically equivalent outputs at fp32 precision" with tolerance 1e-4. For a stochastic algorithm producing different random numbers, the outputs would be completely different, not within 1e-4.
This means the evaluation must either:
1. Not check exact values for stochastic outputs, or
2. Seed both the reference and our model, or
3. Call it multiple times and average, or
4. Use a deterministic test where they somehow make the randomness match.
Looking at the reference code, there's no seeding. The evaluation harness probably seeds PyTorch's global RNG before calling both models with the same seed, or they might check that the outputs have the right structure/shape/types.
Wait, but for correctness they need the same outputs. Since the reference uses `torch.randn_like` and `torch.rand`, which use PyTorch's CUDA RNG, and our kernel uses curand, we'd get different random numbers unless we explicitly call PyTorch's RNG or match it somehow.
I think the safest approach is: DON'T use a custom RNG. Instead, call PyTorch's RNG from within the kernel? That's not possible with `torch.utils.cpp_extension.load_inline` easily.
Alternative: generate the random numbers using PyTorch operations first, then pass them to the kernel. This way we use the same RNG as the reference.
So the approach is:
1. In Python `forward()`, call `torch.randn_like(x)` and `torch.rand(x.shape[0], device=x.device)` just like the reference
2. Pass these to a custom kernel that fuses everything else
This ensures we match PyTorch's RNG exactly! The kernel then only does the arithmetic parts (addition, log_prob computation, comparison, where).
Actually, this is much cleaner and guarantees correctness. We lose some performance from not fusing the RNG, but:
- PyTorch's `torch.randn_like` and `torch.rand` are highly optimized
- The main speedup comes from fusing the log_prob reductions and the element-wise operations into fewer kernel launches
- We still reduce from ~5-6 kernel launches to 1
Let me estimate the reference launches:
1. `torch.randn_like(x)` - CUDA RNG kernel
2. `proposal = x + step_size * noise` - element-wise
3. `_log_prob(proposal)` - element-wise + reduction
4. `_log_prob(x)` - element-wise + reduction (but this could be computed once and cached? No, x changes each call)
5. `log_alpha = lp_prop - lp_x` - element-wise
6. `torch.rand(x.shape[0])` - CUDA RNG kernel
7. `log_u = torch.log(rand).clamp(min=1e-10)` - element-wise
8. `accepted = log_u < log_alpha` - element-wise
9. `x_new = torch.where(...)` - element-wise
That's a lot of kernel launches! Our fused kernel would replace steps 2-9 except the RNG.
Actually, even better: we can generate the random numbers inside the kernel but using the same seed/state as PyTorch. That's hard.
Or: we can pre-generate the random numbers in Python and then fuse all the math. Let me count what we still save:
- Reference: 8-10 kernel launches (including RNG)
- Our approach: 2 RNG kernels + 1 fused math kernel = 3 kernel launches
But we can do even better. The `_log_prob` does reduction over dim. In PyTorch, `(a ** 2).sum(dim=-1)` on a [B, dim] tensor launches a reduction kernel. For B=4096, dim=64, this is a small reduction, which is memory-bandwidth bound.
Our fused kernel can compute both log_probs and do all the rest in one pass.
Actually, let me see if we can do even better by also fusing one of the RNG generations into our kernel while keeping PyTorch's sequence intact? No, that's too risky.
I think 2 PyTorch RNG calls + 1 custom fused kernel is a solid approach.
Wait, actually I just realized: can we avoid generating `torch.rand(x.shape[0])` separately by generating uniforms from the same `torch.randn_like` noise? No, that doesn't make sense.
Or: can we use `torch.rand` to generate both the proposal noise and the acceptance uniforms in one call? Like `torch.rand(B, dim+1)`? But then we'd need to convert to normal distribution for the first dim elements. We could use the Box-Muller transform in the kernel. But that still requires one call to `torch.rand` instead of `torch.randn_like` + `torch.rand`. That might be faster.
Hmm, but generating normals via Box-Muller from uniforms is extra computation. PyTorch's `torch.randn` uses the Ziggurat method or Box-Muller, which is highly optimized.
Let me benchmark mentally: PyTorch's `torch.randn_like` on 262K elements is probably very fast (a few microseconds). The overhead of an extra kernel launch is maybe 5-10 microseconds. So 2 RNG calls + 1 fused kernel might be 20-30 microseconds total, while the reference might be 50-80 microseconds.
But let me think about whether we can do better by implementing our own RNG that matches PyTorch... Actually, PyTorch uses Philox with a specific state. We could read PyTorch's global RNG state and advance it, but that's very hacky and fragile.
OK, I'm overthinking this. Let me go with: use PyTorch's `torch.randn_like` and `torch.rand` in Python, pass results to a fused CUDA kernel.
Wait, one more optimization: `_log_prob(x)` is always computed, and it doesn't depend on randomness. We could cache it between forward calls! But in the reference, it's recomputed each time. The evaluation compares our output to the reference's output for a single forward call. Caching doesn't help for a single call.
For a single forward call, the fused kernel is still a big win.
Let me design the fused kernel that takes `x`, `noise`, `uniforms`, `mu`, `sigma`, `step_size` and produces `x_new` and `accepted`.
```cuda
#include <cuda_runtime.h>
__global__ void mcmc_step_kernel(const float* x, const float* noise, const float* uniforms,
const float* mu, const float* sigma, float step_size, int B, int dim,
float* x_new, bool* accepted) {
int row = blockIdx.x * blockDim.y + threadIdx.y;
int col = threadIdx.x;
if (row >= B) return;
float log_p_x = 0.0f;
float log_p_prop = 0.0f;
for (int c = col; c < dim; c += 32) {
float x_val = x[row * dim + c];
float prop_val = x_val + step_size * noise[row * dim + c];
float diff_x = (x_val - mu[c]) / sigma[c];
log_p_x += -0.5f * diff_x * diff_x;
float diff_prop = (prop_val - mu[c]) / sigma[c];
log_p_prop += -0.5f * diff_prop * diff_prop;
x_new[row * dim + c] = prop_val;
}
__syncwarp();
// Warp reduce
#pragma unroll
for (int offset = 16; offset > 0; offset /= 2) {
log_p_x += __shfl_down_sync(0xffffffff, log_p_x, offset);
log_p_prop += __shfl_down_sync(0xffffffff, log_p_prop, offset);
}
bool acc = false;
if (col == 0) {
float u = uniforms[row];
u = fmaxf(u, 1e-10f);
float log_u = logf(u);
float log_alpha = log_p_prop - log_p_x;
acc = log_u < log_alpha;
accepted[row] = acc;
}
acc = __shfl_sync(0xffffffff, acc, 0);
// Rewrite rejected rows
if (!acc) {
for (int c = col; c < dim; c += 32) {
x_new[row * dim + c] = x[row * dim + c];
}
}
}
```
Wait, I need to be careful. The `__syncwarp()` after writing x_new is important because we want all threads to finish writing proposal before thread 0 starts potentially copying back. But then after thread 0 computes acc, the other threads might start the "if (!acc)" block before thread 0 finishes writing accepted[] and computing acc... but __shfl_sync will stall them. Then once they get acc, all threads in the warp either enter the if block (acc=false) or skip it (acc=true). Since all threads in the warp have the same acc value, there's no divergence in the if block - either all threads execute it or none do. Good.
Actually, one more thing: should I use template parameters for block size? Or just determine at runtime? For dim=64, blockDim=(32, 8) is good. I'll use a launch configuration in the C++ wrapper.
For the wrapper function:
```cpp
torch::Tensor mcmc_step(torch::Tensor x, torch::Tensor noise, torch::Tensor uniforms,
torch::Tensor mu, torch::Tensor sigma, float step_size) {
int B = x.size(0);
int dim = x.size(1);
auto x_new = torch::empty_like(x);
auto accepted = torch::empty({B}, torch::TensorOptions().dtype(torch::kBool).device(x.device()));
int rows_per_block = 8;
dim3 block(32, rows_per_block);
dim3 grid((B + rows_per_block - 1) / rows_per_block);
mcmc_step_kernel<<<grid, block>>>(
x.data_ptr<float>(), noise.data_ptr<float>(), uniforms.data_ptr<float>(),
mu.data_ptr<float>(), sigma.data_ptr<float>(), step_size, B, dim,
x_new.data_ptr<float>(), accepted.data_ptr<bool>());
return x_new, accepted; // Can't return tuple directly like this in C++
}
```
Actually, returning a tuple in the wrapper function... we can return a `std::vector<torch::Tensor>` or a `std::tuple`. PyTorch C++ bindings support `std::tuple<torch::Tensor, torch::Tensor>`.
Let me check the exact syntax. Yes, pybind11 can bind functions returning `std::tuple<at::Tensor, at::Tensor>` or `std::vector<at::Tensor>`.
Actually, in `torch.utils.cpp_extension.load_inline`, the auto-generated pybind11 module will handle returning tuples if the function returns `std::tuple<torch::Tensor, torch::Tensor>`.
Let me write the complete code.
Wait, one optimization: I can avoid the `__syncwarp()` and two-pass write by having all threads write the correct value from the start. But we need acc before writing. We can compute log_probs first (without writing), then reduce, then broadcast acc, then write:
```cuda
float log_p_x = 0.0f;
float log_p_prop = 0.0f;
for (int c = col; c < dim; c += 32) {
float x_val = x[row * dim + c];
float prop_val = x_val + step_size * noise[row * dim + c];
float diff_x = (x_val - mu[c]) / sigma[c];
log_p_x += -0.5f * diff_x * diff_x;
float diff_prop = (prop_val - mu[c]) / sigma[c];
log_p_prop += -0.5f * diff_prop * diff_prop;
}
// Reduce
#pragma unroll
for (int offset = 16; offset > 0; offset /= 2) {
log_p_x += __shfl_down_sync(0xffffffff, log_p_x, offset);
log_p_prop += __shfl_down_sync(0xffffffff, log_p_prop, offset);
}
bool acc = false;
if (col == 0) {
float u = uniforms[row];
u = fmaxf(u, 1e-10f);
float log_u = logf(u);
float log_alpha = log_p_prop - log_p_x;
acc = log_u < log_alpha;
accepted[row] = acc;
}
acc = __shfl_sync(0xffffffff, acc, 0);
for (int c = col; c < dim; c += 32) {
float x_val = x[row * dim + c];
float prop_val = x_val + step_size * noise[row * dim + c];
x_new[row * dim + c] = acc ? prop_val : x_val;
}
```
Wait, this recomputes prop_val in the second loop. We re-read noise, recompute prop_val, but we don't need to recompute log probs. This avoids the need to store prop_val or the sync/copy-back complexity. For dim=64 with 2 elements per thread, the extra computation is negligible. But we do re-read noise from global memory. Since noise is in global memory, reading it twice has bandwidth cost.
Alternatively, we can keep prop_val in registers. Each thread handles `ceil(dim / 32)` elements. For small dim (like 64), this is 2 elements. We could declare `float prop_vals[2];` but for general dim we'd need dynamic allocation.
We could use a small shared memory buffer for proposals. With rows_per_block=8 and dim=64, that's 8*64*4 = 2KB of shared memory. Very manageable. Each block writes all proposals to shared memory, then reads them back after computing acceptance.
```cuda
__shared__ float proposals[8 * 64]; // But 64 is not known at compile time...
```
Hmm, shared memory with dynamic size requires `extern __shared__ float proposals[];` passed as kernel argument. Or we can make it a macro, but dim is runtime.
Actually for simplicity and since dim is usually small, let me just go with the first approach: write proposal to x_new, sync, thread 0 serial copy back if rejected. The bandwidth savings from fewer reads might compensate for the serial copy.
Wait, actually there's an even better approach. We can have all threads help copy by using shared memory to broadcast the acceptance.
```cuda
__shared__ bool s_acc[8]; // one per row in block
// Phase 1: compute and write proposals
for (int c = col; c < dim; c += 32) {
float x_val = x[row * dim + c];
float prop_val = x_val + step_size * noise[row * dim + c];
float diff_x = (x_val - mu[c]) / sigma[c];
log_p_x += -0.5f * diff_x * diff_x;
float diff_prop = (prop_val - mu[c]) / sigma[c];
log_p_prop += -0.5f * diff_prop * diff_prop;
x_new[row * dim + c] = prop_val;
}
__syncwarp();
// Phase 2: decide
if (col == 0) {
float u = uniforms[row];
u = fmaxf(u, 1e-10f);
float log_u = logf(u);
float log_alpha = log_p_prop - log_p_x;
bool acc = log_u < log_alpha;
s_acc[threadIdx.y] = acc;
accepted[row] = acc;
}
__syncwarp();
bool acc = s_acc[threadIdx.y];
// Phase 3: conditional fix
if (!acc) {
for (int c = col; c < dim; c += 32) {
x_new[row * dim + c] = x[row * dim + c];
}
}
```
Here, `__syncwarp()` after phase 2 ensures all threads see the s_acc write before reading it. But wait, `__syncwarp()` memory ordering: it ensures memory operations before the sync are visible to all threads in the warp. Since col==0 writes to s_acc[threadIdx.y], and all threads in the same row (same threadIdx.y) have the same threadIdx.y, threads in the same warp might have different threadIdx.y values.
With blockDim=(32, 8), threads are organized as:
- threadIdx.x = 0..31, threadIdx.y = 0..7
- warp 0: threads with linear id 0..31, which corresponds to (x:0..31, y:0)
- warp 1: (x:0..31, y:1)
- etc.
So each warp contains only threads with the SAME threadIdx.y! This means __syncwarp() only syncs threads within the same row. And s_acc[threadIdx.y] is written by thread 0 of the warp and read by all threads in the same warp. So after __syncwarp(), the write is visible. Then all threads read s_acc[threadIdx.y].
Wait, but after the first __syncwarp(), thread 0 computes acc and writes s_acc. Other threads in the warp are also past the first __syncwarp(). Do they execute the second __syncwarp() immediately or do they wait? They just proceed to the if (col == 0), evaluate false, skip it, and reach the second __syncwarp().
Meanwhile, thread 0 is computing u, log_u, etc. Once thread 0 writes to s_acc and reaches the second __syncwarp(), all threads in the warp are synced. Then everyone reads s_acc.
This works. And in phase 3, all 32 threads in the warp participate in the copy-back, handling dim elements. For dim=64, each thread does 2 writes. Parallel and efficient!
But there's a tiny issue: the if (!acc) causes warp divergence if acc differs across warps. But within a warp, all threads have the same acc (since they all read s_acc[threadIdx.y], and all threads in a warp have the same threadIdx.y). So no intra-warp divergence. Perfect.
Let me verify once more:
- block = (32, 8), grid = ((B + 7) / 8, 1)
- row = blockIdx.x * 8 + threadIdx.y
- Each row has exactly one warp (32 threads with threadIdx.x varying, threadIdx.y fixed)
- __syncwarp() syncs the 32 threads of one row
- s_acc[8] is shared across the block, but each warp only accesses its own entry
- No bank conflicts on s_acc because each warp accesses a different bank (or same bank but broadcast? Actually threadIdx.y is constant within a warp, so all threads in a warp read the SAME address s_acc[threadIdx.y]. This is a broadcast, no bank conflict.)
This looks great. Let me write the final kernel.
Actually, let me also think about further micro-optimizations:
1. `mu` and `sigma` are broadcast across rows. They are read as `mu[c]` and `sigma[c]`. With dim=64 and 32 threads, consecutive threads read the same `mu[c]` and `sigma[c]` values (since c differs by 32 for consecutive threads in x dimension, wait no...).
Wait, thread (x=0, y=0) reads c=0,32
thread (x=1, y=0) reads c=1,33
...
thread (x=31, y=0) reads c=31,63
So consecutive threads in a warp read consecutive c values. For mu[c], this means consecutive threads read consecutive memory locations. Coalesced! Good.
For x[row*dim+c], consecutive threads also read consecutive memory. Coalesced! Good.
2. Division by sigma[c]: since sigma is typically all ones, we could skip it. But let's keep it general.
3. `-0.5f * diff * diff`: fused multiply-add opportunities. The compiler should handle this.
4. The `clamp(min=1e-10)` in the reference: in our kernel, we do `fmaxf(u, 1e-10f)`. Since u is from `torch.rand` which is [0, 1), u can be 0. fmaxf handles this. Then `logf(u)`.
5. One subtle thing: `torch.rand` produces [0, 1), and `curand_uniform` produces (0, 1]. Our reference uses `torch.rand`, so that's fine.
Actually, know what? I just realized: if we're using PyTorch's `torch.randn_like` and `torch.rand`, then `torch.rand` already returns values in [0, 1). Some could be 0. The reference clamps them. In our kernel, we need to apply the same clamp. Good.
Now let me think about whether we can avoid generating `torch.rand(x.shape[0])` by using `torch.rand(B)` and passing it. That's what we planned.
Let me write the full Python module.
One more thing: I should handle the case where dim is not a multiple of 32. With the loop `for (int c = col; c < dim; c += 32)`, if dim=65, thread 0 handles c=0,32,64 while thread 1 handles c=1,33. This naturally handles any dim. Good.
Actually, there's a potential issue: if dim is say 1, then only thread 0 does work, and threads 1-31 return immediately from the loop. The reduction still works. Thread 0 reads uniforms[row], etc. This is fine.
Let me also consider: should we use const __restrict__ pointers for better memory access? Yes, it's a good practice.
```cpp
__global__ void mcmc_step_kernel(const float* __restrict__ x, const float* __restrict__ noise,
const float* __restrict__ uniforms, const float* __restrict__ mu,
const float* __restrict__ sigma, float step_size, int B, int dim,
float* __restrict__ x_new, bool* __restrict__ accepted)
```
Also, I should compile with appropriate flags for H100: `-O3`, `--use_fast_math` maybe? `--use_fast_math` makes logf faster but slightly less accurate. The tolerance is 1e-4 for fp32, so fast math should be fine. But let me be safe and not use it unless needed.
Actually, `logf` with fast math might have accuracy issues near 1e-10. The reference uses `torch.log(torch.rand(...).clamp(min=1e-10))`. PyTorch uses standard accuracy log. Let me also use standard `logf` to be safe.
Let me check: `1e-10` is not that small for log. log(1e-10) = -23.0. Fast math should handle this fine. But I'll stick with regular compilation flags unless we need the speed.
Now, in the Python model wrapper:
```python
import torch
import torch.nn as nn
from torch.utils.cpp_extension import load_inline
# Load inline CUDA
cuda_src = """
#include <cuda_runtime.h>
#include <torch/extension.h>
__global__ void mcmc_step_kernel(const float* __restrict__ x, const float* __restrict__ noise,
const float* __restrict__ uniforms, const float* __restrict__ mu,
const float* __restrict__ sigma, float step_size, int B, int dim,
float* __restrict__ x_new, bool* __restrict__ accepted) {
int row = blockIdx.x * blockDim.y + threadIdx.y;
int col = threadIdx.x;
if (row >= B) return;
float log_p_x = 0.0f;
float log_p_prop = 0.0f;
for (int c = col; c < dim; c += 32) {
float x_val = x[row * dim + c];
float prop_val = x_val + step_size * noise[row * dim + c];
float diff_x = (x_val - mu[c]) / sigma[c];
log_p_x += -0.5f * diff_x * diff_x;
float diff_prop = (prop_val - mu[c]) / sigma[c];
log_p_prop += -0.5f * diff_prop * diff_prop;
x_new[row * dim + c] = prop_val;
}
__syncwarp();
#pragma unroll
for (int offset = 16; offset > 0; offset /= 2) {
log_p_x += __shfl_down_sync(0xffffffff, log_p_x, offset);
log_p_prop += __shfl_down_sync(0xffffffff, log_p_prop, offset);
}
__shared__ bool s_acc[8];
if (col == 0) {
float u = uniforms[row];
u = fmaxf(u, 1e-10f);
float log_u = logf(u);
float log_alpha = log_p_prop - log_p_x;
bool acc = log_u < log_alpha;
s_acc[threadIdx.y] = acc;
accepted[row] = acc;
}
__syncwarp();
bool acc = s_acc[threadIdx.y];
if (!acc) {
for (int c = col; c < dim; c += 32) {
x_new[row * dim + c] = x[row * dim + c];
}
}
}
std::tuple<torch::Tensor, torch::Tensor> mcmc_step(
torch::Tensor x, torch::Tensor noise, torch::Tensor uniforms,
torch::Tensor mu, torch::Tensor sigma, float step_size) {
TORCH_CHECK(x.is_cuda(), "x must be CUDA");
int B = x.size(0);
int dim = x.size(1);
auto x_new = torch::empty_like(x);
auto accepted = torch::empty({B}, torch::TensorOptions().dtype(torch::kBool).device(x.device()));
int rows_per_block = 8;
dim3 block(32, rows_per_block);
dim3 grid((B + rows_per_block - 1) / rows_per_block);
mcmc_step_kernel<<<grid, block>>>(
x.data_ptr<float>(), noise.data_ptr<float>(), uniforms.data_ptr<float>(),
mu.data_ptr<float>(), sigma.data_ptr<float>(), step_size, B, dim,
x_new.data_ptr<float>(), accepted.data_ptr<bool>());
return std::make_tuple(x_new, accepted);
}
"""
mcmc_cuda = load_inline(
name="mcmc_cuda",
cpp_sources="",
cuda_sources=cuda_src,
functions=["mcmc_step"],
extra_cuda_cflags=["-O3"],
verbose=False,
)
class ModelNew(nn.Module):
def __init__(self, dim, step_size=0.1):
super().__init__()
self.dim = dim
self.step_size = step_size
self.register_buffer("mu", torch.zeros(dim))
self.register_buffer("sigma", torch.ones(dim))
def forward(self, x):
proposal_noise = torch.randn_like(x)
uniforms = torch.rand(x.shape[0], device=x.device)
return mcmc_cuda.mcmc_step(
x, proposal_noise, uniforms,
self.mu, self.sigma, self.step_size
)
```
Hmm, I notice a potential issue: in the reference, `torch.rand(x.shape[0], device=x.device).clamp(min=1e-10)` is computed, then `torch.log` is applied. In our code, we pass the raw uniforms to the kernel and do `fmaxf(u, 1e-10f)` and `logf(u)` inside. This should be equivalent.
Wait, the reference does `torch.log(torch.rand(...).clamp(min=1e-10))`. `torch.clamp` on the result of `torch.rand`, then `torch.log`. In our kernel, `fmaxf` is equivalent to `.clamp(min=1e-10)`. Good.
One more thing: `torch.rand` generates floats, so `uniforms` is `torch.float32`. Good.
Let me think about whether `torch.randn_like(x)` gives the same distribution as `x + step_size * noise` in the kernel. Yes, `noise ~ N(0, 1)`, so `step_size * noise ~ N(0, step_size^2)`. Then `proposal = x + step_size * noise ~ N(x, step_size^2)`. Same as reference.
Now, is there any floating point difference? The reference computes `x + self.step_size * noise`. In our kernel, we compute `x_val + step_size * noise[row * dim + c]`. Same computation.
The reference computes `-0.5 * (((proposal - mu) / sigma) ** 2).sum(dim=-1)`. Our kernel computes `-0.5f * diff * diff` and reduces with warp shuffle. The reduction order is different: warp shuffle does tree reduction. The reference likely uses a reduction that may have slightly different ordering. But with 64 elements, the difference should be well within 1e-4.
Actually, one improvement: in our kernel, we do the reduction with warp shuffle. But warp shuffle may not be the most accurate tree reduction. However, for 64 floats, the error should be tiny.
Wait, I just realized: our reduction only works correctly if `dim` is a multiple of 32 or if we properly account for the missing elements. With the loop `for (int c = col; c < dim; c += 32)`, some threads might process fewer elements than others, but they still participate in the shuffle. Since threads with no elements add 0 to log_p_x, the shuffle reduction still works correctly.
Let me think of an example: dim=64.
- Thread 0: c=0, 32 (2 elements)
- Thread 1: c=1, 33 (2 elements)
- ...
- Thread 31: c=31, 63 (2 elements)
- Shuffle down by 16: threads 0-15 add their log_p_x with threads 16-31. Then down by 8, etc. Result in thread 0. Correct.
Example: dim=65.
- Thread 0: c=0, 32, 64 (3 elements)
- Thread 1: c=1, 33 (2 elements)
- ...
- Thread 1-31: some have 2 elements
- Thread 0 has 3 elements
- Shuffle down by 16: thread 0 adds thread 16. Thread 16 has computed c=16, 48 (2 elements). Correct.
- Result in thread 0. Correct.
But what if dim=1?
- Thread 0: c=0 (1 element)
- Thread 1-31: no elements (log_p_x = 0)
- Shuffle down by 16: thread 0 adds thread 16 (0). Correct.
- Eventually thread 0 has the sum. Correct.
So the reduction is correct for any dim.
One more optimization: since `mu` and `sigma` are the same for all rows, we could load them into shared memory once per block. But with dim=64 and 8 warps, that's only 64 floats each = 256 bytes. The benefit is minimal since they're already in L2 cache after the first few blocks. Let me skip this optimization.
What about `uniforms`? It's accessed as `uniforms[row]`, so each warp reads one float. This goes into L1 cache and might be broadcast. Fine.
Actually, a potential issue with our kernel: we have `__shared__ bool s_acc[8];` inside the kernel. If `rows_per_block` changes, this fixed size might be wrong. But in our C++ wrapper, we hardcode `rows_per_block = 8`, so it's fine. Or we can make it dynamic: `extern __shared__ bool s_acc[];` and pass `rows_per_block * sizeof(bool)` as the 3rd kernel launch parameter. But for simplicity and since 8 is fixed, I'll keep `__shared__ bool s_acc[8];`.
Hmm, but C++ doesn't allow variable-length arrays (VLA) in standard C++. CUDA allows static shared memory arrays with compile-time constant sizes. 8 is a compile-time constant, so it's fine.
Actually, I just realized I should verify that the `accepted` output type matches PyTorch's bool tensor. In the reference, `accepted` is a boolean mask of accepted proposals, shape (B,), dtype bool. In our kernel, we write `bool* accepted`. But `torch::kBool` tensors use 1 byte per element. And in CUDA, `bool` might be 1 byte or might cause issues. Let me check...
In PyTorch's `TensorImpl`, `kBool` is `at::kBool`, which is `at::ScalarType::Bool`. The element size is 1. In C++, `sizeof(bool)` is implementation-defined but typically 1. In CUDA, `bool` is also 1 byte.
To be absolutely safe, I could use `uint8_t* accepted_ptr` and write 0 or 1:
```cpp
uint8_t* accepted = accepted_tensor.data_ptr<uint8_t>();
accepted[row] = acc ? 1 : 0;
```
And create the tensor with `torch::kUInt8` in Python. But then the Python return type would be uint8, not bool. We can convert with `.bool()` after the kernel, but that's an extra operation.
Actually, in PyTorch C++ API, `torch::kBool` tensors have `data_ptr<bool>()` available. Let me just use that. If there are issues, we can switch.
Wait, I just had a thought. Instead of using PyTorch's `torch.randn_like` and `torch.rand`, what if I generate the random numbers inside the kernel using a deterministic seed that matches PyTorch's? No, that's too complex and fragile.
OK, let me compile and test.
Actually, one more idea: can we fuse the `torch.clamp(min=1e-10)` and `torch.log` into our kernel while also handling the `torch.rand` generation in Python? Yes, we already do that.
What about `torch.where(accepted.unsqueeze(-1), proposal, x)`? This is handled by our kernel's conditional write.
One more tiny optimization: in the reference, `_log_prob(x)` and `_log_prob(proposal)` both compute `((x - mu) / sigma) ** 2`. In our kernel, we compute both in the same loop, which is good for cache reuse.
OK, I think the kernel is ready. Let me write the full code and try to compile it.
Wait, actually I realize I can simplify even further. Since we pass `noise` from Python, we already have the proposal noise. We also pass `uniforms`. The kernel does:
1. For each element: compute proposal, compute log_prob contributions
2. Reduce log_probs
3. Compare with log(uniform)
4. Write x_new
There's one subtle issue: in the reference, `_log_prob(proposal)` and `_log_prob(x)` both compute `-0.5 * sum(squares)`. The reference subtracts them: `log_alpha = lp_prop - lp_x`. In our kernel, we compute both sums and subtract after reduction. Same thing.
But wait, the reference computes these as two separate reductions. Our kernel fuses them into one pass. This is better.
Let me also make sure we're not doing anything that introduces NaNs or infinities. `sigma` is initialized to ones, so no division by zero. `uniforms` from `torch.rand` might include 0, but we clamp with `fmaxf(u, 1e-10f)`. Good.
I just want to double check: `torch.rand` returns values in `[0, 1)`. Can it return exactly `0.0`? According to PyTorch docs, `torch.rand` returns a tensor filled with random numbers from a uniform distribution on the interval `[0, 1)`. So yes, it can return 0.0. Our `fmaxf(u, 1e-10f)` ensures `log(u)` doesn't give `-inf`. Same as reference.
One more thought: the reference does `torch.log(torch.rand(x.shape[0], device=x.device).clamp(min=1e-10))`. Note that `.clamp(min=1e-10)` operates on the tensor. In fp32, `1e-10` is `0.0000000001`. This is well-representable in fp32 (which has about 7 decimal digits of precision). Actually, `1e-10` in fp32 is approximately `9.999999e-11`. `fmaxf(u, 1e-10f)` should be equivalent.
But wait: `torch.clamp(min=1e-10)` on a float32 tensor uses the exact float32 representation of `1e-10`. And `1e-10f` in our CUDA code is also the float32 representation of `1e-10`. So they should match exactly (assuming the compiler does correct rounding).
Now let me also think: should we use `logf` or `log`? `logf` is the float version. `log` with float argument might be promoted to double on some platforms, but in CUDA `log(float)` is usually the same as `logf(float)`. I'll use `logf` to be explicit.
Hmm, I just thought of another potential issue. In PyTorch, `torch.where(accepted.unsqueeze(-1), proposal, x)` where `accepted` is bool. In our kernel, if `acc` is `false`, we copy `x`. If `acc` is `true`, we keep what we wrote (which is `proposal`). This matches.
Let me write the complete module now and compile it.
Actually, I want to consider: is there a better block size? Maybe `rows_per_block = 16` instead of 8? With 16 warps per block = 512 threads. H100 supports up to 2048 threads per block, so 512 is fine. But shared memory is per block, so `s_acc[16]` would be used. More warps per block might improve occupancy or might hurt due to register pressure.
With our kernel, each thread uses maybe ~20-30 registers. 512 threads * 30 registers = ~15K registers per block. H100 has 64K registers per SM, so we can fit ~4 blocks per SM. That's plenty.
Actually, let me keep it at 8 for simplicity. The grid will have 512 blocks for B=4096, which is more than enough to saturate the GPU.
One more optimization: for H100, we might want to use vectorized loads/stores. With dim=64 and each thread handling 2 elements, we could process them as float2 to load/store 8 bytes at a time. But our accesses within a warp are:
- Thread 0: c=0, 32
- Thread 1: c=1, 33
These are strided by 1 within the first 32 elements, then by 1 within the next 32. So a warp reads x[row*dim + 0:31] together, then x[row*dim + 32:63] together. These are consecutive and can be vectorized! But individual threads don't read consecutive pairs - thread 0 reads 0 and 32, thread 1 reads 1 and 33, etc.
For vectorized loads, we'd need thread 0 to read (0, 1), thread 1 to read (2, 3), etc. That's not our access pattern. Our pattern is strided across threads, which is actually optimal for memory coalescing. Vectorized loads wouldn't help here because each thread already has perfect coalescing for its individual loads.
Wait, but each thread does two loads per loop (c and c+32 if dim=64). These are 128 bytes apart. The first warp loads c=0..31 (one cache line of 128 bytes), then c=32..63 (another cache line). So each warp does 2 loads of 128 bytes = 256 bytes total. These hit L1 cache nicely. Good.
Actually, I realize that with the loop `for (int c = col; c < dim; c += 32)`, for dim=64, thread 0 does c=0 then c=32. The load at c=0 is in cache line 0, and c=32 is in cache line 1. But the warp executes these loads together for all threads. Specifically, all threads do `x[row*dim + threadIdx.x]` first, which is one 128-byte transaction covering 32 floats. Then all threads do `x[row*dim + threadIdx.x + 32]`, which is another 128-byte transaction. Perfect coalescing.
So our memory access pattern is already optimal.
One last thing before I compile: the `__shared__ bool s_acc[8]` should be declared outside the if-block and before any conditional code. Also, `bool` in CUDA shared memory might have alignment issues. But since we're only reading/writing one byte per row, and threads in a warp read the same byte, it should be fine.
Actually, I just realized that `bool` in C++ might pack multiple bools into a byte when used in arrays? No, `bool arr[8]` is an array of 8 separate bool objects, each typically 1 byte. No bit packing.
Let me write the kernel code and compile.
One more issue: in `load_inline`, I need to make sure the source is correct. Also, `torch::Tensor` vs `at::Tensor`. In the C++ API, they're the same thing.
Wait, in the wrapper function, I should also check that all tensors are contiguous and on the same device. But the benchmark probably gives us contiguous tensors. Let's add `.contiguous()` in Python if needed, or checks in C++. Actually, for simplicity, I'll assume inputs are contiguous and on CUDA. If not, the kernel will either crash or produce wrong results. In production code, we'd add checks.
Let me also think about whether `x_new = torch.empty_like(x)` in the C++ wrapper creates a tensor with the same strides as x. If x is non-contiguous, empty_like preserves strides, and our pointer arithmetic `row * dim + c` assumes row-major contiguous. So I should add a check or ensure x is contiguous in Python.
In the `forward` method, I can call `x = x.contiguous()` to ensure this. The reference doesn't require x to be contiguous, but our kernel does. Actually, in the benchmark, x is created with `torch.randn(batch_size, dim)`, which is contiguous. So it's fine. But to be safe, I'll add `x = x.contiguous()` at the start of `forward`.
Actually, x is [B, dim], so if it's contiguous, `row * dim + c` is correct. If it's transposed, this would be wrong. Let me just not worry about it for the benchmark, but I'll be aware.
Let me also consider: do I need to `#include <torch/extension.h>`? Yes, for `torch::Tensor`, `TORCH_CHECK`, etc. In `load_inline`, this header is typically included automatically or available.
Now, let me structure the Python code properly.
Actually, I want to check one more thing. In the reference:
```python
log_u = torch.log(torch.rand(x.shape[0], device=x.device).clamp(min=1e-10))
```
This creates a tensor of shape [B], clamps it, then takes log. Our kernel does:
```cpp
float u = uniforms[row];
u = fmaxf(u, 1e-10f);
float log_u = logf(u);
```
If `torch.rand` returns 0.0, the reference clamps to 1e-10 and log gives log(1e-10). Our kernel does the same. Good.
One potential discrepancy: `torch.log` in PyTorch might handle subnormal numbers or edge cases differently than CUDA `logf`. But since we clamp to [1e-10, 1), both should behave similarly.
OK enough thinking. Let me write and compile the code.
Actually, one final thought on kernel fusion. Can we avoid the `torch.rand(x.shape[0])` call entirely by generating the uniform random inside the kernel but using PyTorch's random state?
Actually, I just remembered: PyTorch exposes C++ APIs for random number generation. We could call `at::rand` or similar from C++, but that would generate the tensor and then we'd still need to pass it to the kernel. Not helpful for fusion.
Alternatively, we could keep the `torch.randn_like(x)` call (which is highly optimized) and generate the uniform inside the kernel reusing a simple hash of the normal random values. For example, we could hash one of the proposal noise values per row to get a uniform. But this breaks the stochastic independence and might not match the reference distribution exactly.
No, let's stick with two PyTorch RNG calls + one fused kernel. This is safe and correct.
Let me now write the complete Python file.
Wait, I want to also consider: could we at least fuse the `step_size * noise` multiplication into the random number generation? No, because `torch.randn_like` generates standard normals and we multiply after.
What about using `torch.normal(mean=0, std=step_size, size=x.shape)` instead of `step_size * torch.randn_like(x)`? That's the same.
OK, writing the code now.
```python
import torch
import torch.nn as nn
from torch.utils.cpp_extension import load_inline
CUDA_SOURCE = """
#include <cuda_runtime.h>
#include <torch/extension.h>
__global__ void mcmc_step_kernel(
const float* __restrict__ x,
const float* __restrict__ noise,
const float* __restrict__ uniforms,
const float* __restrict__ mu,
const float* __restrict__ sigma,
float step_size,
int B,
int dim,
float* __restrict__ x_new,
bool* __restrict__ accepted) {
int row = blockIdx.x * blockDim.y + threadIdx.y;
int col = threadIdx.x;
if (row >= B) return;
float log_p_x = 0.0f;
float log_p_prop = 0.0f;
for (int c = col; c < dim; c += 32) {
float x_val = x[row * dim + c];
float prop_val = x_val + step_size * noise[row * dim + c];
float diff_x = (x_val - mu[c]) / sigma[c];
log_p_x += -0.5f * diff_x * diff_x;
float diff_prop = (prop_val - mu[c]) / sigma[c];
log_p_prop += -0.5f * diff_prop * diff_prop;
x_new[row * dim + c] = prop_val;
}
__syncwarp();
#pragma unroll
for (int offset = 16; offset > 0; offset >>= 1) {
log_p_x += __shfl_down_sync(0xffffffff, log_p_x, offset);
log_p_prop += __shfl_down_sync(0xffffffff, log_p_prop, offset);
}
__shared__ bool s_acc[8];
if (col == 0) {
float u = uniforms[row];
u = fmaxf(u, 1e-10f);
float log_u = logf(u);
float log_alpha = log_p_prop - log_p_x;
bool acc = log_u < log_alpha;
s_acc[threadIdx.y] = acc;
accepted[row] = acc;
}
__syncwarp();
bool acc = s_acc[threadIdx.y];
if (!acc) {
for (int c = col; c < dim; c += 32) {
x_new[row * dim + c] = x[row * dim + c];
}
}
}
std::tuple<torch::Tensor, torch::Tensor> mcmc_step(
torch::Tensor x,
torch::Tensor noise,
torch::Tensor uniforms,
torch::Tensor mu,
torch::Tensor sigma,
float step_size) {
TORCH_CHECK(x.is_cuda(), "x must be a CUDA tensor");
int B = x.size(0);
int dim = x.size(1);
auto x_new = torch::empty_like(x);
auto accepted = torch::empty({B}, torch::TensorOptions().dtype(torch::kBool).device(x.device()));
const int rows_per_block = 8;
dim3 block(32, rows_per_block);
dim3 grid((B + rows_per_block - 1) / rows_per_block);
mcmc_step_kernel<<<grid, block>>>(
x.data_ptr<float>(),
noise.data_ptr<float>(),
uniforms.data_ptr<float>(),
mu.data_ptr<float>(),
sigma.data_ptr<float>(),
step_size,
B,
dim,
x_new.data_ptr<float>(),
accepted.data_ptr<bool>());
return std::make_tuple(x_new, accepted);
}
"""
mcmc_cuda = load_inline(
name="mcmc_cuda",
cpp_sources="",
cuda_sources=CUDA_SOURCE,
functions=["mcmc_step"],
extra_cuda_cflags=["-O3"],
verbose=False,
)
class ModelNew(nn.Module):
def __init__(self, dim, step_size=0.1):
super().__init__()
self.dim = dim
self.step_size = step_size
self.register_buffer("mu", torch.zeros(dim))
self.register_buffer("sigma", torch.ones(dim))
def forward(self, x):
noise = torch.randn_like(x)
uniforms = torch.rand(x.shape[0], device=x.device)
return mcmc_cuda.mcmc_step(
x, noise, uniforms, self.mu, self.sigma, self.step_size
)
```
Wait, I think there might be an issue. The reference code does:
```python
log_u = torch.log(torch.rand(x.shape[0], device=x.device).clamp(min=1e-10))
accepted = log_u < log_alpha
```
In our kernel, we compute `log_u = logf(fmaxf(uniforms[row], 1e-10f))`. In the reference, `torch.clamp(min=1e-10)` on a tensor might have slightly different behavior than `fmaxf` for some edge cases, but for normal values they should be identical.
One more thing: `_log_prob` returns `-0.5 * (((x - self.mu) / self.sigma) ** 2).sum(dim=-1)`. Note the `.sum(dim=-1)`. Since dim is the last dimension, and x is [B, dim], sum(dim=-1) gives [B]. Our reduction does exactly this.
But wait, in the reference, when computing `_log_prob(proposal)`, it first computes `((proposal - mu) / sigma) ** 2` which is [B, dim], then `.sum(dim=-1)` which reduces to [B]. In PyTorch, this reduction might use higher precision internally. The result is still fp32, but intermediate sums might accumulate in fp32 or higher. Our kernel accumulates in fp32 via warp shuffle, which might have a different summation order.
Actually, PyTorch's sum reduction on CUDA also typically uses fp32 accumulation with tree-like reduction. Our warp shuffle does a similar tree reduction. The order might differ slightly, but for 64 elements, the difference is tiny. Let me verify: max possible value of each term is... if x is normal ~ N(0,1), then (x-mu)^2 is chi-squared with mean 1 and variance 2. Summing 64 gives mean 64, sd ~11.3. So the sum is around 64, and adding 64 numbers of size ~1 with different orders gives errors on the order of 64 * epsilon ~ 64 * 1e-7 = 6e-6. Well within 1e-4 tolerance.
If x is large, say 1e4, then x^2 = 1e8, and sum = 6.4e9. Adding 64 numbers of size 1e8 could have errors of ~64 * 1e8 * 1e-7 = 0.64. That's within 1e-4 relative? Wait, absolute tolerance is 1e-4. 0.64 > 1e-4. That's a problem!
Hmm, but the benchmark input is `torch.randn(batch_size, dim)`, so x values are O(1). The reference also initializes x with randn. So x won't be 1e4. In fact, for the specific benchmark, x is standard normal. So we're fine.
But what if the test uses different inputs? The problem says `get_inputs` returns [torch.randn(batch_size, dim)], so x is standard normal. Good.
Actually, let me double-check: even with standard normal, x can occasionally be large (e.g., 5 or more). With x=5, x^2=25. Summing 64 of these: if all are 25, sum=1600. Error from different ordering: 64 * 25 * 1e-7 = 1.6e-4. That's borderline with 1e-4 absolute tolerance!
Hmm, that could be a concern. Let me think about how to match PyTorch's reduction exactly.
Actually, PyTorch's `sum()` on CUDA uses different algorithms depending on tensor size. For a [4096, 64] tensor, it might use a block-level reduction with shared memory where each thread reduces a segment, then the block reduces the partial sums. The exact algorithm depends on the implementation.
But note: our tolerance is 1e-4 absolute. With 64 values of size ~25, the sum is ~1600. PyTorch's reduction and our reduction might differ by a few ULPs. With fp32, ULP at 1600 is about 1600 * 2^-23 ≈ 1.9e-4. So different ordering could give differences of ~1-2 ULPs ≈ 2-4e-4. That's above 1e-4 tolerance!
Wait, this is potentially problematic. If PyTorch's sum uses Kahan summation or some other extended precision, or if it sums in a different order, we might get differences > 1e-4.
Hmm, but PyTorch's `sum()` on CUDA typically uses plain fp32 accumulation for fp32 tensors. Actually, I need to check... PyTorch has a `float32_matmul_precision` setting, but for sum, I think it uses fp32 accumulation. The implementation might use `thrust` or custom reduction kernels.
Actually, there's a simple way to avoid this issue: instead of doing our own warp shuffle reduction, we can use PyTorch's reduction for the log_probs and only fuse the rest!
Let me reconsider the design. We could:
1. Generate proposal = x + step_size * noise
2. Compute log_prob_x and log_prob_prop using PyTorch operations
3. Compute log_alpha, acceptance, and x_new in a fused kernel
But that still leaves multiple kernel launches:
- proposal = x + step_size * noise (1 launch)
- log_prob(x) (2 launches: element-wise + reduction)
- log_prob(proposal) (2 launches: element-wise + reduction)
- log_alpha = lp_prop - lp_x (1 launch)
- log_u (1 launch)
- accepted = log_u < log_alpha (1 launch)
- x_new = where (1 launch)
Even with PyTorch's reductions, that's still ~8 launches. Our fused kernel is 1 launch. The fusion is a big win even if reduction accuracy is slightly different.
Wait, I just realized: for the specific case where mu=0 and sigma=1, `_log_prob(x) = -0.5 * (x^2).sum(dim=-1)`. The sum of x^2 is always non-negative, and with randn inputs, x^2 ~ chi-squared(1), mean=1, var=2. Sum of 64 is chi-squared(64), mean=64, var=128. sd≈11.3.
For a single row, sum(x^2) ≈ 64. ULP at 64 in fp32 is 64 * 2^-23 ≈ 7.6e-6. So even if our reduction differs by a few ULPs, the absolute error is ~2e-5. And log_alpha = lp_prop - lp_x, where each is -0.5 * sum ≈ -32. The difference is between two sums of size ~64, so the error in log_alpha is also ~2e-5. Then accepted = log_u < log_alpha. If log_alpha is near log_u, a 2e-5 difference might flip the acceptance bit!
If the acceptance bit flips, x_new will be completely different for that row (proposal vs x), leading to large errors! This is a real concern.
So we need to ensure our reduction matches PyTorch's very closely, or we need an alternative approach.
Alternative approach: compute log_prob(x) using PyTorch's exact sum, store it, then use it in the fused kernel. But then we lose the fusion benefit for the log_prob computation.
Wait, we don't have to fuse the reduction perfectly. We can:
1. In Python, call `torch.randn_like(x)` and `torch.rand(x.shape[0])` as before.
2. Compute `proposal = x + step_size * noise` using PyTorch (1 kernel).
3. Compute `lp_x` and `lp_prop` using PyTorch's `_log_prob` (2 kernels).
4. Pass `lp_x`, `lp_prop`, `proposal`, `x`, and `uniforms` to a fused kernel that does the rest.
The fused kernel would be tiny: clamp uniforms, take log, compare, and select. That's still a big win from reducing launches, and we guarantee exact log_prob values.
But even simpler: we can write a fused kernel that computes `proposal`, computes element-wise `(proposal - mu)^2 / sigma^2` and `(x - mu)^2 / sigma^2`, stores them in shared memory, and then does the reduction using a block-level algorithm that might match PyTorch more closely.
Actually, the most robust approach: just use PyTorch for the log_probs. Since sigma=1 and mu=0, we can do:
```python
def forward(self, x):
noise = torch.randn_like(x)
proposal = x + self.step_size * noise
lp_x = -0.5 * ((x - self.mu) / self.sigma).pow(2).sum(dim=-1)
lp_prop = -0.5 * ((proposal - self.mu) / self.sigma).pow(2).sum(dim=-1)
log_alpha = lp_prop - lp_x
uniforms = torch.rand(x.shape[0], device=x.device).clamp(min=1e-10)
log_u = torch.log(uniforms)
accepted = log_u < log_alpha
x_new = torch.where(accepted.unsqueeze(-1), proposal, x)
return x_new, accepted
```
This is the reference! We can't beat it this way.
Let me think more carefully about the reduction accuracy. The reference computes `_log_prob` as:
```python
-0.5 * (((x - self.mu) / self.sigma) ** 2).sum(dim=-1)
```
In PyTorch CUDA, `.sum()` on a tensor of shape [4096, 64] uses a reduction algorithm. The exact implementation depends on the PyTorch version, but for small inner dimensions, it might use a simple serial reduction per thread or a warp-based reduction.
Actually, for a [4096, 64] tensor, `sum(dim=-1)`:
1. Each thread loads multiple elements from the last dimension
2. Threads in a block partially reduce
3. The partial results are further reduced
For dim=64, the reduction is very small. PyTorch might use `cub::BlockReduce` or a custom warp shuffle reduction.
Let me check if PyTorch's sum uses pairwise summation or sequential summation. For small reductions, it's likely sequential within each thread, then tree reduction across threads. With 64 elements and float32, if each thread handles 2 elements, it does sequential sum of 2, then warp shuffle tree reduction of 32 partial sums. This is very similar to our approach.
The key question: will our reduction and PyTorch's reduction differ by more than 1-2 ULPs? For 64 positive numbers, sequential summation vs tree summation can differ slightly. But with 64 numbers of size ~1 and sum ~64, the error is bounded by ~64 * eps * max_magnitude ≈ 64 * 1.2e-7 * 1 ≈ 7.7e-6 in the worst case for sequential summation. Tree summation might be slightly better. The difference between sequential and tree is probably < 1e-5.
But here's the thing: our kernel computes both log_p_x and log_p_prop. If we compute them with the SAME reduction pattern, then `log_alpha = log_p_prop - log_p_x` will have correlated errors that partially cancel. The reference also computes both with the SAME PyTorch sum algorithm. So both should be consistent within themselves.
The issue is whether our sum matches PyTorch's sum. If both use tree-like reductions, they might match well. If PyTorch uses a different approach, there could be differences.
I just realized something important: for `sum(dim=-1)` on a [4096, 64] tensor, PyTorch doesn't necessarily use warp shuffles for the entire reduction. It might launch blocks where each block handles one row, with threads sequentially summing elements and then reducing with shared memory. The exact algorithm might involve:
- Block size: maybe 512 threads
- Each thread sums `ceil(64/512) = 1` element (or some handle multiple)
- Shared memory reduction of partial sums
- Final result per row
But 512 threads for 64 elements means most threads do nothing. PyTorch is smarter than that - it probably uses fewer threads per row or handles multiple rows per block.
Actually, for a [4096, 64] tensor summing over dim=1, PyTorch might use TensorIterator with a specific reduction path. For small reduction dimensions, it might use `sum_kernel_impl` which has special handling.
I think the safest approach is: compute everything in our kernel, but make the reduction algorithm explicitly match a common pattern, and then verify correctness through the testing tool. If it passes, great. If not, we'll adjust.
Let me compile and test.
Actually, I just had another idea. What if instead of a warp shuffle reduction, we use `atomicAdd` to a shared memory location per row? With only 32 threads adding to one location, atomicAdd should be fast (no contention across warps, minimal contention within warp since adds are sequentialized but fast).
```cuda
__shared__ float s_log_p_x[8];
__shared__ float s_log_p_prop[8];
if (col == 0) {
s_log_p_x[threadIdx.y] = 0.0f;
s_log_p_prop[threadIdx.y] = 0.0f;
}
__syncwarp();
for (int c = col; c < dim; c += 32) {
// ... compute contributions ...
atomicAdd(&s_log_p_x[threadIdx.y], -0.5f * diff_x * diff_x);
atomicAdd(&s_log_p_prop[threadIdx.y], -0.5f * diff_prop * diff_prop);
x_new[...] = prop_val;
}
__syncwarp();
if (col == 0) {
float log_p_x = s_log_p_x[threadIdx.y];
float log_p_prop = s_log_p_prop[threadIdx.y];
// ... compute acceptance ...
}
```
Wait, `atomicAdd` to shared memory among threads in the same warp has some overhead but it's serialized across the 32 threads. Each thread does one atomicAdd per element it processes. For dim=64 and 32 threads, each thread does 2 atomicAdds. Total 64 atomicAdds per row. An atomicAdd to shared memory takes about 1-2 cycles on modern GPUs? Actually, within a warp, atomic operations to the same address are serialized. If all 32 threads try to atomicAdd to `s_log_p_x[threadIdx.y]` simultaneously, they'll serialize. Since `threadIdx.y` is the same for all threads in the warp, they all contend on the same address!
Shared memory atomicAdd serialization: each atomic takes ~10-20 cycles when contended. With 32 threads doing 2 atomics each = 64 serialized operations per row, that's ~640-1280 cycles. At 3GHz, that's ~0.2-0.4 microseconds per row. For 4096 rows and 512 blocks (8 rows each), that's... well, it's serialized within each warp, but warps run independently. A block has 8 warps, so 8 rows are processed in parallel. Each row takes ~0.3us. 512 blocks each take ~0.3us, but they run in parallel on SMs. With 132 SMs on H100, maybe ~4 blocks per SM at once? Not sure. But the atomics might not be a huge bottleneck.
However, warp shuffle is O(log n) and fully parallel within the warp. Much faster than 64 serialized atomics. So warp shuffle is better.
OK, I'll stick with warp shuffle and test correctness. If it fails due to precision, I'll try alternative reduction ordering.
Let me write the code and compile it.
Actually, I just thought of one more thing. In our kernel:
```cpp
float diff_x = (x_val - mu[c]) / sigma[c];
log_p_x += -0.5f * diff_x * diff_x;
```
This is `-0.5f * (x_val - mu[c]) * (x_val - mu[c]) / (sigma[c] * sigma[c])`. If sigma[c] = 1, this is just `-0.5f * diff * diff`. The compiler might not optimize away the division if it doesn't know sigma=1.
Since the reference always sets sigma=ones, and the benchmark tests this exact case, we could add a fast path:
```cpp
bool sigma_is_one = (sigma[0] == 1.0f); // Check first element
// ...
```
But that's messy. Let me just keep it general. Division by 1.0 is fast anyway.
One more micro-optimization: precompute `-0.5f / (sigma[c] * sigma[c])` or `-0.5f / sigma[c]`? The reference computes `((x - mu) / sigma) ** 2`, which is `(x - mu)^2 / sigma^2`. We compute it the same way.
Actually, wait: `(x - mu) / sigma` first does subtraction and division, then square. We do the same. Good.
Let me compile the code.
Actually, let me reconsider whether we can also fuse the `torch.rand` generation. If we generate the uniform inside the kernel using a simple LCG based on row index and a seed, we don't need `torch.rand` at all. But then the random numbers differ from the reference. However, since the operation is stochastic, the evaluation might just verify that the output has the correct statistical properties and shape.
Hmm, actually for the correctness check, the harness probably seeds the PyTorch RNG, calls both models, and compares the outputs. If our model uses different random numbers, the outputs won't match. So we MUST use PyTorch's random numbers.
Wait, could we seed our custom RNG with the same seed as PyTorch's RNG? PyTorch's global RNG state is accessible via `torch.cuda.get_rng_state()`. But interpreting that state to produce the same sequence as `torch.rand` is extremely complex. So no, we must call PyTorch's RNG functions.
Actually, maybe we can use `torch.rand` to generate not just the uniforms but also proposal noise in a single call? No, because we need normals for proposal and uniforms for acceptance.
One creative idea: we can call `torch.randn_like(x)` to get proposal noise (matching reference). For uniforms, instead of `torch.rand`, we can generate uniforms from the SAME noise tensor using a transform. For example, `uniform = torch.sigmoid(noise[:, 0])` or something. But this destroys the stochastic properties.
Alternatively, `torch.rand` is cheap. Let me not over-optimize that part.
OK, compiling now.
Actually, I want to check if there are any syntax issues with my CUDA code. `__shfl_down_sync` takes a mask, a value, and an offset. I have `__shfl_down_sync(0xffffffff, log_p_x, offset)`. That looks correct.
Also, `bool* accepted` - I should make sure this works with PyTorch's bool tensor. Let me check: in PyTorch C++, `at::kBool` tensors can be accessed with `data_ptr<bool>()`. Yes, this should work.
Let me also check: `torch::empty({B}, torch::TensorOptions().dtype(torch::kBool).device(x.device()))` - this creates a bool tensor on the same device. Good.
One final optimization: since `mu` and `sigma` are module buffers, they might be on CPU if the module was created on CPU and then moved to GPU. But `register_buffer` with `torch.zeros(dim)` creates on CPU by default. Then when the model is moved to GPU with `.cuda()`, they're moved too. In the benchmark, the model is likely created and inputs are provided on GPU. Let me make sure.
Actually, the reference `Model` doesn't explicitly move buffers to CUDA. The benchmark probably creates the model, calls `.to(device)` or something, or inputs are on CPU. Looking at the reference forward: `torch.rand(x.shape[0], device=x.device)` - it uses x's device. So if x is on CUDA, the model should also be on CUDA.
In our `ModelNew`, when the module is moved to a device, `self.mu` and `self.sigma` move with it because they're registered buffers. Good.
But wait, in `forward`, we pass `self.mu` and `self.sigma` to the C++ function. If they're on CPU and x is on CUDA, the C++ function would crash or get wrong results. We should ensure they're on the same device as x. The benchmark probably handles this, but let's be safe: in `forward`, we can add `self.mu = self.mu.to(x.device)` and `self.sigma = self.sigma.to(x.device)`. But that creates copies if they're already on the device. Better: `mu = self.mu.to(x.device, non_blocking=True)`? Hmm, unnecessary overhead for the benchmark.
Actually, let me just assume the benchmark sets everything up correctly.
Let me compile.
Hmm, actually I want to also include `-arch=sm_90` or similar for H100? The `load_inline` might not add it automatically. But the default architecture might be sufficient. If not, we can add it via `extra_cuda_cflags`. H100 is sm_90. Let me add `"-arch=sm_90"`.
Wait, adding `-arch=sm_90` might cause compilation issues if the CUDA toolkit doesn't support it. But for H100, it should. Let me add it.
Hmm, but there's a risk: if the compilation environment uses an older CUDA toolkit that doesn't know sm_90, the compilation will fail. The evaluation environment might use a toolkit that supports H100. Let me check... H100 was released in 2022, so CUDA 11.8+ or 12.0+ should support it. I'll add `"-arch=sm_90"`.
Actually, PyTorch's default compilation flags usually add the appropriate architecture. So I might not need to add it manually. Let me just use `"-O3"` and see.
Now compiling...
Wait, I don't have a code editor in this environment. I'll write the complete Python string and call compile_kernel.
One more thought: should I use `const float* __restrict__` for all input pointers? Yes, this helps the compiler optimize.
Also, I noticed I wrote `#pragma unroll` before the shuffle loop. That's good since the loop has a fixed number of iterations (4: 16, 8, 4, 2, 1... wait, offset goes 16, 8, 4, 2, 1, which is 5 iterations). The `#pragma unroll` will unroll this small loop.
Let me also add `__forceinline__` to the kernel. Not necessary but good practice.
OK, here's the code I'll compile: