reasoning
Let me analyze this task carefully.
The reference implementation is a channel-wise gated delta attention recurrence. Let me understand what it does:
1. For each timestep t in sequence:
- gate_t = gate[:, t] with shape (batch_size, num_heads, 1, head_dim)
- kv_outer = k[:, t].unsqueeze(-1) * v[:, t].unsqueeze(-2)
- k[:, t] has shape (batch_size, num_heads, head_dim)
- k[:, t].unsqueeze(-1) has shape (batch_size, num_heads, head_dim, 1)
- v[:, t].unsqueeze(-2) has shape (batch_size, num_heads, 1, head_dim)
- kv_outer has shape (batch_size, num_heads, head_dim, head_dim) - outer product
- state = gate_t * state + kv_outer (element-wise decay and update)
- y_t = (q[:, t].unsqueeze(-1) * state).sum(dim=-2)
- q[:, t].unsqueeze(-1) has shape (batch_size, num_heads, head_dim, 1)
- state has shape (batch_size, num_heads, head_dim, head_dim)
- q[:, t].unsqueeze(-1) * state has shape (batch_size, num_heads, head_dim, head_dim)
- sum over dim=-2 gives shape (batch_size, num_heads, head_dim)
So the state is a (batch_size, num_heads, head_dim, head_dim) tensor, and for each timestep:
- We compute an outer product of k and v
- We apply a gated decay to the state and add the new kv outer product
- We query the state with q
This is essentially a recurrent attention mechanism where the state accumulates kv outer products with exponential decay controlled by gate.
The output has shape (batch_size, seq_len, num_heads, head_dim). To implement this efficiently in CUDA, I need to consider parallelization strategies. Since the state update is sequential along the sequence dimension, each batch-head combination must process its 128 timesteps in order. I could assign one thread block per (batch, head) pair and process all timesteps sequentially within the block, which would reduce state storage overhead. Alternatively, I could parallelize across batch elements and use sequential processing for each head's timesteps.
The main computational bottlenecks are the outer product computation requiring head_dim^2 operations, the state decay involving head_dim^2 multiply-adds, the query-state multiplication needing head_dim^2 operations, and the final reduction sum across head_dim elements. Given that head_dim is only 24, the state itself is quite compact at 576 elements per (batch, head) pair, making it feasible to store in shared memory or even registers.
To parallelize this, I can assign each thread to compute one output element by maintaining the relevant state slice. With 24 threads per block (one per head dimension), each thread handles one column of the state matrix across all timesteps, accumulating the kv outer products into that column and then applying the query vector to produce its output element. For the 576-element state matrix, I could assign one thread per output element, giving 24 threads that each compute one y_t[i] value. Each thread would need to maintain its own slice of the state vector.
Alternatively, I could parallelize across the entire (batch, heads, head_dim) space with 192 threads total. This gets more complex with synchronization requirements across the sequence dimension.
The simplest approach would assign one thread per output element, keeping the full state vector in shared memory since it's only 24×24 elements. With seq_len=128 timesteps to process, each thread loops through the sequence while maintaining its accumulated state. blockIdx.y * num_heads + blockIdx.z; // head
int d = threadIdx.x; // head_dim index
// State stored in shared memory
__shared__ float state_s[head_dim][head_dim];
// Initialize state to zero
for (int i = 0; i < head_dim; i++) {
state_s[d][i] = 0.0f;
}
__syncthreads();
// Process each timestep
for (int t = 0; t < seq_len; t++) {
float k_t = k[b * seq_len * num_heads * head_dim + t * num_heads * head_dim + h * head_dim + d];
float q_t = q[b * seq_len * num_heads * head_dim + t * num_heads * head_dim + h * head_dim + d];
// Update state with the current key and value
for (int i = 0; i < head_dim; i++) {
float v_t = v[b * seq_len * num_heads * head_dim + t * num_heads * head_dim + h * head_dim + i];
state_s[d][i] += k_t * v_t;
}
// Compute output for this timestep
float y_t = 0.0f;
for (int i = 0; i < head_dim; i++) {
y_t += q_t * state_s[d][i];
}
output[b * seq_len * num_heads * head_dim + t * num_heads * head_dim + h * head_dim + d] = y_t;
}
}
```
I'm simplifying the indexing logic and combining the state update loop with the output computation. The gate is applied to decay the previous state before adding the new key-value contribution. The outer product between k_t and all v values updates each column of the state matrix, then the query vector multiplies with the state to produce the output.
```cuda
__global__ void gated_attention_kernel(
const float* __restrict__ q,
const float* __restrict__ k,
const float* __restrict__ v,
const float* __restrict__ gate,
float* __restrict__ output,
int seq_len, int num_heads, int head_dim
) {
int b = blockIdx.x; // batch
int h = blockIdx.y; // head
int d = threadIdx.x; // head_dim index
__shared__ float state[24][24]; // Fixed size for head_dim=24
// Initialize state to zero
for (int i = 0; i < head_dim; i++) {
state[d][i] = 0.0f;
}
__syncthreads();
// Process each timestep
for (int t = 0; t < seq_len; t++) {
// Load gate
float g = gate[((b * seq_len + t) * num_heads + h) * head_dim + d];
// Decay state
for (int i = 0; i < head_dim; i++) {
state[d][i] *= g;
}
__syncthreads();
// Load k and v for this timestep
float k_t = k[((b * seq_len + t) * num_heads + h) * head_dim + d];
// Update state with kv outer product
for (int i = 0; i < head_dim; i++) {
float v_i = v[((b * seq_len + t) * num_heads + h) * head_dim + i];
state[d][i] += k_t * v_i;
}
__syncthreads();
// Compute output
float q_t = q[((b * seq_len + t) * num_heads + h) * head_dim + d];
float y_t = 0.0f;
for (int i = 0; i < head_dim; i++) {
y_t += q_t * state[i][d];
}
output[((b * seq_len + t) * num_heads + h) * head_dim + d] = y_t;
}
}
```
The kernel processes one batch and head per block, with each thread handling one dimension of the head. The state matrix gets decayed by the gate factor at each timestep, then updated with the outer product of k and v vectors. Each thread accumulates its contribution to the output by multiplying the query vector with the corresponding column of the state matrix. The `gate_t` has shape (batch_size, num_heads, 1, head_dim), so when multiplied with the state tensor of shape (batch_size, num_heads, head_dim, head_dim), it broadcasts across the second-to-last dimension. Each row of the state gets multiplied by the corresponding gate value, meaning `state[b, h, i, j] *= gate[b, t, h, j]` for all i values.
So in the CUDA kernel, I need to apply the gate decay as `state[d][i] *= g` where `g` corresponds to `gate[..., d]`, but I should reconsider the indexing—perhaps using `state[i][d] *= gate[..., d]` would be clearer.
Actually, let me trace through the shapes more carefully. With `state[b,h,:,:]` as a (head_dim, head_dim) matrix and `gate_t[b,h,0,:]` as a (head_dim,) vector, the operation `gate_t * state` multiplies each column j by `gate[..., j]`. In the kernel where d indexes head_dim, I need `state[i][d] *= gate_value` for each row i, meaning the gate at position d scales all rows in column d.
The decay step becomes:
```cuda
for (int i = 0; i < head_dim; i++) {
state[i][d] *= g;
}
```
For the kv outer product, since k_t is indexed at position d and multiplied across all positions i of v, this adds k_t * v_i to each row of column d in the state matrix:
```cuda
for (int i = 0; i < head_dim; i++) {
state[i][d] += k_t * v_i;
}
```
These two operations work on the same column, so I could potentially merge them. Looking at the dimensions: k has shape (batch_size, num_heads, head_dim) and becomes (batch_size, num_heads, head_dim, 1) after unsqueeze, while v becomes (batch_size, num_heads, 1, head_dim). The outer product k.unsqueeze(-1) * v.unsqueeze(-2) produces a (batch_size, num_heads, head_dim, head_dim) matrix, which then gets added to the state.
Each batch and head operates independently, so I can parallelize across those dimensions. Within each operation, the computations for different dimensions don't overlap—each d value updates its own column without interfering with others. This means I can remove synchronization between the decay and outer product steps, though I still need barriers when transitioning between timesteps to ensure all threads have completed their updates before moving forward.
However, I'm reading state[i][d] where i belongs to the current thread but d could come from any other thread. After the outer product update, another thread might modify state[d][i] before I've finished reading, creating a race condition. So I need to keep the __syncthreads() calls to ensure all threads complete their reads and writes before proceeding to the next timestep.
I should test a simpler version first—perhaps one thread per output element with proper synchronization—before trying more aggressive optimizations like eliminating the barriers. Let me start with a straightforward implementation and verify correctness.