Let me analyze this task. I need to implement a custom CUDA kernel for a scalar-gated delta-rule linear attention recurrence. The reference implementation: 1. Initializes a state matrix of shape (batch_size, num_heads, head_dim, head_dim) to zeros 2. For each timestep t in seq_len: - gate_t = gate[:, t] reshaped to (batch_size, num_heads, 1, 1) - beta_t = beta[:, t] reshaped to (batch_size, num_heads, 1, 1) - kv_outer = outer product of k[:, t] and v[:, t], shape (batch_size, num_heads, head_dim, head_dim) - state = gate_t * state + beta_t * kv_outer - y_t = sum over dim=-2 of (q[:, t].unsqueeze(-1) * state), shape (batch_size, num_heads, head_dim) 3. Stack all outputs This is a sequential operation, so I need to think about how to parallelize it effectively. The key insight is that for each batch and head, the computation is independent - so we can parallelize over batch_size * num_heads threads. For each (batch, head) pair, we have a state matrix of size head_dim x head_dim, and we process seq_len timesteps sequentially. Let me design a CUDA kernel: 1. Each thread block handles one (batch, head) pair 2. Within the block, we use shared memory to store the state matrix and cooperate 3. We iterate through seq_len timesteps Given the dimensions: - batch_size = 2 - seq_len = 128 - num_heads = 4 - head_dim = 24 The state matrix is 24x24 = 576 floats per (batch, head). That's 2304 bytes, which fits in shared memory. Let me think about the kernel design: Option 1: Each thread handles one (batch, head) pair and iterates through all timesteps sequentially. Option 2: Warp-level parallelism where threads within a warp cooperate on the matrix operations. Given head_dim = 24, I should consider the best approach. Let me write a kernel where each thread handles one (batch, head) pair, with threads cooperating via shared memory for the state matrix operations. The inputs are q: (B, T, H, D), k: (B, T, H, D), v: (B, T, H, D), gate: (B, T, H), and beta: (B, T, H), and the output is y: (B, T, H, D). The computation loops through each timestep, multiplying state by the gate value, adding the beta-scaled outer product of k and v, then computing y as the dot product of q with state. For parallelization, I could have each thread block process one (batch, head) combination, using shared memory to store the state matrix and coordinate updates across threads. Since head_dim = 24, a single thread could manage the entire state for its (batch, head) pair without needing extensive shared memory coordination. Each thread would maintain a 24×24 state matrix in registers, iterating through 128 timesteps and computing outer products and dot products along the way. This straightforward approach might work well given the small head dimension. However, I should consider whether a cooperative approach could yield better performance. Instead of one thread handling all 576 elements of the state matrix, multiple threads could divide the work. For instance, 24 threads could each manage one row of the state matrix, computing the outer product and applying the gate and beta coefficients in parallel before writing back the result. With 24 threads per thread block handling one (batch, head) pair, this could potentially improve throughput through better register utilization and parallel computation. _t[b * num_heads + h] = gate[b, t, h] float beta_t = beta[b * num_heads * seq_len + h * seq_len + t]; // Load k and v for this timestep for (int d = 0; d < head_dim; ++d) { k_local[d] = k[b * num_heads * seq_len * head_dim + h * seq_len * head_dim + t * head_dim + d]; v_local[d] = v[b * num_heads * seq_len * head_dim + h * seq_len * head_dim + t * head_dim + d]; } // Load q for this timestep for (int d = 0; d < head_dim; ++d) { q_local[d] = q[b * num_heads * seq_len * head_dim + h * seq_len * head_dim + t * head_dim + d]; } // Update state: state = gate_t * state + beta_t * k * v^T for (int i = 0; i < head_dim; ++i) { for (int j = 0; j < head_dim; ++j) { state[i][j] = gate_t * state[i][j] + beta_t * k_local[i] * v_local[j]; } } // Compute output: y = q @ state for (int j = 0; j < head_dim; ++j) { float sum = 0.0f; for (int i = 0; i < head_dim; ++i) { sum += q_local[i] * state[i][j]; } y[b * num_heads * seq_len * head_dim + h * seq_len * head_dim + t * head_dim + j] = sum; } } ``` The kernel processes each batch and head combination independently. For each timestep, it loads the gate and beta scalars, reads the key, value, and query vectors, then updates the recurrent state through the gated outer product formulation before computing the output as a matrix-vector product. I need to reconsider the memory access patterns. The tensor dimensions follow (B, T, H, D) ordering, so for gate with shape (B, T, H), indexing should be `gate[b, t, h] = gate[b * T * H + t * H + h]`. The current implementation has this backward for the gate and beta indexing—it should stride by T*H for the batch dimension, not H*T. For the full 4D tensors like q with shape (B, T, H, D), the correct offset is `b * T * H * D + t * H * D + h * D + d`. The state matrix grows to (B, H, D, D), which is 4608 elements total. Since head_dim = 24 and num_heads = 4, that's only 576 floats per head pair—manageable in registers across a thread block. A kernel where each thread handles one (batch, head) pair, iterating through all timesteps sequentially, should work well here. I need to verify the computation matches the reference: at each step, gate_t scales the existing state, then I add the outer product of k and v scaled by beta_t, and finally multiply by q to get y_t. The shapes check out—gate_t and beta_t are both (B, H, 1, 1), the outer product creates (B, H, D, D), and the query multiplication produces (B, H, D). Now I'll implement this with proper CUDA indexing. beta[b * seq_len * num_heads + t * num_heads + h]; // Load k[t], v[t], q[t] for this (batch, head) for (int d = 0; d < head_dim; ++d) { float k_val = k[b * seq_len * num_heads * head_dim + t * num_heads * head_dim + h * head_dim + d]; float v_val = v[b * seq_len * num_heads * head_dim + t * num_heads * head_dim + h * head_dim + d]; float q_val = q[b * seq_len * num_heads * head_dim + t * num_heads * head_dim + h * head_dim + d]; k_local[d] = k_val; v_local[d] = v_val; q_local[d] = q_val; } // Update state: state = gate_t * state + beta_t * (k x v^T) for (int i = 0; i < head_dim; ++i) { for (int j = 0; j < head_dim; ++j) { state[i][j] = gate_t * state[i][j] + beta_t * k_local[i] * v_local[j]; } } // Compute output: y = q * state for (int j = 0; j < head_dim; ++j) { float sum = 0.0f; for (int i = 0; i < head_dim; ++i) { sum += q_local[i] * state[i][j]; } y[b * seq_len * num_heads * head_dim + t * num_heads * head_dim + h * head_dim + j] = sum; } } } ``` I need to verify the indexing is correct for each dimension. For the gate tensor with shape (B, T, H), each element should be accessed as gate[b][t][h], which maps to offset b*T*H + t*H + h. Similarly for beta. The 4D tensors like k follow the pattern k[b][t][h][d] = b*T*H*D + t*H*D + h*D + d. The indexing I'm using matches these patterns correctly.

Now let me analyze the computation: For each timestep t: 1. gate_t = gate[:, t].unsqueeze(-1).unsqueeze(-1) -> shape (batch, num_heads, 1, 1) 2. beta_t = beta[:, t].unsqueeze(-1).unsqueeze(-1) -> shape (batch, num_heads, 1, 1) 3. kv_outer = k[:, t].unsqueeze(-1) * v[:, t].unsqueeze(-2) -> outer product, shape (batch, num_heads, head_dim, head_dim) 4. state = gate_t * state + beta_t * kv_outer -> update state (batch, num_heads, head_dim, head_dim) 5. y_t = (q[:, t].unsqueeze(-1) * state).sum(dim=-2) -> compute output, shape (batch, num_heads, head_dim) The state is a 4D tensor of shape (batch, num_heads, head_dim, head_dim). For the given dimensions: - batch_size = 2 - seq_len = 128 - num_heads = 4 - head_dim = 24 State size: 2 * 4 * 24 * 24 = 4608 floats = 18KB, which easily fits in shared memory. Key observations: 1. The computation for different batches and heads is independent - we can parallelize over (batch, heads). 2. The sequential dependency is across timesteps, so we need to iterate through them within each thread block. 3. We can use shared memory to store the state matrix. 4. We can have multiple threads cooperate on the matrix operations. Let me think about parallelization: - We could have one thread block per (batch, head) pair, or - We could have warps working together within thread blocks Given the small head_dim (24), each state matrix is 24x24 = 576 elements. Strategy: - Launch a kernel with thread blocks for each (batch, head) pair - Each block has multiple threads that cooperate on the state - Use shared memory for the state matrix - Iterate through timesteps sequentially within each block Let me design the kernel: Option 1: One thread per (batch, head) handles all computations - Simple but each thread does 24*24 work per timestep - No parallelism within the state computation Option 2: Warp per (batch, head) - 32 threads for 24*24 = 576 elements - Each thread handles 18 elements on average - Can parallelize the outer product and the query-state multiplication Option 3: Thread block per (batch, head) - Similar to warp but more threads For head_dim=24 and seq_len=128: - Total elements in output: 2 * 128 * 4 * 24 = 24,576 - State updates per (batch, head): 128 * 24 * 24 = 73,728 operations Let me write a first version: ```python import torch import torch.utils.cpp_extension as cpp_extension cuda_source = """ #include <torch/extension.h> #include <cuda.h> #include <cuda_runtime.h> template <typename scalar_t> __global__ void linear_attention_kernel( const scalar_t* __restrict__ q, const scalar_t* __restrict__ k, const scalar_t* __restrict__ v, const scalar_t* __restrict__ gate, const scalar_t* __restrict__ beta, scalar_t* __restrict__ output, int batch_size, int seq_len, int num_heads, int head_dim) { // Map block to (batch, head) int batch = blockIdx.x / num_heads; int head = blockIdx.x % num_heads; int tid = threadIdx.x; int block_size = blockDim.x; // Shared memory for state (head_dim x head_dim) extern __shared__ char shared_mem[]; scalar_t* state = reinterpret_cast<scalar_t*>(shared_mem); // Initialize state to zero for (int i = tid; i < head_dim * head_dim; i += block_size) { state[i] = scalar_t(0); } __syncthreads(); // Base indices for this (batch, head) int bh_offset = batch * num_heads * seq_len + head * seq_len; int bh_state_offset = batch * num_heads * head_dim * head_dim + head * head_dim * head_dim; // Iterate through sequence for (int t = 0; t < seq_len; t++) { scalar_t gate_t = gate[batch * num_heads * seq_len + head * seq_len + t]; scalar_t beta_t = beta[batch * num_heads * seq_len + head * seq_len + t]; // Get k[t] and v[t] for this (batch, head) const scalar_t* k_t = k + (batch * seq_len * num_heads * head_dim + t * num_heads * head_dim + head * head_dim); const scalar_t* v_t = v + (batch * seq_len * num_heads * head_dim + t * num_heads * head_dim + head * head_dim); const scalar_t* q_t = q + (batch * seq_len * num_heads * head_dim + t * num_heads * head_dim + head * head_dim); // Update state: state = gate_t * state + beta_t * k[t] * v[t]^T // Each thread handles some elements of the state matrix for (int i = tid; i < head_dim * head_dim; i += block_size) { int row = i / head_dim; int col = i % head_dim; state[i] = gate_t * state[i] + beta_t * k_t[row] * v_t[col]; } __syncthreads(); // Compute output: y[t] = q[t]^T @ state -> sum over rows // y[j] = sum_i q[i] * state[i, j] scalar_t* y_t = output + (batch * seq_len * num_heads * head_dim + t * num_heads * head_dim + head * head_dim); for (int j = tid; j < head_dim; j += block_size) { scalar_t sum = scalar_t(0); for (int i = 0; i < head_dim; i++) { sum += q_t[i] * state[i * head_dim + j]; } y_t[j] = sum; } __syncthreads(); } } torch::Tensor linear_attention_cuda( torch::Tensor q, torch::Tensor k, torch::Tensor v, torch::Tensor gate, torch::Tensor beta) { int batch_size = q.size(0); int seq_len = q.size(1); int num_heads = q.size(2); int head_dim = q.size(3); auto output = torch::zeros_like(q); int blocks = batch_size * num_heads; int threads = 128; // More than enough for head_dim=24 int shared_mem_size = head_dim * head_dim * sizeof(float); AT_DISPATCH_FLOATING_TYPES(q.scalar_type(), "linear_attention_kernel", ([&] { linear_attention_kernel<scalar_t><<<blocks, threads, shared_mem_size>>>( q.data_ptr<scalar_t>(), k.data_ptr<scalar_t>(), v.data_ptr<scalar_t>(), gate.data_ptr<scalar_t>(), beta.data_ptr<scalar_t>(), output.data_ptr<scalar_t>(), batch_size, seq_len, num_heads, head_dim); })); return output; } """ module = cpp_extension.load_inline( name='linear_attention', cpp_sources=['torch::Tensor linear_attention_cuda(torch::Tensor q, torch::Tensor k, torch::Tensor v, torch::Tensor gate, torch::Tensor beta);'], cuda_sources=[cuda_source], functions=['linear_attention_cuda'], extra_cuda_cflags=['-O3'], ) class ModelNew(nn.Module): def __init__(self): super().__init__() self.module = module def forward(self, q, k, v, gate, beta): return self.module.linear_attention_cuda(q, k, v, gate, beta) ``` Wait, I need to be more careful about the tensor layout. Let me check: q.shape = (batch_size, seq_len, num_heads, head_dim) So for q[:, t, h, :], the index would be: - batch * seq_len * num_heads * head_dim + t * num_heads * head_dim + h * head_dim + d Let me rewrite more carefully. Actually, let me trace through more carefully. The reference does: - k[:, t] has shape (batch, num_heads, head_dim) - k[:, t].unsqueeze(-1) -> (batch, num_heads, head_dim, 1) - v[:, t] has shape (batch, num_heads, head_dim) - v[:, t].unsqueeze(-2) -> (batch, num_heads, 1, head_dim) - kv_outer = k[:, t].unsqueeze(-1) * v[:, t].unsqueeze(-2) -> (batch, num_heads, head_dim, head_dim) So kv_outer[b, h, i, j] = k[b, t, h, i] * v[b, t, h, j] And the query computation: - q[:, t] has shape (batch, num_heads, head_dim) - q[:, t].unsqueeze(-1) -> (batch, num_heads, head_dim, 1) - q[:, t].unsqueeze(-1) * state -> (batch, num_heads, head_dim, head_dim) - sum(dim=-2) -> (batch, num_heads, head_dim) So y[b, t, h, j] = sum_i q[b, t, h, i] * state[b, h, i, j] Let me write the kernel more carefully.

The static check passes. Now let me submit the kernel. But before that, let me think about potential optimizations: 1. Current implementation: Each block handles one (batch, head) pair, iterates through sequence 2. With head_dim=24, state is 24x24 = 576 elements = 2.25KB for fp32 3. For each timestep, we: - Load gate and beta (1 value each) - Load k, v, q vectors (24 values each) - Update state matrix (576 FMA operations) - Compute output (24 * 24 = 576 multiplications + 24 * 23 additions = 24 dot products) This should be quite efficient. Let me submit and see how it performs.