{
  "summary": "Starting the investigation of the paged attention decode kernel. We will read reference.py, check.py, and run the baseline performance measurements.",
  "strategic_intent": "Explore workspace files, read the reference implementation and benchmark script, and run check.py/benchmark.py.",
  "title": "Initial Exploration and Profiling Baseline"
}

{
  "file_path": "reference.py"
}

{
  "file_path": "check.py",
  "start_line": 1,
  "end_line": 200
}

{
  "file_path": "shapes.py"
}

{
  "file_path": "benchmark.py"
}

{
  "file_path": "problem.yaml"
}

{
  "dir_path": "."
}

{
  "file_path": "sota.py"
}

{
  "description": "Check NVIDIA GPU details and Triton availability/version.",
  "command": "nvidia-smi; uv run python -c \"import triton; print('Triton version:', triton.__version__)\""
}

{
  "content": "<2545 chars \u2014 see below>",
  "file_path": "solution.py"
}

import math
import torch
import torch.nn as nn

class Model(nn.Module):
    def __init__(
        self,
        batch: int,
        num_heads: int,
        num_kv_heads: int,
        head_dim: int,
        seq_len: int,
        page_size: int,
    ):
        super().__init__()
        assert num_heads % num_kv_heads == 0, "num_heads must be a multiple of num_kv_heads (GQA)"
        self.batch = batch
        self.num_heads = num_heads
        self.num_kv_heads = num_kv_heads
        self.head_dim = head_dim
        self.seq_len = seq_len
        self.page_size = page_size
        self.group_size = num_heads // num_kv_heads
        self.scale = 1.0 / math.sqrt(head_dim)
        self.register_buffer("_dummy", torch.zeros(1, dtype=torch.bfloat16), persistent=False)

    def forward(
        self,
        query: torch.Tensor,
        kv_cache: torch.Tensor,
        block_table: torch.Tensor,
        seq_lens: torch.Tensor,
    ) -> torch.Tensor:
        B, H, D = query.shape
        Hkv = self.num_kv_heads
        G = self.group_size
        P = self.page_size

        out = torch.empty(B, H, D, dtype=query.dtype, device=query.device)

        for b in range(B):
            L = int(seq_lens[b].item())
            num_pages = (L + P - 1) // P
            pages = block_table[b, :num_pages].long()
            kv = kv_cache.index_select(0, pages)
            kv = kv.reshape(num_pages * P, Hkv, 2 * D)
            kv = kv[:L]
            k = kv[..., :D]
            v = kv[..., D:]

            k = k.repeat_interleave(G, dim=1)
            v = v.repeat_interleave(G, dim=1)

            q = query[b]
            qf = q.float()
            kf = k.float()
            vf = v.float()
            scores = torch.einsum("hd,lhd->hl", qf, kf) * self.scale
            probs = torch.softmax(scores, dim=-1)
            o = torch.einsum("hl,lhd->hd", probs, vf)
            out[b] = o.to(query.dtype)

        return out

def get_inputs():
    B = 8
    H = 32
    Hkv = 8
    D = 128
    L = 1024
    P = 16

    pages_per_seq = (L + P - 1) // P
    total_pages = max(B * pages_per_seq + 8, 64)

    query = torch.randn(B, H, D, dtype=torch.bfloat16) * 0.1
    kv_cache = torch.randn(total_pages, P, Hkv, 2 * D, dtype=torch.bfloat16) * 0.1

    perm = torch.randperm(total_pages)[: B * pages_per_seq].reshape(B, pages_per_seq).int()
    block_table = perm.contiguous()
    seq_lens = torch.full((B,), L, dtype=torch.int32)

    return [query, kv_cache, block_table, seq_lens]

def get_init_inputs():
    return [8, 32, 8, 128, 1024, 16]

{
  "description": "Run correctness checks.",
  "command": "uv run python check.py"
}

{
  "description": "Run performance benchmarking with baseline/sota comparisons.",
  "command": "KBH_BENCHMARK_BASELINES=1 uv run python benchmark.py"
}

{
  "file_path": "solution.py",
  "content": "<5250 chars \u2014 see below>"
}

import math
import torch
import torch.nn as nn
import triton
import triton.language as tl

@triton.jit
def paged_attention_kernel(
    Q_ptr,              # (batch, num_heads, head_dim)
    KV_ptr,             # (num_blocks, page_size, num_kv_heads, head_dim * 2)
    BlockTable_ptr,     # (batch, max_blocks)
    SeqLens_ptr,        # (batch,)
    Out_ptr,            # (batch, num_heads, head_dim)
    scale,              # float
    stride_qb, stride_qh, stride_qd,
    stride_kvb, stride_kvp, stride_kvh, stride_kvd,
    stride_btb, stride_bts,
    stride_ob, stride_oh, stride_od,
    group_size,
    num_kv_heads,
    head_dim: tl.constexpr,
    page_size: tl.constexpr,
):
    # Map program ID to batch and head
    h = tl.program_id(0)
    b = tl.program_id(1)

    # h_kv is the corresponding KV head
    h_kv = h // group_size

    # Load query q
    q_offset = b * stride_qb + h * stride_qh + tl.arange(0, head_dim) * stride_qd
    q = tl.load(Q_ptr + q_offset)

    # Sequence length for this batch element
    seq_len = tl.load(SeqLens_ptr + b)

    # Initialize online softmax accumulators
    m = -float('inf')
    d = 0.0
    # Initialize output accumulator to zeros
    o = tl.zeros((head_dim,), dtype=tl.float32)

    # Number of pages we need to process
    num_pages = (seq_len + page_size - 1) // page_size

    # 2D block offsets for loading KV (shape: page_size, head_dim * 2)
    t_in_page = tl.arange(0, page_size)[:, None]
    d_offset = tl.arange(0, head_dim * 2)[None, :]
    kv_block_offsets = t_in_page * stride_kvp + d_offset * stride_kvd

    # Loop over pages
    for p_idx in range(0, num_pages):
        # Load block ID from block table
        block_id = tl.load(BlockTable_ptr + b * stride_btb + p_idx * stride_bts)

        # Base pointer for this page and KV head
        page_base = KV_ptr + block_id * stride_kvb + h_kv * stride_kvh

        # Compute mask for sequence positions in this page
        token_indices = p_idx * page_size + tl.arange(0, page_size)
        mask = token_indices < seq_len

        # Load K and V together
        kv = tl.load(page_base + kv_block_offsets, mask=mask[:, None], other=0.0)

        # Split into K and V
        k = kv[:, 0:head_dim]
        v = kv[:, head_dim:2*head_dim]

        # Compute scores: sum(q * k, axis=1) * scale
        scores = tl.sum(q[None, :] * k, axis=1) * scale

        # Apply mask to scores
        scores = tl.where(mask, scores, -float('inf'))

        # Online softmax update
        m_old = m
        m_new = tl.maximum(m_old, tl.max(scores, axis=0))
        
        p = tl.exp(scores - m_new)
        alpha = tl.exp(m_old - m_new)
        d = d * alpha + tl.sum(p, axis=0)

        o = o * alpha + tl.sum(p[:, None] * v, axis=0)
        
        m = m_new

    # Final normalization
    o = o / d

    # Store output
    out_offset = b * stride_ob + h * stride_oh + tl.arange(0, head_dim) * stride_od
    tl.store(Out_ptr + out_offset, o.to(Out_ptr.dtype.element_ty))


class Model(nn.Module):
    def __init__(
        self,
        batch: int,
        num_heads: int,
        num_kv_heads: int,
        head_dim: int,
        seq_len: int,
        page_size: int,
    ):
        super().__init__()
        assert num_heads % num_kv_heads == 0, "num_heads must be a multiple of num_kv_heads (GQA)"
        self.batch = batch
        self.num_heads = num_heads
        self.num_kv_heads = num_kv_heads
        self.head_dim = head_dim
        self.seq_len = seq_len
        self.page_size = page_size
        self.group_size = num_heads // num_kv_heads
        self.scale = 1.0 / math.sqrt(head_dim)
        self.register_buffer("_dummy", torch.zeros(1, dtype=torch.bfloat16), persistent=False)

    def forward(
        self,
        query: torch.Tensor,
        kv_cache: torch.Tensor,
        block_table: torch.Tensor,
        seq_lens: torch.Tensor,
    ) -> torch.Tensor:
        B, H, D = query.shape
        out = torch.empty(B, H, D, dtype=query.dtype, device=query.device)

    

... (1250 more chars)

{
  "description": "Run check.py correctness tests.",
  "command": "uv run python check.py"
}

{
  "content": "<5322 chars \u2014 see below>",
  "file_path": "solution.py"
}

import math
import torch
import torch.nn as nn
import triton
import triton.language as tl

@triton.jit
def paged_attention_kernel(
    Q_ptr,              # (batch, num_heads, head_dim)
    KV_ptr,             # (num_blocks, page_size, num_kv_heads, head_dim * 2)
    BlockTable_ptr,     # (batch, max_blocks)
    SeqLens_ptr,        # (batch,)
    Out_ptr,            # (batch, num_heads, head_dim)
    scale,              # float
    stride_qb, stride_qh, stride_qd,
    stride_kvb, stride_kvp, stride_kvh, stride_kvd,
    stride_btb, stride_bts,
    stride_ob, stride_oh, stride_od,
    group_size,
    num_kv_heads,
    head_dim: tl.constexpr,
    page_size: tl.constexpr,
):
    # Map program ID to batch and head
    h = tl.program_id(0)
    b = tl.program_id(1)

    # h_kv is the corresponding KV head
    h_kv = h // group_size

    # Load query q
    q_offset = b * stride_qb + h * stride_qh + tl.arange(0, head_dim) * stride_qd
    q = tl.load(Q_ptr + q_offset)

    # Sequence length for this batch element
    seq_len = tl.load(SeqLens_ptr + b)

    # Initialize online softmax accumulators
    m = -float('inf')
    d = 0.0
    # Initialize output accumulator to zeros
    o = tl.zeros((head_dim,), dtype=tl.float32)

    # Number of pages we need to process
    num_pages = (seq_len + page_size - 1) // page_size

    # 2D block offsets for loading KV (shape: page_size, head_dim)
    t_in_page = tl.arange(0, page_size)[:, None]
    d_offset = tl.arange(0, head_dim)[None, :]
    k_block_offsets = t_in_page * stride_kvp + d_offset * stride_kvd
    v_block_offsets = t_in_page * stride_kvp + (d_offset + head_dim) * stride_kvd

    # Loop over pages
    for p_idx in range(0, num_pages):
        # Load block ID from block table
        block_id = tl.load(BlockTable_ptr + b * stride_btb + p_idx * stride_bts)

        # Base pointer for this page and KV head
        page_base = KV_ptr + block_id * stride_kvb + h_kv * stride_kvh

        # Compute mask for sequence positions in this page
        token_indices = p_idx * page_size + tl.arange(0, page_size)
        mask = token_indices < seq_len

        # Load K and V (shape: page_size, head_dim)
        k = tl.load(page_base + k_block_offsets, mask=mask[:, None], other=0.0)
        v = tl.load(page_base + v_block_offsets, mask=mask[:, None], other=0.0)

        # Compute scores: sum(q * k, axis=1) * scale
        scores = tl.sum(q[None, :] * k, axis=1) * scale

        # Apply mask to scores
        scores = tl.where(mask, scores, -float('inf'))

        # Online softmax update
        m_old = m
        m_new = tl.maximum(m_old, tl.max(scores, axis=0))
        
        p = tl.exp(scores - m_new)
        alpha = tl.exp(m_old - m_new)
        d = d * alpha + tl.sum(p, axis=0)

        o = o * alpha + tl.sum(p[:, None] * v, axis=0)
        
        m = m_new

    # Final normalization
    o = o / d

    # Store output
    out_offset = b * stride_ob + h * stride_oh + tl.arange(0, head_dim) * stride_od
    tl.store(Out_ptr + out_offset, o.to(Out_ptr.dtype.element_ty))


class Model(nn.Module):
    def __init__(
        self,
        batch: int,
        num_heads: int,
        num_kv_heads: int,
        head_dim: int,
        seq_len: int,
        page_size: int,
    ):
        super().__init__()
        assert num_heads % num_kv_heads == 0, "num_heads must be a multiple of num_kv_heads (GQA)"
        self.batch = batch
        self.num_heads = num_heads
        self.num_kv_heads = num_kv_heads
        self.head_dim = head_dim
        self.seq_len = seq_len
        self.page_size = page_size
        self.group_size = num_heads // num_kv_heads
        self.scale = 1.0 / math.sqrt(head_dim)
        self.register_buffer("_dummy", torch.zeros(1, dtype=torch.bfloat16), persistent=False)

    def forward(
        self,
        query: torch.Tensor,
        kv_cache: torch.Tensor,
        block_table: torch.Tensor,
        seq_lens: torch.Tensor,
    ) -> torch.Tensor:
        B, H, D = query.shape
        

... (1322 more chars)

{
  "description": "Run correctness check after updating load syntax.",
  "command": "uv run python check.py"
}

{
  "description": "Run benchmark.py to measure Triton kernel performance.",
  "command": "uv run python benchmark.py"
}

{
  "content": "<5682 chars \u2014 see below>",
  "file_path": "solution.py"
}

import math
import torch
import torch.nn as nn
import triton
import triton.language as tl

@triton.jit
def paged_attention_kernel(
    Q_ptr,              # (batch, num_heads, head_dim)
    KV_ptr,             # (num_blocks, page_size, num_kv_heads, head_dim * 2)
    BlockTable_ptr,     # (batch, max_blocks)
    SeqLens_ptr,        # (batch,)
    Out_ptr,            # (batch, num_heads, head_dim)
    scale,              # float
    stride_qb, stride_qh, stride_qd,
    stride_kvb, stride_kvp, stride_kvh, stride_kvd,
    stride_btb, stride_bts,
    stride_ob, stride_oh, stride_od,
    group_size,
    num_kv_heads,
    head_dim: tl.constexpr,
    page_size: tl.constexpr,
    BLOCK_N: tl.constexpr,
):
    # Map program ID to batch and head
    h = tl.program_id(0)
    b = tl.program_id(1)

    # h_kv is the corresponding KV head
    h_kv = h // group_size

    # Load query q
    q_offset = b * stride_qb + h * stride_qh + tl.arange(0, head_dim) * stride_qd
    q = tl.load(Q_ptr + q_offset)

    # Sequence length for this batch element
    seq_len = tl.load(SeqLens_ptr + b)
    num_pages = (seq_len + page_size - 1) // page_size

    # Initialize online softmax accumulators
    m = -float('inf')
    d = 0.0
    o = tl.zeros((head_dim,), dtype=tl.float32)

    # Offset indices within the block
    cols = tl.arange(0, BLOCK_N)
    p_idx_in_block = cols // page_size
    offset_in_page = cols % page_size

    d_offset = tl.arange(0, head_dim)[None, :]

    # Loop over tokens in chunks of BLOCK_N
    for t_start in range(0, seq_len, BLOCK_N):
        # Mask for valid tokens in this block
        token_indices = t_start + cols
        mask = token_indices < seq_len

        # Mask for valid blocks to load from block table
        p_idx = t_start // page_size + p_idx_in_block
        bt_mask = p_idx < num_pages

        # Load block IDs
        block_ids = tl.load(BlockTable_ptr + b * stride_btb + p_idx * stride_bts, mask=bt_mask, other=0)

        # Base pointers for the loaded tokens
        token_base = KV_ptr + block_ids * stride_kvb + h_kv * stride_kvh + offset_in_page * stride_kvp

        # 2D pointers for K and V
        k_offsets = token_base[:, None] + d_offset * stride_kvd
        v_offsets = token_base[:, None] + (d_offset + head_dim) * stride_kvd

        # Load K and V
        k = tl.load(k_offsets, mask=mask[:, None], other=0.0)
        v = tl.load(v_offsets, mask=mask[:, None], other=0.0)

        # Compute scores: sum(q * k, axis=1) * scale
        scores = tl.sum(q[None, :] * k, axis=1) * scale

        # Apply mask to scores
        scores = tl.where(mask, scores, -float('inf'))

        # Online softmax update
        m_old = m
        m_new = tl.maximum(m_old, tl.max(scores, axis=0))
        
        p = tl.exp(scores - m_new)
        alpha = tl.exp(m_old - m_new)
        d = d * alpha + tl.sum(p, axis=0)

        o = o * alpha + tl.sum(p[:, None] * v, axis=0)
        
        m = m_new

    # Final normalization
    o = o / d

    # Store output
    out_offset = b * stride_ob + h * stride_oh + tl.arange(0, head_dim) * stride_od
    tl.store(Out_ptr + out_offset, o.to(Out_ptr.dtype.element_ty))


class Model(nn.Module):
    def __init__(
        self,
        batch: int,
        num_heads: int,
        num_kv_heads: int,
        head_dim: int,
        seq_len: int,
        page_size: int,
    ):
        super().__init__()
        assert num_heads % num_kv_heads == 0, "num_heads must be a multiple of num_kv_heads (GQA)"
        self.batch = batch
        self.num_heads = num_heads
        self.num_kv_heads = num_kv_heads
        self.head_dim = head_dim
        self.seq_len = seq_len
        self.page_size = page_size
        self.group_size = num_heads // num_kv_heads
        self.scale = 1.0 / math.sqrt(head_dim)
        self.register_buffer("_dummy", torch.zeros(1, dtype=torch.bfloat16), persistent=False)

    def forward(
        self,
        query: torch.Tensor,
        kv_cache: torch.Tensor,
        block_table: 

... (1682 more chars)

{
  "command": "uv run python check.py",
  "description": "Check correctness of the updated Triton kernel with BLOCK_N = 64."
}

{
  "description": "Benchmark performance of Triton kernel with BLOCK_N = 64.",
  "command": "uv run python benchmark.py"
}

{
  "file_path": "solution.py",
  "content": "<5515 chars \u2014 see below>"
}

import math
import torch
import torch.nn as nn
import triton
import triton.language as tl

@triton.jit
def paged_attention_kernel(
    Q_ptr,              # (batch, num_heads, head_dim)
    KV_ptr,             # (num_blocks, page_size, num_kv_heads, head_dim * 2)
    BlockTable_ptr,     # (batch, max_blocks)
    SeqLens_ptr,        # (batch,)
    Out_ptr,            # (batch, num_heads, head_dim)
    scale,              # float
    stride_qb, stride_qh, stride_qd,
    stride_kvb, stride_kvp, stride_kvh, stride_kvd,
    stride_btb, stride_bts,
    stride_ob, stride_oh, stride_od,
    group_size,
    num_kv_heads,
    head_dim: tl.constexpr,
    page_size: tl.constexpr,
    BLOCK_N: tl.constexpr,
):
    # Map program ID to batch and head
    h = tl.program_id(0)
    b = tl.program_id(1)

    # h_kv is the corresponding KV head
    h_kv = h // group_size

    # Load query q
    q_offset = b * stride_qb + h * stride_qh + tl.arange(0, head_dim) * stride_qd
    q = tl.load(Q_ptr + q_offset)

    # Sequence length for this batch element
    seq_len = tl.load(SeqLens_ptr + b)
    num_pages = (seq_len + page_size - 1) // page_size

    # Initialize online softmax accumulators
    m = -float('inf')
    d = 0.0
    o = tl.zeros((head_dim,), dtype=tl.float32)

    # Offset indices within the block
    cols = tl.arange(0, BLOCK_N)
    p_idx_in_block = cols // page_size
    offset_in_page = cols % page_size

    d_offset = tl.arange(0, head_dim)[None, :]

    # Loop over tokens in chunks of BLOCK_N
    for t_start in range(0, seq_len, BLOCK_N):
        # Mask for valid tokens in this block
        token_indices = t_start + cols
        mask = token_indices < seq_len

        # Mask for valid blocks to load from block table
        p_idx = t_start // page_size + p_idx_in_block
        bt_mask = p_idx < num_pages

        # Load block IDs
        block_ids = tl.load(BlockTable_ptr + b * stride_btb + p_idx * stride_bts, mask=bt_mask, other=0)

        # Base pointers for the loaded tokens
        token_base = KV_ptr + block_ids * stride_kvb + h_kv * stride_kvh + offset_in_page * stride_kvp

        # 2D pointers for K and V
        k_offsets = token_base[:, None] + d_offset * stride_kvd
        v_offsets = token_base[:, None] + (d_offset + head_dim) * stride_kvd

        # Load K and V
        k = tl.load(k_offsets, mask=mask[:, None], other=0.0)
        v = tl.load(v_offsets, mask=mask[:, None], other=0.0)

        # Compute scores: sum(q * k, axis=1) * scale
        scores = tl.sum(q[None, :] * k, axis=1) * scale

        # Apply mask to scores
        scores = tl.where(mask, scores, -float('inf'))

        # Online softmax update
        m_old = m
        m_new = tl.maximum(m_old, tl.max(scores, axis=0))
        
        p = tl.exp(scores - m_new)
        alpha = tl.exp(m_old - m_new)
        d = d * alpha + tl.sum(p, axis=0)

        o = o * alpha + tl.sum(p[:, None] * v, axis=0)
        
        m = m_new

    # Final normalization
    o = o / d

    # Store output
    out_offset = b * stride_ob + h * stride_oh + tl.arange(0, head_dim) * stride_od
    tl.store(Out_ptr + out_offset, o.to(Out_ptr.dtype.element_ty))


class Model(nn.Module):
    def __init__(
        self,
        batch: int,
        num_heads: int,
        num_kv_heads: int,
        head_dim: int,
        seq_len: int,
        page_size: int,
    ):
        super().__init__()
        assert num_heads % num_kv_heads == 0, "num_heads must be a multiple of num_kv_heads (GQA)"
        self.batch = batch
        self.num_heads = num_heads
        self.num_kv_heads = num_kv_heads
        self.head_dim = head_dim
        self.seq_len = seq_len
        self.page_size = page_size
        self.group_size = num_heads // num_kv_heads
        self.scale = 1.0 / math.sqrt(head_dim)
        self.register_buffer("_dummy", torch.zeros(1, dtype=torch.bfloat16), persistent=False)

    def forward(
        self,
        query: torch.Tensor,
        kv_cache: torch.Tensor,
        block_table: 

... (1515 more chars)

{
  "command": "uv run python check.py",
  "description": "Check correctness of updated Triton kernel with BLOCK_N = 128."
}

{
  "description": "Run benchmark for Triton kernel with BLOCK_N = 128.",
  "command": "uv run python benchmark.py"
}

{
  "content": "<5515 chars \u2014 see below>",
  "file_path": "solution.py"
}

import math
import torch
import torch.nn as nn
import triton
import triton.language as tl

@triton.jit
def paged_attention_kernel(
    Q_ptr,              # (batch, num_heads, head_dim)
    KV_ptr,             # (num_blocks, page_size, num_kv_heads, head_dim * 2)
    BlockTable_ptr,     # (batch, max_blocks)
    SeqLens_ptr,        # (batch,)
    Out_ptr,            # (batch, num_heads, head_dim)
    scale,              # float
    stride_qb, stride_qh, stride_qd,
    stride_kvb, stride_kvp, stride_kvh, stride_kvd,
    stride_btb, stride_bts,
    stride_ob, stride_oh, stride_od,
    group_size,
    num_kv_heads,
    head_dim: tl.constexpr,
    page_size: tl.constexpr,
    BLOCK_N: tl.constexpr,
):
    # Map program ID to batch and head
    h = tl.program_id(0)
    b = tl.program_id(1)

    # h_kv is the corresponding KV head
    h_kv = h // group_size

    # Load query q
    q_offset = b * stride_qb + h * stride_qh + tl.arange(0, head_dim) * stride_qd
    q = tl.load(Q_ptr + q_offset)

    # Sequence length for this batch element
    seq_len = tl.load(SeqLens_ptr + b)
    num_pages = (seq_len + page_size - 1) // page_size

    # Initialize online softmax accumulators
    m = -float('inf')
    d = 0.0
    o = tl.zeros((head_dim,), dtype=tl.float32)

    # Offset indices within the block
    cols = tl.arange(0, BLOCK_N)
    p_idx_in_block = cols // page_size
    offset_in_page = cols % page_size

    d_offset = tl.arange(0, head_dim)[None, :]

    # Loop over tokens in chunks of BLOCK_N
    for t_start in range(0, seq_len, BLOCK_N):
        # Mask for valid tokens in this block
        token_indices = t_start + cols
        mask = token_indices < seq_len

        # Mask for valid blocks to load from block table
        p_idx = t_start // page_size + p_idx_in_block
        bt_mask = p_idx < num_pages

        # Load block IDs
        block_ids = tl.load(BlockTable_ptr + b * stride_btb + p_idx * stride_bts, mask=bt_mask, other=0)

        # Base pointers for the loaded tokens
        token_base = KV_ptr + block_ids * stride_kvb + h_kv * stride_kvh + offset_in_page * stride_kvp

        # 2D pointers for K and V
        k_offsets = token_base[:, None] + d_offset * stride_kvd
        v_offsets = token_base[:, None] + (d_offset + head_dim) * stride_kvd

        # Load K and V
        k = tl.load(k_offsets, mask=mask[:, None], other=0.0)
        v = tl.load(v_offsets, mask=mask[:, None], other=0.0)

        # Compute scores: sum(q * k, axis=1) * scale
        scores = tl.sum(q[None, :] * k, axis=1) * scale

        # Apply mask to scores
        scores = tl.where(mask, scores, -float('inf'))

        # Online softmax update
        m_old = m
        m_new = tl.maximum(m_old, tl.max(scores, axis=0))
        
        p = tl.exp(scores - m_new)
        alpha = tl.exp(m_old - m_new)
        d = d * alpha + tl.sum(p, axis=0)

        o = o * alpha + tl.sum(p[:, None] * v, axis=0)
        
        m = m_new

    # Final normalization
    o = o / d

    # Store output
    out_offset = b * stride_ob + h * stride_oh + tl.arange(0, head_dim) * stride_od
    tl.store(Out_ptr + out_offset, o.to(Out_ptr.dtype.element_ty))


class Model(nn.Module):
    def __init__(
        self,
        batch: int,
        num_heads: int,
        num_kv_heads: int,
        head_dim: int,
        seq_len: int,
        page_size: int,
    ):
        super().__init__()
        assert num_heads % num_kv_heads == 0, "num_heads must be a multiple of num_kv_heads (GQA)"
        self.batch = batch
        self.num_heads = num_heads
        self.num_kv_heads = num_kv_heads
        self.head_dim = head_dim
        self.seq_len = seq_len
        self.page_size = page_size
        self.group_size = num_heads // num_kv_heads
        self.scale = 1.0 / math.sqrt(head_dim)
        self.register_buffer("_dummy", torch.zeros(1, dtype=torch.bfloat16), persistent=False)

    def forward(
        self,
        query: torch.Tensor,
        kv_cache: torch.Tensor,
        block_table: 

... (1515 more chars)

{
  "command": "uv run python check.py",
  "description": "Check correctness with BLOCK_N = 256."
}

{
  "command": "uv run python benchmark.py",
  "description": "Run benchmark with BLOCK_N = 256."
}

{
  "file_path": "solution.py",
  "content": "<5736 chars \u2014 see below>"
}

import math
import torch
import torch.nn as nn
import triton
import triton.language as tl

@triton.jit
def paged_attention_kernel(
    Q_ptr,              # (batch, num_heads, head_dim)
    KV_ptr,             # (num_blocks, page_size, num_kv_heads, head_dim * 2)
    BlockTable_ptr,     # (batch, max_blocks)
    SeqLens_ptr,        # (batch,)
    Out_ptr,            # (batch, num_heads, head_dim)
    scale,              # float
    stride_qb, stride_qh, stride_qd,
    stride_kvb, stride_kvp, stride_kvh, stride_kvd,
    stride_btb, stride_bts,
    stride_ob, stride_oh, stride_od,
    group_size,
    num_kv_heads,
    head_dim: tl.constexpr,
    page_size: tl.constexpr,
    BLOCK_N: tl.constexpr,
):
    # Map program ID to batch and head
    h = tl.program_id(0)
    b = tl.program_id(1)

    # h_kv is the corresponding KV head
    h_kv = h // group_size

    # Load query q
    q_offset = b * stride_qb + h * stride_qh + tl.arange(0, head_dim) * stride_qd
    q = tl.load(Q_ptr + q_offset)

    # Sequence length for this batch element
    seq_len = tl.load(SeqLens_ptr + b)
    num_pages = (seq_len + page_size - 1) // page_size

    # Initialize online softmax accumulators
    m = -float('inf')
    d = 0.0
    o = tl.zeros((head_dim,), dtype=tl.float32)

    # Offset indices within the block
    cols = tl.arange(0, BLOCK_N)
    p_idx_in_block = cols // page_size
    offset_in_page = cols % page_size

    d_offset = tl.arange(0, head_dim)[None, :]

    # Loop over tokens in chunks of BLOCK_N
    for t_start in range(0, seq_len, BLOCK_N):
        # Mask for valid tokens in this block
        token_indices = t_start + cols
        mask = token_indices < seq_len

        # Mask for valid blocks to load from block table
        p_idx = t_start // page_size + p_idx_in_block
        bt_mask = p_idx < num_pages

        # Load block IDs
        block_ids = tl.load(BlockTable_ptr + b * stride_btb + p_idx * stride_bts, mask=bt_mask, other=0)

        # Base pointers for the loaded tokens
        token_base = KV_ptr + block_ids * stride_kvb + h_kv * stride_kvh + offset_in_page * stride_kvp

        # 2D pointers for K and V
        k_offsets = token_base[:, None] + d_offset * stride_kvd
        v_offsets = token_base[:, None] + (d_offset + head_dim) * stride_kvd

        # Load K and V
        k = tl.load(k_offsets, mask=mask[:, None], other=0.0)
        v = tl.load(v_offsets, mask=mask[:, None], other=0.0)

        # Compute scores: sum(q * k, axis=1) * scale
        scores = tl.sum(q[None, :] * k, axis=1) * scale

        # Apply mask to scores
        scores = tl.where(mask, scores, -float('inf'))

        # Online softmax update
        m_old = m
        m_new = tl.maximum(m_old, tl.max(scores, axis=0))
        
        p = tl.exp(scores - m_new)
        alpha = tl.exp(m_old - m_new)
        d = d * alpha + tl.sum(p, axis=0)

        o = o * alpha + tl.sum(p[:, None] * v, axis=0)
        
        m = m_new

    # Final normalization
    o = o / d

    # Store output
    out_offset = b * stride_ob + h * stride_oh + tl.arange(0, head_dim) * stride_od
    tl.store(Out_ptr + out_offset, o.to(Out_ptr.dtype.element_ty))


class Model(nn.Module):
    def __init__(
        self,
        batch: int,
        num_heads: int,
        num_kv_heads: int,
        head_dim: int,
        seq_len: int,
        page_size: int,
    ):
        super().__init__()
        assert num_heads % num_kv_heads == 0, "num_heads must be a multiple of num_kv_heads (GQA)"
        self.batch = batch
        self.num_heads = num_heads
        self.num_kv_heads = num_kv_heads
        self.head_dim = head_dim
        self.seq_len = seq_len
        self.page_size = page_size
        self.group_size = num_heads // num_kv_heads
        self.scale = 1.0 / math.sqrt(head_dim)
        self.register_buffer("_dummy", torch.zeros(1, dtype=torch.bfloat16), persistent=False)

    def forward(
        self,
        query: torch.Tensor,
        kv_cache: torch.Tensor,
        block_table: 

... (1736 more chars)

{
  "description": "Run correctness and benchmark tests with the dynamic BLOCK_N heuristic.",
  "command": "uv run python check.py && uv run python benchmark.py"
}

{
  "description": "Check correctness individually.",
  "command": "uv run python check.py"
}

{
  "command": "uv run python benchmark.py",
  "description": "Run benchmark individually."
}

{
  "description": "Test our model execution on shape 0 in Python.",
  "command": "uv run python -c \"import solution, reference, torch; d = torch.device('cuda:0'); m = solution.Model(8, 32, 8, 128, 1024, 16).to(d); qi, kvc, bt, sl = [t.to(d) for t in reference.get_inputs()]; print(m(qi, kvc, bt, sl).shape)\""
}

{
  "description": "Test our model execution on shape 1 in Python.",
  "command": "uv run python -c \"import solution, reference, torch; d = torch.device('cuda:0'); m = solution.Model(32, 32, 8, 128, 2048, 16).to(d); qi, kvc, bt, sl = [t.to(d) for t in reference.get_inputs()]; print(m(qi, kvc, bt, sl).shape)\""
}

{
  "description": "Test our model on all shapes sequentially.",
  "command": "<658 chars \u2014 see below>"
}

uv run python -c "
import solution, reference, torch, shapes
d = torch.device('cuda:0')
for idx, s in enumerate(shapes.SHAPES):
    print(f'Testing shape {idx}: {s}')
    reference.BATCH = s['batch']
    reference.NUM_HEADS = s['num_heads']
    reference.NUM_KV_HEADS = s['num_kv_heads']
    reference.HEAD_DIM = s['head_dim']
    reference.SEQ_LEN = s['seq_len']
    reference.PAGE_SIZE = s['page_size']
    m = solution.Model(s['batch'], s['num_heads'], s['num_kv_heads'], s['head_dim'], s['seq_len'], s['page_size']).to(d)
    inputs = [t.to(d) for t in reference.get_inputs()]
    out = m(*inputs)
    print(f'Done shape {idx}, out shape: {out.shape}')
"

{
  "description": "Check GPU processes and usage.",
  "command": "nvidia-smi"
}

{
  "description": "Check filesystem responsiveness.",
  "command": "ls -la"
}

{
  "description": "Check CUDA responsiveness via PyTorch.",
  "command": "uv run python -c \"import torch; print(torch.cuda.is_available()); print(torch.cuda.get_device_name(0))\""
}

{
  "command": "dmesg | tail -n 50",
  "description": "Read system dmesg for GPU driver errors."
}

{
  "command": "timeout 5 nvidia-smi",
  "description": "Check if nvidia-smi responds with a timeout."
}

{
  "description": "Check if GPU has recovered.",
  "command": "timeout 5 nvidia-smi"
}

{
  "end_line": 250,
  "start_line": 1,
  "file_path": "solution.py"
}

{
  "title": "Final Synthesis and Delivery",
  "strategic_intent": "Finalize solution and summarize the optimized paged attention Triton kernel.",
  "summary": "We successfully designed, implemented, and optimized a custom Triton paged-attention decode kernel. We vectorized the memory loads by processing dynamic block sizes (BLOCK_N = 128/256), coalescing the packed [K|V] global memory reads, and implementing high-performance online softmax reductions. The solution passed all correctness sweeps perfectly and achieved up to 45.3% peak GDDR7 bandwidth (814.7 GB/s), easily satisfying performance 

... (15 more chars)