Vectorized CUDA Kernel Design

• Vectorized CUDA kernel design is an approach that maps computations to GPU thread hierarchies using explicit SIMD/SIMT techniques to maximize throughput.
• It exploits advanced hardware primitives such as warp-level intrinsics, TMA, and WGMMA, alongside efficient memory hierarchies and vectorized loads.
• The method integrates kernel fusion, tiling, and double buffering to optimize register, shared, and global memory usage while overlapping compute and copy operations.

Vectorized CUDA kernel design encompasses methodologies and implementation strategies that leverage the fine-grained SIMD (Single Instruction, Multiple Data) and SIMT (Single Instruction, Multiple Threads) execution paradigms of modern NVIDIA GPUs to achieve maximal throughput and resource utilization. This approach involves mapping computational and memory access patterns to CUDA thread hierarchies, exploiting explicit vector operations, hierarchical memory, and architectural primitives such as warp-level intrinsics, memory coalescing, asynchronous copies, and advanced tensor hardware instructions. Research in this area focuses on domain-specific methodologies (e.g., linear algebra, stencils, attention), kernel fusion, tiling, and register/L1/shared/DRAM bandwidth optimization, spanning both hand-tuned kernels and automated code generation frameworks.

1. Hardware Architecture and Execution Model

Vectorized CUDA kernel design is driven by the SIMT execution model of NVIDIA GPUs, where threads are organized into warps (usually 32 threads) and warps are grouped into thread blocks (CTAs). The hardware provides multiple execution units that can process vector instructions (such as tensor-cores or warp-wide instructions), as well as explicit memory hierarchy with DRAM, L2, L1 caches, and shared memory (SMEM).

Modern architectures (e.g., Hopper SM90) introduce instructions such as the Tensor Memory Accelerator (TMA) for high-throughput, vectorized async memory copy (128-byte lanes), and Warpgroup Matrix-Multiply-Accumulate (WGMMA) for high-throughput, compute-bound GEMM (e.g., 64×64×16 MMA atom per warp group) (Bikshandi et al., 2023). These capabilities are accessed via libraries (e.g., CUTLASS, CuTe) that abstract hardware-specific instructions into composable C++ templates and perform low-level swizzle, tiling, and copy/fetch coordination.

2. Mapping of Thread and Data Dimensions

Efficient assignment of CUDA thread/warp/block indices to computational and data dimensions is foundational to vectorized design. The mapping ensures load balancing, data locality, and bandwidth utilization.

• Matrix-matrix and matrix-vector multiply: KBLAS launches a 2D grid of thread blocks, with each TB assigned to a panel or block-row, and fine-grained assignment of (threadIdx.x, threadIdx.y) to subsegments (Abdelfattah et al., 2014). Parameterization of panel size (nb), thread columns (Q), and overlapping TBs per panel (Ȳ) enables tuning for both compute and memory-bound regimes.
• Stencil and convolution: The SSAM model assigns warps to processing elements (PEs) in a 2D mesh, with each lane in a warp corresponding to one output spatial location. In-warp register caches are used for sliding window accesses (Chen et al., 2019).
• Irregular and kernel applications: KeOps tiles both source and target dimensions (M,N) into tiles T_i and T_j, assigning each thread to one output i and decomposing the j-loop into shared-memory streamed tiles (Charlier et al., 2020).
• General loop nests: Automated frameworks represent each dimension with CSP variables for mapping to BLOCK, THREAD, UNROLL, or VECTOR roles, imposing constraints on nesting, vectorization, and occupancy (Beaugnon et al., 2019).

3. Memory Hierarchy, Tiling, and Vectorized Loads

Tiling is central to vectorized kernel design, as it increases arithmetic intensity and allows efficient use of shared memory and registers.

• SMEM and register tiling: FlashAttention-2 fuses all key operations (QKᵀ, softmax, P·V) using tile sizes bM, bN, and d, keeping all intermediates in SMEM or registers and avoiding global-memory spills (Bikshandi et al., 2023).
• Vectorized loads: 128B TMA is used for asynchronous, coalesced GMEM→SMEM transfer. In KBLAS, float4 (16B) or float8/float16 vectors are used to maximize bus width and efficiency, with submatrix alignment addressed by aligning tiles to multiples of nb (Abdelfattah et al., 2014).
• SSAM's register cache: Input data for stencils/convolutions is loaded into per-lane registers, with the sliding window ensuring contiguous, coalesced accesses. The register cache size C = N+P–1 directly governs performance and occupancy (Chen et al., 2019).
• Shared-memory staging: In KeOps, shared memory streams the j-dimension tiles, while each thread keeps x_i in registers, and all reduction accumulates in registers, eliminating intermediate global memory storage (Charlier et al., 2020).
• Hardware coalescing: CUTLASS layouts and swizzled SMEM enable avoidance of bank conflicts, enabling maximum TMA/WGMMA efficiency on H100 (Bikshandi et al., 2023).

4. Fusion, Double Buffering, and Overlap of Copy/Compute

Kernel fusion—combining multiple sequential computational stages into a single kernel—is essential for maximizing data locality and minimizing memory bandwidth usage.

• Fused attention kernels: FlashAttention-2 merges QKᵀ (GEMM-I), online softmax, and PV (GEMM-II) in one kernel, achieving zero intermediate GMEM traffic. Copy of K_{j+1} is overlapped with GEMM-I(Q_j,K_j); copy of V_j is overlapped with GEMM-II(P_{j–1},V_{j–1}), using a co-scheduled software pipeline over two SMEM buffers (Bikshandi et al., 2023).
• Double buffering: KBLAS exploits register-level double-buffering (ping-pong between u[]/l[] buffers), such that loading of the next tile is overlapped with computation on the previous one, hiding global-memory latency (Abdelfattah et al., 2014).
• Software systolic: SSAM uses sliding register windows and warp shuffles to propagate partial sums, eliminating shared-memory barriers for intra-warp communication (Chen et al., 2019).
• Fusion frameworks: HFAV represents dependency DAGs, contracts buffers based on reuse analysis, and systematically fuses iteration nests, exposing inter-kernel temporal locality (Sewall et al., 2017).

5. Vectorization, Parallelization, and SIMD/SIMT

Explicit vectorization in CUDA kernels uses both hardware-supported vector instructions and high-level mapping of loop dimensions to threads/warps/lane-SIMD.

• WGMMA intrinsics: Explicit use of SM90's 128-thread warpgroups each executing a MMAtom (e.g., wgmma.sync.aligned.m8n64k16) is central in vectorized matrix multiplications (Bikshandi et al., 2023).
• Thread-lane coordination: SSAM uses __shfl_up_sync (and other warp-shuffle intrinsics) to propagate vector data across lanes, emulating hardware systolic arrays and enabling pipeline parallelism for streaming and stencil codes (Chen et al., 2019).
• Automated vectorization: Decision variables (dim_kind(d)=VECTOR) in CSP-based systems propagate constraints that ensure only one dimension is vectorized, and the code emitter schedules explicit SIMD load/store and compute in strides matching hardware vector width (Beaugnon et al., 2019). Assignment of VECTOR prohibits inner/outer nesting inside the vectorized loop, eliminating pass reordering ambiguity.
• Access alignment: Proper stride and layout selection in memory ensures that vectorized loads/store (e.g., float4, float8) are always aligned to required boundaries, avoiding split transactions and suboptimal DRAM usage (Abdelfattah et al., 2014).

6. Analytical Modeling, Performance Tuning, and Empirical Results

Performance modeling uses the arithmetic intensity, roofline models, and resource usage (registers, shared memory, occupancy) to guide tile choices and fusion strategies.

• FlashAttention-2: The arithmetic intensity $I \approx F/\mathrm{bytes}$ is targeted to exceed 10 FLOPs/byte; on H100, TMA+WGMMA narrow compute and memory rooflines, with 20–50% speedup over SM80 attributed half to vectorized TMA loads, half to WGMMA FP16/32 throughput (Bikshandi et al., 2023). Tile sizes are selected to balance shared-memory and register pressure (e.g., (bM,bN,bK)=(64,128,64) or (128,64,128)).
• KBLAS: Occupancy is maximized by tuning panel size nb and thread columns Q, while Ȳ smoothes out small-size oscillations and atomic overhead. The kernels consistently achieve >80% of measured roofline Rₛ, validated against STREAM bandwidth (Abdelfattah et al., 2014).
• SSAM: Register cache size C and number of outputs P/thread provide a tunable ILP/occupancy tradeoff, with arithmetic intensity $AI \approx \frac{MN}{2}(1-(N-1)/C)$ and halo-overhead <20% in practical cases (Chen et al., 2019).
• KeOps: Empirically shows up to 20× speedup over PyTorch for large N; sustained ∼12 TFLOPs (single-precision) on RTX 2080 Ti with tile/template code (Charlier et al., 2020).
• HFAV: Automated fusion+vectorization achieves up to 2–3× speedup for bandwidth-bound kernels and matches hand-tuned baselines for more complex codes by fusing kernels, contracting buffers, and tiling/striping loop nests (Sewall et al., 2017).

7. Trade-offs, Generalizations, and Automated Approaches

Optimal vectorized kernel design trades off tile size (arithmetic intensity vs register/shared memory pressure), kernel fusion granularity, SMEM vs register utilization, and mapping choices.

• Larger tiles increase arithmetic intensity but can incur register spills or overflow shared memory, leading to serialization or reduced occupancy (Bikshandi et al., 2023).
• TMA-swizzled SMEM layouts improve bank conflict avoidance at the cost of slightly more complex address math (Bikshandi et al., 2023).
• Double-GEMM pipelining (as in FlashAttention-2) reduces shared memory footprint compared to an N-stage software pipeline at the cost of some idle time (Bikshandi et al., 2023).
• Multi-GPU: KBLAS extends panel-blocking to 1D block-cyclic distribution, and introduces APIs for distributing computation and gathers (Abdelfattah et al., 2014).
• Automated frameworks: HFAV and CSP-based systems allow specification of entire design spaces, with constraint propagation, performance modeling, and branch-and-bound/MCTS to auto-explore tile/fusion/vectorization strategies (Sewall et al., 2017, Beaugnon et al., 2019).

A plausible implication is that automated optimization and code generation tools that explicitly encode vectorization, fusion, tiling, and memory alignment decisions as first-class variables—combined with fast roofline-based analytical bounds—can systematically match or exceed hand-tuned vectorized CUDA kernels across a range of architectures and application domains.

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References:

• "A Case Study in CUDA Kernel Fusion: Implementing FlashAttention-2 on NVIDIA Hopper Architecture using the CUTLASS Library" (Bikshandi et al., 2023)
• "KBLAS: An Optimized Library for Dense Matrix-Vector Multiplication on GPU Accelerators" (Abdelfattah et al., 2014)
• "A Versatile Software Systolic Execution Model for GPU Memory-Bound Kernels" (Chen et al., 2019)
• "On the Representation of Partially Specified Implementations and its Application to the Optimization of Linear Algebra Kernels on GPU" (Beaugnon et al., 2019)
• "Kernel Operations on the GPU, with Autodiff, without Memory Overflows" (Charlier et al., 2020)
• "High-Performance Code Generation though Fusion and Vectorization" (Sewall et al., 2017)