🤖 AI Summary
This study addresses the significant performance gap between theoretically optimal memory-efficient Transformer kernels and their practical hardware deployment. We propose a method grounded in Mathematics of Arrays (MoA) that models hardware adaptation as Operand Normal Form (ONF) rewriting under a fixed Disjunctive Normal Form (DNF), enabling migration to novel architectures without re-deriving correctness guarantees. Validation is conducted via PyTorch integration and multi-platform profiling across two HPC clusters. The proposed approach achieves up to 2.5× speedup, resolves a regression involving GPU atomic races, quantifies NUMA topology penalties reaching 535×, and identifies the root cause of C/Fortran cross-platform performance inversion.
📝 Abstract
We validate memory-optimal cost functions for transformer kernels derived via the Mathematics of Arrays (MoA). Companion Papers I-IV formally derive kernels for attention forward, backward, fused forward+backward, decode, and the complete block (RMSNorm, gated MLP) as a hardware-independent specification (DNF) transformed to a machine-specific realization (ONF) via gamma, with verification to machine precision against PyTorch. This paper checks those predictions against measured performance on two HPC clusters (Purdue Anvil, NCSA Delta) across CPU and GPU. Three results stand out. (1) We identify and fix a GPU regression: fusing forward+backward, proven to avoid materializing an O(n^2) intermediate, initially ran slower than naive on GPU due to atomic contention. Profiling confirmed 2.00x more atomic instructions; a targeted ONF rewrite reversed it, yielding up to 2.5x speedup. (2) Identical derivations produce markedly different real costs by topology: 535x NUMA-locality penalty on one cluster vs<3x oversubscription on another, showing optimal deployment is a function of the machine's array structure. (3) We report a partially resolved anomaly: identical denotational computations run faster in C than Fortran on CPU but faster in Fortran than C on GPU, narrowed to one dominant kernel and one memory-latency stall mechanism (3.17x time gap matches 3.35x stall gap). We treat hardware-specific optimization as a routine ONF rewrite with fixed, verified DNF, a candidate methodology for scaling AI onto evolving hardware without re-deriving correctness.