GOMA: Geometrically Optimal Mapping via Analytical Modeling for Spatial Accelerators

2026-03-09

AI Summary

This work addresses the combinatorial explosion in GEMM mapping space on spatial accelerators, which hinders efficient discovery of globally optimal mappings. For the first time, the authors derive an analytical energy model from first principles using geometric abstraction, formulating mapping selection as an integer optimization problem subject to hardware constraints. This approach enables exact energy evaluation and global optimum search with O(1) complexity. Evaluated across diverse accelerators and large language model prefill tasks, the method achieves 2.24–4.24× improvement in energy-delay product and 3.83–73.6× faster solution times compared to state-of-the-art mappers.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems

Abstract

General matrix multiplication (GEMM) on spatial accelerators is highly sensitive to mapping choices in both execution efficiency and energy consumption. However, the mapping space exhibits combinatorial explosion, which makes it extremely challenging to obtain optimal mappings within an acceptable time budget. Existing approaches typically face challenges: They often lack global-optimality guarantees and become prohibitively slow as the mapping space grows. To address these limitations, we propose \textsc{GOMA}, a geometric-abstraction-based, globally optimal GEMM mapping framework via analytical modeling, which achieves efficient solving while guaranteeing optimality. \textsc{GOMA} introduces, from first principles, a geometric abstraction for GEMM mapping, yielding an exact analytical energy objective with $O(1)$ evaluation for any given mapping. The objective is highly accurate. \textsc{GOMA} then formulates mapping selection as an integer optimization problem under hardware and mapping constraints, using the analytical energy model as the objective to automate mapping search. \textsc{GOMA} can quickly compute a global-optimal mapping for any (GEMM workload, target hardware) pair, achieving this for the first time in mapping space exploration. Experiments confirm that across representative accelerators and large language model prefill workloads, \textsc{GOMA} improves the energy--delay product (EDP) by $2.24$--$4.24\times$ over SOTA mappers, while accelerating time-to-solution by $3.83$--$73.6\times$.

Problem

• GEMM
• spatial accelerators
• mapping optimization
• combinatorial explosion
• energy efficiency

Innovation

• geometric abstraction
• analytical modeling
• global optimality
• GEMM mapping
• spatial accelerators