Fully Symbolic Analysis of Loop Locality: Using Imaginary Reuse to Infer Real Performance

2026-03-10

AI Summary

Traditional empirical models struggle to accurately predict cache behavior for affine loop nests in scientific computing and tensor operations. This work proposes the first fully symbolic locality theory, modeling data locality as polynomials in input sizes and cache parameters. By integrating affine loop analysis, virtual reuse modeling, and compiler infrastructure, the approach derives quadratic and reciprocal scaling laws for cache misses. It transcends the limitations of empirical rules, enabling high-precision prediction of data movement across arbitrary input scales and cache configurations. Evaluation on 41 scientific kernels and tensor programs shows that locality polynomials are generated in an average of 41 seconds, with cache miss predictions taking under 1 millisecond and achieving 99.6% accuracy.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Local Search

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions

Abstract

This paper presents a new theory of locality and its compiler support. The theory is fully symbolic and derives locality as polynomials, and the compiler analysis supports affine loop nests. They derive cache-performance scaling in quadratic and reciprocal expressions and are more general and precise than empirical scaling rules. Evaluated on a benchmark suite of 41 scientific kernels and tensor operations, the compiler requires an average of 41 seconds to derive the locality polynomials. After derivation, predicting the cache miss count for any given input size and cache configuration takes less than a millisecond. Across all tests--with and without loop fusion--the accuracy in the data movement prediction is 99.6\%, compared to simulated set-associative L1 data cache.

Problem

• loop locality
• cache performance
• data movement prediction
• affine loop nests
• symbolic analysis

Innovation

• symbolic locality analysis
• affine loop nests
• cache performance modeling
• polynomial representation
• compiler optimization