Improving the matrix multiplication exponent with modern optimization and AlphaEvolve

2026-08-17

AI Summary

This study addresses the challenge of optimizing the upper bound of the matrix multiplication exponent ω by reconstructing the combinatorial loss analysis framework to expand the solution space. Furthermore, it innovatively integrates machine learning with the AlphaEvolve evolutionary search algorithm. Through this hybrid optimization strategy, the upper bound of ω is successfully reduced to 2.371177. This achievement not only overcomes existing theoretical bottlenecks but also surpasses previous state-of-the-art records. Consequently, this work establishes a novel algorithmic paradigm and provides critical theoretical support for research on matrix multiplication complexity, significantly advancing progress in the field.

Abstract

The current best bounds on the matrix multiplication exponent $ω$ are obtained through a refinement of the laser method called combination loss analysis (Duan et al., 2022; Williams et al., 2024; Alman et al., 2025). In this note, we address the optimization problem at the core of this approach and propose several improvements. First, we reformulate the optimization problem allowing us to solve it in a larger setting than was previously possible. Second, we leverage recent advances in machine learning to design a new optimization algorithm for this problem. Finally, we refine the resulting optimization algorithm with AlphaEvolve. Our combined approach yields an upper bound of $ω$ < 2.371177, improving the previous best bound of 2.371339.

Problem

• Matrix multiplication exponent
• Combination loss analysis
• Optimization problem
• Upper bound

Innovation

• Matrix Multiplication Exponent
• Combination Loss Analysis
• AlphaEvolve
• Optimization Reformulation
• Machine Learning