New Upper Bound on the Matrix Multiplication Exponent via Modern Optimization and AlphaEvolve

Forum topic · 小凯 · 2026-08-19

Summary

Researchers including Emilien Dupont, Marvin Eisenberger, and Borislav Kozlovskii report an improved upper bound on the matrix multiplication exponent ω, showing ω < 2.371177, down from the previous best of 2.371339. The best current bounds on ω come from combination loss analysis, a refinement of the laser method introduced in work by Duan et al. (2022), Williams et al. (2024), and Alman et al. (2025). The paper tackles the optimization problem at the core of this approach with three contributions: it reformulates the optimization so it can be solved in a larger setting than previously possible; it designs a new optimization algorithm using recent advances in machine learning; and it further refines the resulting algorithm with AlphaEvolve, DeepMind's evolutionary code-optimization system. The combined approach yields the improved bound, as detailed in the arXiv preprint 2608.16884, published August 17, 2026.

Paper Overview

Research area: Machine Learning Authors: Emilien Dupont, Marvin Eisenberger, Borislav Kozlovskii et al. (10 authors) Published: 2026-08-17 arXiv: 2608.16884

Abstract (translated)

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, the authors address the optimization problem at the core of this approach and propose several improvements:

1. They reformulate the optimization problem, allowing it to be solved in a larger setting than was previously possible. 2. They leverage recent advances in machine learning to design a new optimization algorithm for this problem. 3. They refine the resulting optimization algorithm with AlphaEvolve.

The combined approach yields an upper bound of ω < 2.371177, improving the previous best bound of 2.371339.

Original 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.

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