Improving the matrix multiplication exponent with modern optimization and AlphaEvolve

Summary

A research note improving the best known upper bound on the matrix multiplication exponent ω. The authors attack the optimization problem at the core of combination loss analysis (the refinement of the laser method behind the current records): they reformulate the problem so it can be solved in a larger setting than previously possible, design a new optimization algorithm using recent machine-learning advances, and refine that algorithm with AlphaEvolve. The combined approach yields ω < 2.371177, improving the previous best bound of 2.371339. Notably, the author list joins the DeepMind AlphaEvolve team with the complexity theorists whose hand-derived analyses held the prior records (Alman, Vassilevska Williams, Zhou).

Contribution

Claimed and supported by the abstract: a new state-of-the-art bound on a decades-old open problem, obtained with an agentic evolutionary coding system in the loop. The improvement is small in absolute terms (fifth decimal place), as is typical for this line of work.

Method

Theoretical-computer-science note: reformulation of the combination loss optimization problem, a bespoke ML-designed optimization algorithm, and AlphaEvolve refinement. The abstract does not state how much of the gain each of the three steps contributes.

Relevance to RISE

Catalog slug alphaevolve (formal-modeling stage). Together with [georgiev2025alphaevolvemath] this is the second peer-checkable mathematical advance with AlphaEvolve in the loop, and the clearest kind of evidence in the catalog that agentic systems can contribute to results whose correctness is externally verifiable — the opposite pole from research tasks with no machine-checkable correctness signal.

Critique / open questions

The division of labor between the human theorists and the automated components cannot be judged from the abstract, so what "AlphaEvolve refined the algorithm" means in practice — search over parameters vs. discovery of new structure — is open. As a note, it has not been peer-reviewed yet.

Key quotes

"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." (abstract)

"Our combined approach yields an upper bound of ω < 2.371177, improving the previous best bound of 2.371339." (abstract)