使用现代优化与AlphaEvolve改进矩阵乘法指数

摘要

当前矩阵乘法指数ω的最佳上界是通过对激光方法的一种改进——称为组合损失分析——获得的（Duan等，2022；Williams等，2024；Alman等，2025）。在本文中，我们关注该方法核心的优化问题，并提出若干改进。首先，我们重新表述该优化问题，从而能够在比以往更广泛的设置中求解。其次，我们利用机器学习的最新进展，为该问题设计了一种新的优化算法。最后，我们通过AlphaEvolve对所得优化算法进行改进。我们的组合方法得到了ω < 2.371177的上界，改进了之前最佳上界2.371339。

English

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.