利用現代最佳化與AlphaEvolve改進矩陣乘法指數

摘要

目前關於矩陣乘法指數 ω 的最佳上界，是透過雷射方法的一項改良——稱為組合損失分析——所獲得（Duan et al., 2022; Williams et al., 2024; Alman et al., 2025）。在本文中，我們處理此方法核心的最佳化問題，並提出若干改進。首先，我們重新表述此最佳化問題，使我們能在比以往更廣泛的情境下求解。其次，我們利用機器學習的最新進展，為此問題設計了一個新的最佳化演算法。最後，我們使用 AlphaEvolve 對所得的最佳化演算法進行精煉。我們的綜合方法得出 ω < 2.371177 的上界，改善了先前的最佳上界 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.