Improving the matrix multiplication exponent with modern optimization and AlphaEvolve

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The matrix multiplication exponent omega measures the theoretical efficiency limit of matrix multiplication — a fundamental operation underlying scientific computing, linear algebra solvers, and the training of neural networks. Reducing omega tightens algorithmic bounds on what computations are theoretically efficient.\n\nThis paper advances the state of the art in the combination loss analysis approach (building on Duan et al. 2022, Williams et al. 2024, and Alman et al. 2025) through three contributions: a reformulated optimization problem that scales to larger settings than previously possible, a new ML-designed optimization algorithm for this specific problem structure, and final refinement using AlphaEvolve.\n\nThe combined approach yields omega < 2.371177, improving the previous best bound of 2.371339. Authors include Virginia Vassilevska Williams and Josh Alman — two of theoretical CS's most prominent researchers in this area — alongside DeepMind collaborators including Emilien Dupont and Abigail See.\n\nBeyond the specific result, this paper adds to a growing body of evidence that AlphaEvolve can contribute meaningfully to open problems in mathematics and algorithm design, following prior work in combinatorics and geometry.