Improved Convergence Rates of Muon Optimizer for Nonconvex Optimization
Abstract
The Muon optimizer has recently attracted attention due to its orthogonalized first-order updates, and a deeper theoretical understanding of its convergence behavior is essential for guiding practical applications; however, existing convergence guarantees are either coarse or obtained under restrictive analytical settings. In this work, we establish sharper convergence guarantees for the Muon optimizer through a direct and simplified analysis that does not rely on restrictive assumptions on the update rule. Our results improve upon existing bounds by achieving faster convergence rates while covering a broader class of problem settings. These findings provide a more accurate theoretical characterization of Muon and offer insights applicable to a broader class of orthogonalized first-order methods.
Cite
@article{arxiv.2601.19400,
title = {Improved Convergence Rates of Muon Optimizer for Nonconvex Optimization},
author = {Shuntaro Nagashima and Hideaki Iiduka},
journal= {arXiv preprint arXiv:2601.19400},
year = {2026}
}