Improved Convergence Rates of Muon Optimizer for Nonconvex Optimization

Shuntaro Nagashima; Hideaki Iiduka

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arXiv · 2601.19400

Improved Convergence Rates of Muon Optimizer for Nonconvex Optimization

Shuntaro Nagashima·Hideaki Iiduka

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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.

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Shuntaro Nagashima, Hideaki Iiduka. 2026-01-27. Improved Convergence Rates of Muon Optimizer for Nonconvex Optimization. https://arxiv.org/abs/2601.19400

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