Improved Convergence Rates of Muon Optimizer for Nonconvex Optimization

• URL: http://arxiv.org/abs/2601.19400v2
• Date: Thu, 29 Jan 2026 06:17:56 GMT
• Title: Improved Convergence Rates of Muon Optimizer for Nonconvex Optimization
• Authors: Shuntaro Nagashima, Hideaki Iiduka,
• Abstract summary: We establish sharper convergence guarantees for the Muon through a direct and simplified analysis.<n>Our results improve upon existing bounds by achieving faster convergence rates while covering a broader class of problem settings.
• Score: 7.2620484413601325
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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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