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arXiv · 2603.15059

Muon Converges under Heavy-Tailed Noise: Nonconvex H\"{o}lder-Smooth Empirical Risk Minimization

Hideaki Iiduka
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Abstract

Muon is a recently proposed optimizer that enforces orthogonality in parameter updates by projecting gradients onto the Stiefel manifold, leading to stable and efficient training in large-scale deep neural networks. Meanwhile, the previously reported results indicated that stochastic noise in practical machine learning may exhibit heavy-tailed behavior, violating the bounded-variance assumption. In this paper, we consider the problem of minimizing a nonconvex H\"{o}lder-smooth empirical risk that works well with the heavy-tailed stochastic noise. We then show that Muon converges to a stationary point of the empirical risk under the boundedness condition accounting for heavy-tailed stochastic noise. In addition, we show that Muon converges faster than mini-batch SGD.

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BibTeXRIS

Hideaki Iiduka. 2026-03-16. Muon Converges under Heavy-Tailed Noise: Nonconvex H\"{o}lder-Smooth Empirical Risk Minimization. https://arxiv.org/abs/2603.15059

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