Muon优化器在矩阵分解中学习平衡解：避免慢速鞍点动力学

Computer Science > Machine Learning

arXiv:2606.30509v1 (cs) [Submitted on 29 Jun 2026]

Title:Muon learns balanced solutions in matrix factorization without slow saddle-to-saddle dynamics

Authors:Mark Rhee, Jamie Simon, Dhruva Karkada View a PDF of the paper titled Muon learns balanced solutions in matrix factorization without slow saddle-to-saddle dynamics, by Mark Rhee and 2 other authors View PDF HTML (experimental)

Abstract:Matrix factorization (i.e., problems of the form $\min_{\mathbf{P},\mathbf{Q}} \|\mathbf{M}^\star - \mathbf{P}^\top\mathbf{Q}\|_\mathrm{F}^2$) is a minimal learning problem that exhibits both nonlinear parameter dynamics and representation learning. In this setting, we study how parameter trajectories under the Muon optimizer differ from those of gradient descent. We identify three main dynamical differences: 1) Muon avoids the slow saddle-to-saddle dynamics from small initialization. Muon instead learns all the top modes of $\mathbf{M}^\star$ at the same rate, with the smaller modes converging first. 2) Muon remains stable even when the learning rate exceeds the critical threshold set by the local loss sharpness. This frees the learning rate from the condition number of the problem, enabling rapid convergence via exponential learning rate annealing. 3) Once the weights are aligned with each other and the target, Muon flow conserves the matrix quantity $\sqrt{\mathbf{P}^\top \mathbf{P}}-\sqrt{\mathbf{Q}^\top \mathbf{Q}}$, while gradient flow is known to conserve the matrix $\mathbf{P}^\top\mathbf{P} - \mathbf{Q}^\top\mathbf{Q}$. Despite having distinct conserved quantities, both optimizers find the so-called \textit{balanced} solution from vanishing initialization. When training from small random initialization, the weights spontaneously align early in training. We derive the alignment rates in simple settings and show that they predict the empirical alignment rates in general. Finally, we exploit structural properties of Muon to construct a learning rate schedule that achieves near-perfect alignment in only two optimization steps.

Subjects:	Machine Learning (cs.LG)
Cite as:	arXiv:2606.30509 [cs.LG]
	(or arXiv:2606.30509v1 [cs.LG] for this version)
	https://doi.org/10.48550/arXiv.2606.30509 Focus to learn more arXiv-issued DOI via DataCite

Submission history

From: Mark Rhee [view email]
[v1] Mon, 29 Jun 2026 16:17:38 UTC (887 KB)
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