Muon Learns Balanced Solutions in Matrix Factorization without Slow Saddle-to-Saddle Dynamics

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)
Full-text links:

Access Paper:

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)
• TeX Source

view license

Current browse context:

cs.LG < prev   |   next >
new | recent | 2026-06 Change to browse by: cs

References & Citations

• NASA ADS
• Google Scholar
• Semantic Scholar

export BibTeX citation Loading...

BibTeX formatted citation

× Data provided by:

Bookmark

Bibliographic Tools

Bibliographic and Citation Tools

Bibliographic Explorer Toggle Bibliographic Explorer (What is the Explorer?) Connected Papers Toggle Connected Papers (What is Connected Papers?) Litmaps Toggle Litmaps (What is Litmaps?) scite.ai Toggle scite Smart Citations (What are Smart Citations?) Code, Data, Media

Code, Data and Media Associated with this Article

alphaXiv Toggle alphaXiv (What is alphaXiv?) Links to Code Toggle CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub Toggle DagsHub (What is DagsHub?) GotitPub Toggle Gotit.pub (What is GotitPub?) Huggingface Toggle Hugging Face (What is Huggingface?) ScienceCast Toggle ScienceCast (What is ScienceCast?) Demos

Demos

Replicate Toggle Replicate (What is Replicate?) Spaces Toggle Hugging Face Spaces (What is Spaces?) Spaces Toggle TXYZ.AI (What is TXYZ.AI?) Related Papers

Recommenders and Search Tools

Link to Influence Flower Influence Flower (What are Influence Flowers?) Core recommender toggle CORE Recommender (What is CORE?) IArxiv recommender toggle IArxiv Recommender (What is IArxiv?)

• Author
• Venue
• Institution
• Topic

About arXivLabs

arXivLabs: experimental projects with community collaborators

arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.

Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.

Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)