Muon learns balanced solutions in matrix factorization without slow saddle-to-saddle dynamics
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...
| Authors | Mark Rhee, Jamie Simon, Dhruva Karkada |
|---|---|
| Primary category | cs.LG |
| Published | 29 June 2026 |
| Last updated at source | 29 June 2026 |
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