Phases of Muon: When Muon Eclipses SignSGD
Recently, Muon and related spectral optimizers have demonstrated strong empirical performance as scalable stochastic methods, often outperforming Adam. Yet their behaviour remains poorly understood. We analyze stochastic spectral optimizers, including Muon, on a high-dimensional matrix-valued least squares problem. We derive explicit deterministic dynamics that provide a tractable framework for studying learning behaviour with a focus on (stochastic) SignSVD, which Muon approximates, and (stochastic) SignSGD, the latter serving as a proxy for Adam. Our analysis shows that for large batch size, SignSVD performs a square-root preconditioning with respect to the data covariance spectrum, whi...
| Authors | Elliot Paquette, Noah Marshall, Lucas Benigni et al. |
|---|---|
| Primary category | math.OC |
| Published | 10 May 2026 |
| Last updated at source | 10 May 2026 |
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