Spectral Scaling Laws of Muon
Orthonormalized update rules have rapidly become a leading choice of optimizer for training large language models, with recent open-source state-of-the-art models adopting Muon. To keep these updates tractable, Muon performs the orthonormalization with the Newton--Schulz (NS) iteration. Since NS is only approximate, directions with small singular values fail to be orthonormalized. In Muon, NS is applied to the momentum matrix at every step, yet little is known about how the singular value spectrum of these momentum matrices behaves during training, or how that behavior changes with model size. We present the first systematic study of this question. Tracking singular value quantiles of the...
| Authors | Gagik Magakyan, Pablo Parrilo, Asuman Ozdaglar |
|---|---|
| Primary category | cs.LG |
| Published | 2 June 2026 |
| Last updated at source | 5 June 2026 |
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