Muon$^2$: Boosting Muon via Adaptive Second-Moment Preconditioning
Muon has emerged as a promising optimizer for large-scale foundation model pre-training by exploiting the matrix structure of neural network updates through iterative orthogonalization. However, the orthogonalization quality of Muon hinges on the number of Newton--Schulz (NS) iterations performed, which poses efficiency challenges due to its non-trivial computation and communication cost. We propose Muon$^2$, an extension of Muon, to improve both quality and efficiency by applying Adam-style adaptive second-moment preconditioning before orthogonalization. Our key insight is that the core challenge of polar approximation in Muon lies in the ill-conditioned momentum matrix, of which the spe...
| Authors | Ziyue Liu, Ruijie Zhang, Zhengyang Wang et al. |
|---|---|
| Primary category | cs.LG |
| Published | 11 April 2026 |
| Last updated at source | 5 June 2026 |
Silytics shows public metadata from arXiv and links to the original. Information may be incomplete or out of date. Last refreshed 17 October 2026. Licenses and terms of the original apply to the work itself. Not financial, legal or professional advice.