Spectral Allocation: Why Muon Outperforms Adam, and How to Improve Muon
Orthogonal optimisers such as Muon can substantially accelerate large language model pretraining relative to Adam, yet the mechanism remains incompletely understood. We investigate this through an out-of-sample spectral probing analysis of Transformer loss landscapes. At checkpoints along real training trajectories, we decompose each momentum buffer into its singular directions and estimate the loss-optimal step size along each direction on held-out data. The resulting spectral profile is anisotropic yet stable across batches and training stages, and consistent across the optimisers and model scales: a volatile head operating at the Edge-of-Stability supports a much smaller step size than...
| Authors | Xiaodong Wu, Wenyi Yu, Chao Zhang et al. |
|---|---|
| Primary category | cs.LG |
| Published | 26 August 2026 |
| Last updated at source | 26 August 2026 |
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