Muown Implicitly Performs Angular Step-size Decay
Matrix-aware optimizers such as Muon and Muown have recently shown strong empirical performance for pre-training Transformers. In particular, Muown separates each weight matrix into row magnitudes and an un-normalized direction variable, updating the former with Adam and the latter with Muon. We show that the directional update of Muown is equivalent to a Riemannian step on the normalized directions, while the magnitude of the un-normalized parameterization only modulates the angular step size. This explains the step-size stability of Muown and suggests making the angular step size explicit. The resulting method, AngularMuown, optimizes directly over the normalized directions and uses a s...
| Authors | Florian Hübler, Kai Lion, Antonio Orvieto et al. |
|---|---|
| Primary category | cs.LG |
| Published | 22 June 2026 |
| Last updated at source | 22 June 2026 |
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