Muown Implicitly Performs Angular Step-size Decay

arXiv·medium signal

Hübler, Lion, and Orvieto analyze the matrix-aware optimizers Muon and Muown, showing that Muown implicitly performs an angular step-size decay during Transformer pretraining. The result gives a mechanistic explanation for why this optimizer family outperforms standard methods and has practical implications for learning-rate scheduling. Useful for practitioners experimenting with Muon-style optimizers in pretraining runs.