Angle-Scheduled Muon: Replacing Implicit Step Decay with an Explicit Schedule

Muon orthogonalizes momentum so that the norm of each matrix update is fixed by the learning rate and the matrix shape. However, we observe that weight norms keep growing during training while update norms remain nearly constant, so each update moves the weights by a shrinking fraction even at a constant learning rate. We refer to this as implicit step decay: the angular step, i.e., the update norm relative to the weight norm, decays at a rate dictated by weight growth rather than by the learning-rate schedule. Based on this insight, we propose Angle-Scheduled Muon (ASM), which replaces implicit step decay with an explicit schedule. ASM keeps Muon's orthogonalized update direction and sets each update's length to a prescribed fraction of the current weight norm, so that every weight matrix follows the same angular-step schedule. Crucially, ASM is a drop-in modification of Muon that adds only norm computations, requiring no additional orthogonalization or gradient evaluations. Experiments on language modeling and image classification show that ASM consistently improves over Muon and recent Muon variants with limited overhead.