AMO: Operator-level Adaptive Muon Orthogonalization

Muon has recently emerged as a competitive alternative to AdamW for large-scale pre-training, with orthogonalization via Newton-Schulz (NS) iteration as its core operation. Standard Muon applies a uniform NS schedule to all parameter matrices, overlooking possible differences in orthogonalization difficulty and its impact on performance. Through a systematic empirical study, we show that this per-matrix heterogeneity is pervasive and strongly associated with matrix geometry, which evolves dynamically across operator types, training stages, and network depths. Therefore, uniform NS schedules can lead to uneven orthogonalization quality across the model. Motivated by these findings, we propose Operator-level Adaptive Muon Orthogonalization (AMO), an observe-then-commit method that measures weight geometry by operator type early in training and then uses these signals to allocate the NS budget for the remainder of training. AMO delivers consistent improvements over uniform-schedule Muon across standard, prolonged, and continual pre-training, surpassing the strongest baseline by +0.76 on Llama3.1-1.4B and +0.51 on Qwen3-1.7B in average downstream performance of 12 evaluation tasks, with gains persisting at Llama3.1-4B scale.