Muon Under Gradient Noise and the Limits of Orthogonalization Near Optima

arXiv:2609.32861 · cs.LG · Submitted 2026-09-26 · Read on arXiv

cs.LG

Submitted: 2026-09-26

Updated: 2026-09-26

Terminology

• Transformer — 10× in this paper · explained in 2 episodes
• Calibration — 2× in this paper · explained in 7 episodes
• Homogeneity — 2× in this paper · explained in 2 episodes
• Trace — 2× in this paper · explained in 2 episodes
• Adam — 1× in this paper · explained in 2 episodes
• Anisotropy — 1× in this paper · explained in 3 episodes
• Bayesian Inference — 1× in this paper · explained in 5 episodes
• Exponential Moving Average (EMA) — 1× in this paper · explained in 2 episodes

Sources

• Old Optimizer, New Norm: An Anthology
• On the Convergence of Muon and Beyond
• Muon with Nesterov Momentum: Heavy-Tailed Noise and (Randomized) Inexact Polar Decomposition
• Understanding Gradient Orthogonalization for Deep Learning via Non-Euclidean Trust-Region Optimization
• A Note on the Convergence of Muon
• Denoise First, Orthogonalize Later: Understanding Momentum in Muon via Spectral Filtering
• Muon is Scalable for LLM Training
• On the Convergence Analysis of Muon

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