A Quadratic Lens on Muon's Polar Update: Benefits, Pitfalls, and the Role of Newton-Schulz
arXiv:2610.30280v1 Announce Type: new Abstract: Muon updates matrix-valued parameters along approximately orthogonalized directions and often trains large language models faster than AdamW, yet it remains unclear when and why orthogonalization helps. We study PolarGD, the orthogonalized update at the core of Muon, on matrix quadratics, where optimizers can be analyzed in closed form, and compare update directions under line search to remove the effect of learning-rate tuning. Orthogonalization pays off when the gradient is far from low rank and its largest singular directions see high curvatur
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