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Second-Order Muon Done Right: A Principled Marriage of Spectral Geometry and Curvature

2026-08-10 15:55:07
Tong Che
arXiv_AI
       
Abstract

Muon's polar update is exact for an unweighted spectral geometry. We introduce GO-MUON, which uses a matched data-dependent geometry and reuses it across several optimization steps. Conditioned on any positive-definite left and right maps, its raw update exactly solves the corresponding weighted spectral oracle; this statement is independent of how the maps are estimated or how recently they were refreshed. For softmax cross-entropy, we quantify when the observed-label backward factor approaches the model Fisher and generalized Gauss--Newton factor. We also show that four-step refresh nearly preserves the tracking delay of slowly changing geometry while increasing stationary factor noise, making lazy geometry a compute--statistics tradeoff rather than a denoising mechanism.

Abstract (translated)

URL

https://arxiv.org/abs/2608.09763

PDF

https://arxiv.org/pdf/2608.09763.pdf

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