When Is Muon Optimal? Radial Fragility and Minimax Geometry
Abstract
Muon replaces a matrix gradient by an approximate polar factor and has shown strong empirical performance in neural-network training. The Isotropic Curvature Model (ICM) provides a curvature-based explanation for this update by asking when gradient orthogonalization minimizes a one-step local-loss surrogate. We investigate the stability of this explanation when the fixed input radius of the ICM is replaced by a general radial law. We introduce the Random-Radius Isotropic Curvature Model (RR-ICM) and the Polar-Factor Admissibility Set, which encodes exact scaled-polar optimality as a geometric condition on the positive polar factor of the gradient. We derive an exact characterization of this set for arbitrary radial laws and show that all nonscalar admissible responses arise from radial atoms aligned with curvature kinks. Consequently, every atomless radial law exhibits zero-capacity rigidity: exact scaled-polar optimality is possible only when the gradient is already a scaled column isometry. To connect the curvature-based framework with the established spectral geometry of Muon, we introduce the Minimax RR-ICM by replacing directional averaging with worst-case optimization over probability laws on the sphere. We prove that every nonzero global minimizer lies on the polar ray of the gradient, while its scale is determined by a one-dimensional convex problem. The corresponding admissibility geometry is a positive-semidefinite trace slab, in sharp contrast to the spherical-zonoid geometry of the RR-ICM. Together, these results expose the radial fragility of average-curvature explanations and provide a new minimax variational characterization of Muon's polar direction.
Related
- The Fragile Coupon Collector: Random Deadlines, Critical Fragility, and Optimal Sampling
- Discrete Diffusion Models are Minimax Optimal Under Sparsity
- Minimax-Optimal Static Estimators Under Interference
- When Is a Relevance Threshold Statistically Resolvable? Minimax Limits for Effect Classification
- Nonparametric In-Context Learning under Growing Geometric Complexity: Minimax Optimality and Local Geometry-Adaptivity of Transformers
BibTeX
@misc{xu2026when,
title = {{When Is Muon Optimal? Radial Fragility and Minimax Geometry}},
author = {Ao Xu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.21549},
eprint = {2610.21549},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.21549},
}
RIS
TY - UNPB
TI - When Is Muon Optimal? Radial Fragility and Minimax Geometry
AU - Ao Xu
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.21549
UR - https://arxiv.org/abs/2610.21549
AB - Muon replaces a matrix gradient by an approximate polar factor and has shown strong empirical performance in neural-network training. The Isotropic Curvature Model (ICM) provides a curvature-based explanation for this update by asking when gradient orthogonalization minimizes a one-step local-loss surrogate. We investigate the stability of this explanation when the fixed input radius of the ICM is replaced by a general radial law. We introduce the Random-Radius Isotropic Curvature Model (RR-ICM) and the Polar-Factor Admissibility Set, which encodes exact scaled-polar optimality as a geometric condition on the positive polar factor of the gradient. We derive an exact characterization of this set for arbitrary radial laws and show that all nonscalar admissible responses arise from radial atoms aligned with curvature kinks. Consequently, every atomless radial law exhibits zero-capacity rigidity: exact scaled-polar optimality is possible only when the gradient is already a scaled column isometry. To connect the curvature-based framework with the established spectral geometry of Muon, we introduce the Minimax RR-ICM by replacing directional averaging with worst-case optimization over probability laws on the sphere. We prove that every nonzero global minimizer lies on the polar ray of the gradient, while its scale is determined by a one-dimensional convex problem. The corresponding admissibility geometry is a positive-semidefinite trace slab, in sharp contrast to the spherical-zonoid geometry of the RR-ICM. Together, these results expose the radial fragility of average-curvature explanations and provide a new minimax variational characterization of Muon's polar direction.
ER -