Muon meets Tamed Langevin: Momentum Preconditioning beyond Convex and gradient-Lipschitz Potentials
Abstract
We consider the problem of sampling from Gibbs distributions on matrix spaces whose potential energies are neither convex nor globally gradient-Lipschitz. We introduce a family of non-quadratic kinetic energies that lead to a new underdamped Langevin system with momentum preconditioning, in which the gradient of the kinetic energy acts as a smooth spectral taming of the momentum. We prove that, under these relaxed assumptions on the potential, the resulting dynamics leaves the target Gibbs measure invariant, and we establish exponential convergence to equilibrium in a weighted total variation distance. Finally, we show that the corresponding Euler-Maruyama discretization admits moment bounds that are uniform in time, without any modification of the potential gradient, which ensures the stability of the resulting sampling algorithm.
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BibTeX
@misc{makras2026muon,
title = {{Muon meets Tamed Langevin: Momentum Preconditioning beyond Convex and gradient-Lipschitz Potentials}},
author = {Nikolaos Makras and Sotirios Sabanis},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.02158},
eprint = {2610.02158},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.02158},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Muon meets Tamed Langevin: Momentum Preconditioning beyond Convex and gradient-Lipschitz Potentials
AU - Nikolaos Makras
AU - Sotirios Sabanis
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.02158
UR - https://arxiv.org/abs/2610.02158
AB - We consider the problem of sampling from Gibbs distributions on matrix spaces whose potential energies are neither convex nor globally gradient-Lipschitz. We introduce a family of non-quadratic kinetic energies that lead to a new underdamped Langevin system with momentum preconditioning, in which the gradient of the kinetic energy acts as a smooth spectral taming of the momentum. We prove that, under these relaxed assumptions on the potential, the resulting dynamics leaves the target Gibbs measure invariant, and we establish exponential convergence to equilibrium in a weighted total variation distance. Finally, we show that the corresponding Euler-Maruyama discretization admits moment bounds that are uniform in time, without any modification of the potential gradient, which ensures the stability of the resulting sampling algorithm.
ER -