# Muon meets Tamed Langevin: Momentum Preconditioning beyond Convex and gradient-Lipschitz Potentials

arXiv: https://arxiv.org/abs/2610.02158 · HTML: https://mlsanity.com/paper/2610.02158/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 29 of its 53 references that are in the map.
- First posted on arXiv 1 Oct 2026.

## Builds on (most cited first, 29 of 29)

- [Kimi K2: Open Agentic Intelligence](https://mlsanity.com/paper/2507.20534/) (2025)
- [Muon is Scalable for LLM Training](https://mlsanity.com/paper/2502.16982/) (2025)
- [What Are Bayesian Neural Network Posteriors Really Like?](https://mlsanity.com/paper/2104.14421/) (2021)
- [Non-convex learning via Stochastic Gradient Langevin Dynamics: a nonasymptotic analysis](https://mlsanity.com/paper/1702.03849/) (2017)
- [Theoretical guarantees for approximate sampling from smooth and log-concave densities](https://mlsanity.com/paper/1412.7392/) (2014)
- [Non-asymptotic convergence analysis for the Unadjusted Langevin Algorithm](https://mlsanity.com/paper/1507.05021/) (2015)
- [Training Deep Learning Models with Norm-Constrained LMOs](https://mlsanity.com/paper/2502.07529/) (2025)
- [A Complete Recipe for Stochastic Gradient MCMC](https://mlsanity.com/paper/1506.04696/) (2015)
- [Underdamped Langevin MCMC: A non-asymptotic analysis](https://mlsanity.com/paper/1707.03663/) (2017)
- [Efficient Bayesian computation by proximal Markov chain Monte Carlo: when Langevin meets Moreau](https://mlsanity.com/paper/1612.07471/) (2016)
- [Practical Efficiency of Muon for Pretraining](https://mlsanity.com/paper/2505.02222/) (2025)
- [Sampling Can Be Faster Than Optimization](https://mlsanity.com/paper/1811.08413/) (2018)
- [The Tamed Unadjusted Langevin Algorithm](https://mlsanity.com/paper/1710.05559/) (2017)
- [The Langevin Monte Carlo algorithm in the non-smooth log-concave case](https://mlsanity.com/paper/2101.10695/) (2021)
- [Kinetic Langevin MCMC Sampling Without Gradient Lipschitz Continuity -- the Strongly Convex Case](https://mlsanity.com/paper/2301.08039/) (2023)
- [HTMuon: Improving Muon via Heavy-Tailed Spectral Correction](https://mlsanity.com/paper/2603.10067/) (2026)
- [Polygonal Unadjusted Langevin Algorithms: Creating stable and efficient adaptive algorithms for neural networks](https://mlsanity.com/paper/2105.13937/) (2021)
- [Subgradient Langevin Methods for Sampling from Non-smooth Potentials](https://mlsanity.com/paper/2308.01417/) (2023)
- [Non-asymptotic convergence bounds for modified tamed unadjusted Langevin algorithm in non-convex setting](https://mlsanity.com/paper/2207.02600/) (2022)
- [Delving into Muon and Beyond: Deep Analysis and Extensions](https://mlsanity.com/paper/2602.04669/) (2026)
- [Muon Does Not Converge on Convex Lipschitz Functions](https://mlsanity.com/paper/2605.08980/) (2026)
- [Fractal Landscapes in Policy Optimization](https://mlsanity.com/paper/2310.15418/) (2023)
- [The Performance Of The Unadjusted Langevin Algorithm Without Smoothness Assumptions](https://mlsanity.com/paper/2502.03458/) (2025)
- [Softsign: Smooth Sign in Your Optimizer For Better Parameter Heterogeneity Handling](https://mlsanity.com/paper/2605.31371/) (2026)
- [Spectral Scaling Laws of Muon](https://mlsanity.com/paper/2606.04058/) (2026)
- [Move on Muon : A Hamiltonian probability gradient flow perspective of Muon optimizer](https://mlsanity.com/paper/2605.23871/) (2026)
- [On MUON optimization: From non-convergence to an error analysis with Polar Express and the Newton-Schulz polynomial from implementations](https://mlsanity.com/paper/2608.04607/) (2026)
- [Musec: MomentUm SpEctral Clipping for Stable Muon-type Training](https://mlsanity.com/paper/2609.11655/) (2026)
- [The Tamed Subgradient Unadjusted Langevin Algorithm beyond Convexity](https://mlsanity.com/paper/2608.06283/) (2026)

## Abstract (arXiv)

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.