# Move on Muon : A Hamiltonian probability gradient flow perspective of Muon optimizer

arXiv: https://arxiv.org/abs/2605.23871 · HTML: https://mlsanity.com/paper/2605.23871/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 18 references that are in the map.
- First posted on arXiv 22 May 2026.

## Builds on (most cited first, 8 of 8)

- [Muon is Scalable for LLM Training](https://mlsanity.com/paper/2502.16982/) (2025)
- [Training Deep Learning Models with Norm-Constrained LMOs](https://mlsanity.com/paper/2502.07529/) (2025)
- [Understanding Gradient Orthogonalization for Deep Learning via Non-Euclidean Trust-Region Optimization](https://mlsanity.com/paper/2503.12645/) (2025)
- [Muon Optimizes Under Spectral Norm Constraints](https://mlsanity.com/paper/2506.15054/) (2025)
- [Orthogonalising gradients to speed up neural network optimisation](https://mlsanity.com/paper/2202.07052/) (2022)
- [Adaptive Matrix Online Learning through Smoothing with Guarantees for Nonsmooth Nonconvex Optimization](https://mlsanity.com/paper/2602.08232/) (2026)
- [Sharp Capacity Scaling of Spectral Optimizers in Learning Associative Memory](https://mlsanity.com/paper/2603.26554/) (2026)
- [Muon Dynamics as a Spectral Wasserstein Flow](https://mlsanity.com/paper/2604.04891/) (2026)

## Built on by (most cited first, 3 of 3)

- [Muon with Finite Newton-Schulz: The Smoothing Benefit in Nonsmooth Nonconvex Optimization](https://mlsanity.com/paper/2608.26288/) (2026)
- [Physical Muon: Orthogonalization as an Equilibrium Computation](https://mlsanity.com/paper/2609.37525/) (2026)
- [Muon meets Tamed Langevin: Momentum Preconditioning beyond Convex and gradient-Lipschitz Potentials](https://mlsanity.com/paper/2610.02158/) (2026)

## Abstract (arXiv)

We develop a gradient flow on the space of probability measures defined on matrix-valued parameters induced by regularized Muon, an analytically smoothed version of the idealized Muon optimizer. The key observation is that the regularized orthogonalization map is the gradient of a smooth Fenchel-dual smoothing of the nuclear norm. This identifies the (regularized) Muon update as a mirror/prox step in the update variable, with momentum acting as the dual coordinate. We use this structure to lift Muon from a single matrix parameter to finite-particle probability objectives of the form J(ρ)=Rleft(int F d ρright), a setting motivated by mean-field descriptions of neural-network training, and derive the inertial continuous-time limit. Using this structure, we derive the finite-particle continuous-time limit under the inertial scaling of step size and momentum, and then pass to a phase-space mean-field equation over probability laws on parameter-momentum pairs. The resulting flow can be shown to be a damped Hamiltonian probability dynamics whose kinetic energy is induced by the regularized Muon mirror potential. We prove an exact Hamiltonian dissipation identity, showing that the Hamiltonian energy decreases monotonically. While the target objective itself need not be monotone along the inertial Muon dynamics, under additional gradient-dominance, bounded-momentum, and curvature/alignment assumptions, we obtain continuous and discrete-time exponential convergence rates for the objective gap. We also study the well-posedness of the mean-field limit equation and establish propagation of chaos guarantees for the interacting particle system. Finally, we extend the formulation to Hilbert-valued feature maps on product matrix spaces, yielding a blockwise Muon probability flow applicable to smooth transformer mixture-of-experts models.