Move on Muon : A Hamiltonian probability gradient flow perspective of Muon optimizer
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 $J(ρ)=R\left(\int F d ρ\right)$, a setting motivated by mean-field descriptions of neural-network training,...
| Authors | Aratrika Mustafi, Soumya Mukherjee, Bharath K. Sriperumbudur |
|---|---|
| Primary category | stat.ML |
| Published | 22 May 2026 |
| Last updated at source | 22 May 2026 |
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