When Muon Becomes Gradient Descent: Finite-Time Flows and Noise Smoothing Near the Optimum

Muon is a recently proposed optimizer whose matrix update applies the polar factor (matrix sign) of a gradient or momentum matrix. We study an idealized no-momentum, no-weight-decay Muon update on matrix parameters $W\in {\mathbb{R}}^{m\times n}$ in both deterministic and stochastic regimes. In the deterministic continuous-time limit, Muon is steepest descent with respect to the spectral norm and obeys the dissipation identity $\frac{d}{dt}F(W_t)=-\Vert \nabla F(W_t){\Vert }_{\ast }$; under a nuclear Łojasiewicz-type error bound this yields *finite-time* convergence, in contrast to the merely exponential rate of gradient flow. In the minibatch case, the correct continuous model is an Itô SDE whose drift is the noise-smoothed polar map $\Psi (W)=\mathbb{E}[\mathrm{msign}(\nabla F(W)+\xi )]$, not $\mathrm{msign}(\nabla F(W))$. For Gaussian noise $\xi =sZ$, the drift linearizes near stationarity as ${\Psi }_s(Y)=(c_{m,n}/s)Y+o(\Vert Y{\Vert }_F/s)$ with $c_{m,n}=\mathbb{E}\Vert Z{\Vert }_{\ast }/(mn)$, and $c_{N,N}\sim 8/(3\pi \sqrt{N})$ for square $N\times N$ matrices. Thus stochasticity smooths the matrix sign and locally turns Muon into a scaled gradient method with an Ornstein–Uhlenbeck noise neighborhood. This suggests a simple hybrid: use Muon away from stationarity to remove anisotropy and switch to SGD near the optimum, where Muon's finite-time advantage disappears and, at a matched local contraction rate, its stationary noise floor is never lower than that of SGD.