On the Convergence of Muon and Beyond

The Muon optimizer has demonstrated remarkable empirical success in handling matrix-structured parameters for training neural networks. However, a significant gap remains between its practical performance and theoretical understanding. Existing analyses show that the Muon variants achieve only a suboptimal ergodic convergence rate of $\mathcal{O}(T^{-1/4})$ in stochastic non-convex settings, where $T$ denotes the number of iterations. To study the theoretical limits of Muon, we analyze two momentum-based variance-reduced variants: the one-batch Muon-MVR1 and the two-batch Muon-MVR2. We provide the first rigorous proof that, under \textbf{horizon-free} learning-rate schedules, variance reduction enables Muon-MVR2 to attain the optimal anytime convergence rate $\widetilde{\mathcal{O}}(T^{-1/3})$, matching the lower bound for this problem class. Under the Polyak--\L{}ojasiewicz (PL) condition, we establish anytime guarantees for Muon-MVR1 and Muon-MVR2: they attain best-iterate rates of $\widetilde{\mathcal{O}}(T^{-1/4})$ and $\widetilde{\mathcal{O}}(T^{-1/3})$ for the expected square-root suboptimality, and, given an additional uniform gradient bound along the iterates, achieve last-iterate rates of $\mathcal{O}(T^{-1/4})$ and $\mathcal{O}(T^{-1/3})$ for the objective gap, respectively. Experiments on CIFAR-10 and C4 support the practical effectiveness of the proposed variance-reduced Muon variants. Code is available at \href{https://github.com/MaeChd/MUON-MVR}{Muon-MVR} Codebase.