On the Convergence of Muon and Beyond

arXiv:2509.15816v4 Announce Type: replace Abstract: 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 iteration complexity 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 horizon-free learning-rate schedules, variance reduction enables Muon-MVR2 to attain the optimal anytime convergence rate $\tilde{\mathcal{O}}(T^{-1/3})$, matching the lower bound for this problem class. Under the Polyak--\L{}ojasiewicz (PL) condition, we further establish anytime best-iterate guarantees for the expected square-root suboptimality: Muon-MVR1 achieves $\widetilde{\mathcal{O}}(T^{-1/4})$, while Muon-MVR2 achieves $\widetilde{\mathcal{O}}(T^{-1/3})$. Experiments on CIFAR-10 and C4 support the practical effectiveness of the proposed variance-reduced Muon variants.