On the Convergence of Muon and Beyond

• URL: http://arxiv.org/abs/2509.15816v3
• Date: Sun, 09 Nov 2025 08:11:57 GMT
• Title: On the Convergence of Muon and Beyond
• Authors: Da Chang, Yongxiang Liu, Ganzhao Yuan,
• Abstract summary: We provide the first proof that variance reduction enables Muon-MVR2 to attain the optimal complexity.<n>Overall, this work offers the first proof of optimality for a Muon-style.
• Score: 31.900178928104648
• License: http://creativecommons.org/licenses/by/4.0/
• 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 explore the theoretical limits of the Muon framework, we analyze two Momentum-based Variance-Reduced variants: a one-batch version (Muon-MVR1) and a two-batch version (Muon-MVR2). We provide the first rigorous proof that incorporating variance reduction enables Muon-MVR2 to attain the optimal iteration complexity of $\tilde{\mathcal{O}}(T^{-1/3})$, thereby matching the theoretical lower bound for this class of problems. Furthermore, our analysis establishes last-iterate convergence guarantees for Muon variants under the Polyak-{\L}ojasiewicz (P{\L}) condition. Extensive experiments on vision (CIFAR-10) and language (C4) benchmarks corroborate our theoretical findings on per-iteration convergence. Overall, this work offers the first proof of optimality for a Muon-style optimizer and clarifies the path toward developing more practically efficient, accelerated variants.

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