On the Convergence Analysis of Muon

[2505.23737] On the Convergence Analysis of Muon Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Statistics > Machine Learning arXiv:2505.23737 (stat) [Submitted on 29 May 2025 ( v1 ), last revised 28 Jul 2026 (this version, v3)] Title: On the Convergence Analysis of Muon Authors: Wei Shen , Ruichuan Huang , Minhui Huang , Cong Shen , Jiawei Zhang View a PDF of the paper titled On the Convergence Analysis of Muon, by Wei Shen and 4 other authors View PDF HTML (experimental) Abstract: The majority of parameters in neural networks are naturally represented as matrices. However, most commonly used optimizers treat these matrix parameters as flattened vectors during optimization, potentially overlooking their inherent structural properties. Recently, an optimizer called Muon has been proposed, specifically designed to optimize matrix-structured parameters. Extensive empirical evidence shows that Muon can significantly outperform traditional optimizers when training neural networks. Nonetheless, the theoretical understanding of Muon's convergence behavior and the reasons behind its superior performance remain limited. In this work, we present a comprehensive convergence rate analysis of Muon and its comparison with Gradient Descent (GD). We characterize the conditions under which Muon can outperform GD. Our theoretical results reveal that Muon can benefit from the low-rank structure of Hessian matrices, a phenomenon widely observed in practical neural network training. Our experimental results support and corroborate the theoretical findings. Subjects: Machine Learning (stat.ML) ; Information Theory (cs.IT); Machine Learning (cs.LG); Optimization and Control (math.OC) Cite as: arXiv:2505.23737 [stat.ML] (or arXiv:2505.23737v3 [stat.ML] for this version) https://doi.org/10.48550/arXiv.2505.23737 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Wei Shen [ view email ] [v1] Thu, 29 May 2025 17:58:01 UTC (1,223 KB) [v2] Tue, 14 Apr 2026 01:27:05 UTC (2,856 KB) [v3] Tue, 28 Jul 2026 01:40:23 UTC (2,878 KB) Full-text links: Access Paper: View a PDF of the paper titled On the Convergence Analysis of Muon, by Wei Shen and 4 other authors View PDF HTML (experimental) TeX Source view license Current browse context: stat.ML < prev | next > new | recent | 2025-05 Change to browse by: cs cs.IT cs.LG math math.IT math.OC stat References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from

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