A unified convergence theory for adaptive first-order methods in the nonconvex case, including AdaNorm, full and diagonal AdaGrad and Muon

[2604.17423] A unified convergence theory for adaptive first-order methods in the nonconvex case, including AdaNorm, full and diagonal AdaGrad and Muon Skip to main content Search Submit Donate Log in Computer Science > Machine Learning arXiv:2604.17423 (cs) [Submitted on 19 Apr 2026 ( v1 ), last revised 27 Aug 2026 (this version, v3)] Title: A unified convergence theory for adaptive first-order methods in the nonconvex case, including AdaNorm, full and diagonal AdaGrad and Muon Authors: S. Gratton , Ph. L. Toint View PDF HTML (experimental) Abstract: A unified framework for first-order optimization algorithms fornonconvex unconstrained optimization is proposed that uses adaptivelypreconditioned gradients and includes popular methods such as full anddiagonal AdaGrad, AdaNorm, as well as an adpative variant of Muon. This framework also allows combining heterogeneous geometries across different groups of variables while preserving a unified convergence analysis. A fully stochastic global rate-of-convergence analysis is conducted for all methods in the framework, with andwithout two types of momentum, using reasonable assumptions on the variance of the gradient oracle and without assuming bounded stochastic gradients or small enough stepsize. Subjects: Machine Learning (cs.LG) MSC classes: 90C26, 90C06, 90C15, 90C30 ACM classes: I.2.6; F.2.1; G.1.6 Cite as: arXiv:2604.17423 [cs.LG] (or arXiv:2604.17423v3 [cs.LG] for this version) https://doi.org/10.48550/arXiv.2604.17423 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Philippe Toint [ view email ] [v1] Sun, 19 Apr 2026 13:07:51 UTC (30 KB) [v2] Fri, 1 May 2026 14:46:49 UTC (30 KB) [v3] Thu, 27 Aug 2026 08:59:07 UTC (33 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Additional Features Audio Summary Current browse context: cs.LG < prev | next > new | recent | 2026-04 Change to browse by: cs References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation 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? ) IArxiv recommender toggle IArxiv Recommender ( What is IArxiv? ) 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)

Related documents

• A unified convergence theory for adaptive first-order methods in the nonconvex case, including AdaNorm, full and diagonal AdaGrad and Muon#677577
• Can Adaptive Gradient Methods Converge under Heavy-Tailed Noise? A Case Study of AdaGrad#200482
• An Adaptive Augmented Lagrangian Method for Deterministic and Stochastic Nonconvex Optimization#780975
• An Adaptive Augmented Lagrangian Method for Deterministic and Stochastic Nonconvex Optimization#779600
• SGHA: A Single-Loop Fully First-Order Algorithm for Nonconvex-Strongly-Convex Bilevel Optimization#482143