Stochastic Dynamic Barrier Perturbed Gradient Methods for Nonconvex Simple Bilevel Optimization

[2607.10957] Stochastic Dynamic Barrier Perturbed Gradient Methods for Nonconvex Simple Bilevel Optimization Skip to main content Search Submit Donate Log in Mathematics > Optimization and Control arXiv:2607.10957 (math) [Submitted on 12 Jul 2026 ( v1 ), last revised 21 Aug 2026 (this version, v2)] Title: Stochastic Dynamic Barrier Perturbed Gradient Methods for Nonconvex Simple Bilevel Optimization Authors: Mohammad Mahdi Ahmadi , Jincheng Cao , Aryan Mokhtari , Erfan Yazdandoost Hamedani View PDF HTML (experimental) Abstract: We study stochastic simple bilevel optimization with smooth, possibly nonconvex upper- and lower-level objectives accessed only through stochastic gradient oracles. A key challenge is that the dual multiplier induced by the lower-level constraint may become unbounded near lower-level stationary points, invalidating bounded-dual analyses and destabilizing stochastic gradient estimates. To address this, we propose \emph{Stochastic Dynamic Barrier Perturbed Gradient} (SDBPG), a single-loop method that adaptively perturbs the dual formulation to regularize this degeneracy. The perturbation stabilizes the multiplier and yields controlled bias and variance even near the lower-level stationarity region. Under a rare-visit assumption governed by a parameter $\delta \in (0, \tfrac{1}{2}]$, SDBPG finds an $(\epsilon, \epsilon)$-stationary point in $\mathcal{O}(\epsilon^{-1/\delta})$ iterations, with sample gradient complexities $\mathcal{O}(\epsilon^{-2/\delta})$ and $\mathcal{O}(\epsilon^{-3/\delta})$ for the upper- and lower-level objectives, where larger $\delta$ corresponds to rarer visits to the bad region describing the negative alignment between the two objectives when the lower-level gradient is small. We further develop PR-SDBPG, a penalty-regularized variant that eliminates the rare-visit assumption, and VR-PR-SDBPG, which improves the resulting sample complexities entirely through variance reduction. To our knowledge, these are the first explicit $(\epsilon_f,\epsilon_g)$-stationarity guarantees for stochastic nonconvex-nonconvex simple bilevel optimization. Comments: The new version resolves the issue in the proof of Theorem 3.3 and modifies Assumption 3.1 to impose the rare visit assumption on the true bad region only Subjects: Optimization and Control (math.OC) Cite as: arXiv:2607.10957 [math.OC] (or arXiv:2607.10957v2 [math.OC] for this version) https://doi.org/10.48550/arXiv.2607.10957 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Erfan Yazdandoost Hamedani [ view email ] [v1] Sun, 12 Jul 2026 23:23:02 UTC (955 KB) [v2] Fri, 21 Aug 2026 20:26:30 UTC (956 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.OC < prev | next > new | recent | 2026-07 Change to browse by: math 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? ) 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)

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