[2301.00712] On Finding Small Hyper-Gradients in Bilevel Optimization: Hardness Results and Improved Analysis Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Optimization and Control arXiv:2301.00712 (math) [Submitted on 2 Jan 2023 ( v1 ), last revised 28 Apr 2026 (this version, v9)] Title: On Finding Small Hyper-Gradients in Bilevel Optimization: Hardness Results and Improved Analysis Authors: Lesi Chen , Jing Xu , Jingzhao Zhang View a PDF of the paper titled On Finding Small Hyper-Gradients in Bilevel Optimization: Hardness Results and Improved Analysis, by Lesi Chen and 1 other authors View PDF HTML (experimental) Abstract: Bilevel optimization reveals the inner structure of otherwise oblique optimization problems, such as hyperparameter tuning, neural architecture search, and meta-learning. A common goal in bilevel optimization is to minimize a hyper-objective that implicitly depends on the solution set of the lower-level function. Although this hyper-objective approach is widely used, its theoretical properties have not been thoroughly investigated in cases where the lower-level functions lack strong convexity. In this work, we first provide hardness results to show that the goal of finding stationary points of the hyper-objective for nonconvex-convex bilevel optimization can be intractable for zero-respecting algorithms. Then we study a class of tractable nonconvex-nonconvex bilevel problems when the lower-level function satisfies the Polyak-Łojasiewicz (PL) condition. We show a simple first-order algorithm can achieve better complexity bounds of $\tilde{\mathcal{O}}(\epsilon^{-2})$, $\tilde{\mathcal{O}}(\epsilon^{-4})$ and $\tilde{\mathcal{O}}(\epsilon^{-6})$ in the deterministic, partially stochastic, and fully stochastic setting respectively. The complexities in the first two cases are optimal up to logarithmic factors. Comments: Published in COLT 2024. This arXiv version refines Assumption 4.1 (d); adds discussions on related works in Appendix A; and corrects the kappa dependency in the upper bounds Subjects: Optimization and Control (math.OC) ; Artificial Intelligence (cs.AI); Machine Learning (cs.LG) Cite as: arXiv:2301.00712 [math.OC] (or arXiv:2301.00712v9 [math.OC] for this version) https://doi.org/10.48550/arXiv.2301.00712 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Lesi Chen [ view email ] [v1] Mon, 2 Jan 2023 15:09:12 UTC (45 KB) [v2] Fri, 26 May 2023 08:59:55 UTC (798 KB) [v3] Mon, 29 May 2023 01:07:24 UTC (798 KB) [v4] Thu, 8 Feb 2024 07:49:07 UTC (835 KB) [v5] Tue, 14 May 2024 10:32:46 UTC (75 KB) [v6] Sat, 28 Dec 2024 05:44:48 UTC (75 KB) [v7] Sun, 5 Jan 2025 06:43:46 UTC (75 KB) [v8] Wed, 18 Jun 2025 06:53:36 UTC (75 KB) [v9] Tue, 28 Apr 2026 06:26:34 UTC (59 KB) Full-text links: Access Paper: View a PDF of the paper titled On Finding Small Hyper-Gradients in Bilevel Optimization: Hardness Results and Improved Analysis, by Lesi Chen and 1 other authors View PDF HTML (experimental) TeX Source view license Current browse context: math.OC < prev | next > new | recent | 2023-01 Change to browse by: cs cs.AI cs.LG math References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... 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