{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/bilevel-optimization-with-a-lower-level","title":"Bilevel Optimization with a Lower-level Contraction: Optimal Sample Complexity without Warm-start","arxiv_id":"2202.03397","date":"2022-02-07","proceeding":"NeurIPS 2023 11","authors":["Riccardo Grazzi","Massimiliano Pontil","Saverio Salzo"],"abstract":"We analyse a general class of bilevel problems, in which the upper-level problem consists in the minimization of a smooth objective function and the lower-level problem is to find the fixed point of a smooth contraction map. This type of problems include instances of meta-learning, equilibrium models, hyperparameter optimization and data poisoning adversarial attacks. Several recent works have proposed algorithms which warm-start the lower-level problem, i.e.~they use the previous lower-level approximate solution as a staring point for the lower-level solver. This warm-start procedure allows one to improve the sample complexity in both the stochastic and deterministic settings, achieving in some cases the order-wise optimal sample complexity. However, there are situations, e.g., meta learning and equilibrium models, in which the warm-start procedure is not well-suited or ineffective. In this work we show that without warm-start, it is still possible to achieve order-wise (near) optimal sample complexity. In particular, we propose a simple method which uses (stochastic) fixed point iterations at the lower-level and projected inexact gradient descent at the upper-level, that reaches an $\\epsilon$-stationary point using $O(\\epsilon^{-2})$ and $\\tilde{O}(\\epsilon^{-1})$ samples for the stochastic and the deterministic setting, respectively. Finally, compared to methods using warm-start, our approach yields a simpler analysis that does not need to study the coupled interactions between the upper-level and lower-level iterates.","url_abs":"https://arxiv.org/abs/2202.03397v4","url_pdf":"https://arxiv.org/pdf/2202.03397v4.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"abstracts"},"code_links":[{"paper_slug":"bilevel-optimization-with-a-lower-level","repo_url":"https://github.com/csml-iit-ucl/bioptexps","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"pytorch","reach":null},{"paper_slug":"bilevel-optimization-with-a-lower-level","repo_url":"https://github.com/prolearner/hypertorch","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"pytorch","reach":null}],"tasks":[{"task_slug":"bilevel-optimization","task_name":"Bilevel Optimization"},{"task_slug":"data-poisoning","task_name":"Data Poisoning"},{"task_slug":"hyperparameter-optimization","task_name":"Hyperparameter Optimization"},{"task_slug":"meta-learning","task_name":"Meta-Learning"}],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=2202.03397","mcp":{"get_harvested_code_for_paper":{"arxiv_id":"2202.03397"}},"developers":"https://syntology.ai/developers","read_at":"2026-09-24T18:15:14+00:00","read_at_is":"when the build read Syntology's graph, not when any sample ran","claim":"Per-sample execution status on synthesized fixtures; not a correctness claim about the paper. 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