{"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-under-unbounded","title":"Bilevel Optimization under Unbounded Smoothness: A New Algorithm and Convergence Analysis","arxiv_id":"2401.09587","date":"2024-01-17","proceeding":null,"authors":["Jie Hao","Xiaochuan Gong","Mingrui Liu"],"abstract":"Bilevel optimization is an important formulation for many machine learning problems. Current bilevel optimization algorithms assume that the gradient of the upper-level function is Lipschitz. However, recent studies reveal that certain neural networks such as recurrent neural networks (RNNs) and long-short-term memory networks (LSTMs) exhibit potential unbounded smoothness, rendering conventional bilevel optimization algorithms unsuitable. In this paper, we design a new bilevel optimization algorithm, namely BO-REP, to address this challenge. This algorithm updates the upper-level variable using normalized momentum and incorporates two novel techniques for updating the lower-level variable: \\textit{initialization refinement} and \\textit{periodic updates}. Specifically, once the upper-level variable is initialized, a subroutine is invoked to obtain a refined estimate of the corresponding optimal lower-level variable, and the lower-level variable is updated only after every specific period instead of each iteration. When the upper-level problem is nonconvex and unbounded smooth, and the lower-level problem is strongly convex, we prove that our algorithm requires $\\widetilde{\\mathcal{O}}(1/\\epsilon^4)$ iterations to find an $\\epsilon$-stationary point in the stochastic setting, where each iteration involves calling a stochastic gradient or Hessian-vector product oracle. Notably, this result matches the state-of-the-art complexity results under the bounded smoothness setting and without mean-squared smoothness of the stochastic gradient, up to logarithmic factors. Our proof relies on novel technical lemmas for the periodically updated lower-level variable, which are of independent interest. Our experiments on hyper-representation learning, hyperparameter optimization, and data hyper-cleaning for text classification tasks demonstrate the effectiveness of our proposed algorithm.","url_abs":"https://arxiv.org/abs/2401.09587v1","url_pdf":"https://arxiv.org/pdf/2401.09587v1.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-under-unbounded","repo_url":"https://github.com/mingruiliu-ml-lab/bilevel-optimization-under-unbounded-smoothness","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"pytorch","reach":null}],"tasks":[{"task_slug":"bilevel-optimization","task_name":"Bilevel Optimization"},{"task_slug":"hyperparameter-optimization","task_name":"Hyperparameter Optimization"},{"task_slug":"representation-learning","task_name":"Representation Learning"},{"task_slug":"text-classification","task_name":"Text Classification"},{"task_slug":"text-classification-1","task_name":"text-classification"}],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=2401.09587","mcp":{"get_harvested_code_for_paper":{"arxiv_id":"2401.09587"}},"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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