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Penalty-based methods have become popular for solving bilevel optimization (BLO) problems, thanks to their effective first-order nature. However, they often require inner-loop iterations to solve the lower-level (LL) problem and small…

Optimization and Control · Mathematics 2025-11-24 Liuyuan Jiang , Quan Xiao , Lisha Chen , Tianyi Chen

Bilevel optimization is characterized by a two-level optimization structure, where the upper-level problem is constrained by optimal lower-level solutions, and such structures are prevalent in real-world problems. The constraint by optimal…

Machine Learning · Computer Science 2025-05-29 Ruth Wan Theng Chew , Quoc Phong Nguyen , Bryan Kian Hsiang Low

In this paper we study convex bi-level optimization problems for which the inner level consists of minimization of the sum of smooth and nonsmooth functions. The outer level aims at minimizing a smooth and strongly convex function over the…

Optimization and Control · Mathematics 2017-02-15 Shoham Sabach , Shimrit Shtern

Bilevel optimization (BLO) is a popular approach with many applications including hyperparameter optimization, neural architecture search, adversarial robustness and model-agnostic meta-learning. However, the approach suffers from time and…

Machine Learning · Computer Science 2021-06-08 Valerii Likhosherstov , Xingyou Song , Krzysztof Choromanski , Jared Davis , Adrian Weller

While stochastic bilevel optimization methods have been extensively studied for addressing large-scale nested optimization problems in machine learning, it remains an open question whether the optimal complexity bounds for solving bilevel…

Optimization and Control · Mathematics 2026-03-24 Tianshu Chu , Dachuan Xu , Wei Yao , Jin Zhang

We develop and analyze stochastic approximation algorithms for solving nested compositional bi-level optimization problems. These problems involve a nested composition of $T$ potentially non-convex smooth functions in the upper-level, and a…

Optimization and Control · Mathematics 2023-07-12 Xuxing Chen , Krishnakumar Balasubramanian , Saeed Ghadimi

Stochastic bilevel optimization (SBO) has become a standard framework for hyperparameter learning, data reweighting, representation learning, and data-mixture optimization in deep learning. Existing exact single-loop SBO methods and…

Machine Learning · Computer Science 2026-05-13 Hengrui Zhang , Boao Kong , Engao Zhang , Kun Yuan

Bilevel optimization problems arise in many applications where decisions must account for the optimal response of another system, such as in game-theoretic settings. However, these problems are notoriously challenging, as even linear…

Optimization and Control · Mathematics 2026-05-21 Eliabelle Mauduit , Eloïse Berthier , Andrea Simonetto

We introduce a framework based on bilevel programming that unifies gradient-based hyperparameter optimization and meta-learning. We show that an approximate version of the bilevel problem can be solved by taking into explicit account the…

Machine Learning · Statistics 2018-07-04 Luca Franceschi , Paolo Frasconi , Saverio Salzo , Riccardo Grazzi , Massimilano Pontil

Bilevel optimization problems, encountered in fields such as economics, engineering, and machine learning, pose significant computational challenges due to their hierarchical structure and constraints at both upper and lower levels.…

Optimization and Control · Mathematics 2024-08-21 Mengwei Xu , Yu-Hong Dai , Xin-Wei Liu , Bo Wang

The subgradient method is one of the most fundamental algorithmic schemes for nonsmooth optimization. The existing complexity and convergence results for this method are mainly derived for Lipschitz continuous objective functions. In this…

Optimization and Control · Mathematics 2024-11-01 Xiao Li , Lei Zhao , Daoli Zhu , Anthony Man-Cho So

We consider a generic min-max multi-objective bilevel optimization problem with applications in robust machine learning such as representation learning and hyperparameter optimization. We design MORBiT, a novel single-loop gradient…

Machine Learning · Computer Science 2023-03-08 Alex Gu , Songtao Lu , Parikshit Ram , Lily Weng

We present in this paper novel accelerated fully first-order methods in \emph{Bilevel Optimization} (BLO). Firstly, for BLO under the assumption that the lower-level functions admit the typical strong convexity assumption, the…

Optimization and Control · Mathematics 2024-07-10 Chris Junchi Li

Variational regularization techniques are dominant in the field of mathematical imaging. A drawback of these techniques is that they are dependent on a number of parameters which have to be set by the user. A by now common strategy to…

Optimization and Control · Mathematics 2020-12-10 Matthias J. Ehrhardt , Lindon Roberts

Bilevel optimization (BO) is widely applicable to many machine learning problems. Scaling BO, however, requires repeatedly computing hypergradients, which involves solving inverse Hessian-vector products (IHVPs). In practice, these…

Machine Learning · Computer Science 2026-04-01 Disen Liao , Felix Dangel , Yaoliang Yu

Bilevel optimization has recently attracted significant attention in machine learning due to its wide range of applications and advanced hierarchical optimization capabilities. In this paper, we propose a plug-and-play framework, named…

Optimization and Control · Mathematics 2025-05-05 Tianshu Chu , Dachuan Xu , Wei Yao , Chengming Yu , Jin Zhang

Hyperparameter optimization in machine learning is often achieved using naive techniques that only lead to an approximate set of hyperparameters. Although techniques such as Bayesian optimization perform an intelligent search on a given…

Machine Learning · Computer Science 2023-06-21 Ankur Sinha , Satender Gunwal , Shivam Kumar

Functional bilevel optimization (FBO) provides a powerful framework for hierarchical learning in function spaces, yet current methods are limited to static offline settings and perform suboptimally in online, non-stationary scenarios. We…

Machine Learning · Statistics 2026-01-23 Jason Bohne , Ieva Petrulionyte , Michael Arbel , Julien Mairal , Paweł Polak

Bilevel optimization has arisen as a powerful tool in modern machine learning. However, due to the nested structure of bilevel optimization, even gradient-based methods require second-order derivative approximations via Jacobian- or/and…

Machine Learning · Computer Science 2022-06-07 Daouda Sow , Kaiyi Ji , Yingbin Liang

In this paper, we study the problem of solving a simple bilevel optimization problem, where the upper-level objective is minimized over the solution set of the lower-level problem. We focus on the general setting in which both the upper-…

Optimization and Control · Mathematics 2025-08-01 Jincheng Cao , Ruichen Jiang , Erfan Yazdandoost Hamedani , Aryan Mokhtari