It is well known that bilevel optimization problems are hard to solve both in theory and practice. In this paper, we highlight a further computational difficulty when it comes to solving bilevel problems with continuous but nonconvex lower levels. Even if the lower-level problem is solved to $\varepsilon$-feasibility regarding its nonlinear constraints for an arbitrarily small but positive $\varepsilon$, the obtained bilevel solution as well as its objective value may be arbitrarily far away from the actual bilevel solution and its actual objective value. This result even holds for bilevel problems for which the nonconvex lower level is uniquely solvable, for which the strict complementarity condition holds, for which the feasible set is convex, and for which Slater's constraint qualification is satisfied for all feasible upper-level decisions. Since the consideration of $\varepsilon$-feasibility cannot be avoided when solving nonconvex problems to global optimality, our result shows that computational bilevel optimization with continuous and nonconvex lower levels needs to be done with great care. Finally, we illustrate that the nonlinearities in the lower level are the key reason for the observed bad behavior by showing that linear bilevel problems behave much better at least on the level of feasible solutions.
SUN-DSBO: A Structured Unified Framework for Nonconvex Decentralized Stochastic Bilevel Optimization
Southern University of Science and Technology（南方科技大学）；Pengcheng Laboratory（鹏城实验室）；Sun Yat-sen University（中山大学）；National Center for Applied Mathematics Shenzhen（深圳应用数学国家中心）
Adaptive Algorithms for Nonconvex Bilevel Optimization under PŁ Conditions
School of Data Science, Fudan University（复旦大学数据科学学院）；Shanghai Key Laboratory for Contemporary Applied Mathematics, Fudan University（复旦大学当代应用数学上海市重点实验室）
A Hybrid Subgradient Method for Nonsmooth Nonconvex Bilevel Optimization
The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳））；Shenzhen University（深圳大学）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；University of Chinese Academy of Sciences（中国科学院大学）；National University of Singapore（新加坡国立大学）
Bilevel optimization has been successfully applied to many important machine learning problems. Algorithms for solving bilevel optimization have been studied under various settings. In this paper, we study the nonconvex-strongly-convex bilevel optimization under a decentralized setting. We design decentralized algorithms for both deterministic and stochastic bilevel optimization problems. Moreover, we analyze the convergence rates of the proposed algorithms in difference scenarios including the case where data heterogeneity is observed across agents. Numerical experiments on both synthetic and…
We propose a bilevel optimization strategy for selecting the best hyperparameter value for the nonsmooth $\ell_p$ regularizer with $0<p\le 1$. The concerned bilevel optimization problem has a nonsmooth, possibly nonconvex, $\ell_p$-regularized problem as the lower-level problem. Despite the recent popularity of nonconvex $\ell_p$-regularizer and the usefulness of bilevel optimization for selecting hyperparameters, algorithms for such bilevel problems have not been studied because of the difficulty of $\ell_p$-regularizer. Our contribution is the proposal of the first algorithm equipped with a …
In this paper we study constraint qualifications and optimality conditions for bilevel programming problems. We strive to derive checkable constraint qualifications in terms of problem data and applicable optimality conditions. For the bilevel program with convex lower level program we discuss drawbacks of reformulating a bilevel programming problem by the mathematical program with complementarity constraints and present a new sharp necessary optimality condition for the reformulation by the mathematical program with a generalized equation constraint. For the bilevel program with a nonconvex l…
In this paper, we focus on the nonconvex-strongly-convex bilevel optimization problem (BLO). In this BLO, the objective function of the upper-level problem is nonconvex and possibly nonsmooth, and the lower-level problem is smooth and strongly convex with respect to the underlying variable $y$. We show that the feasible region of BLO is a Riemannian manifold. Then we transform BLO to its corresponding unconstrained constraint dissolving problem (CDB), whose objective function is explicitly formulated from the objective functions in BLO. We prove that BLO is equivalent to the unconstrained opti…
A geometric reformulation of the bilevel parameter optimization problem to a single level non-linear programming problem with applications to phase equilibria
University of Bremen（不来梅大学）；Leibniz Institute for Materials Engineering - IWT（莱布尼兹材料工程研究所）
On the Complexity of Finding Stationary Points in Nonconvex Simple Bilevel Optimization
Department of Electrical and Computer Engineering, The University of Texas at Austin（电气与计算机工程系，德克萨斯大学奥斯汀分校）；Department of Systems and Industrial Engineering, The University of Arizona（系统与工业工程系，亚利桑那大学）
Bilevel Learning with Inexact Stochastic Gradients
Department of Mathematical Sciences, University of Bath（巴斯大学数学科学系）；Department of Electronics & Electrical Communication Engineering, Indian Institute of Technology (IIT) Kharagpur（印度理工学院Kharagpur电子与电气通信工程系）；School of Mathematics and Statistics, University of Sydney（悉尼大学数学与统计学学院）
Directional first order approach for a class of bilevel programs
School of Mathematics, Hunan University（湖南大学数学学院）；School of Artificial Intelligence, Southern University of Science and Technology（南方科技大学人工智能学院）；Department of Mathematics and Statistics, University of Victoria（维多利亚大学数学与统计系）；Department of Mathematics, Southern University of Science and Technology（南方科技大学数学系）
Hessian-Free Distributed Bilevel Optimization via Penalization with Time-Scale Separation
State Key Laboratory of Industrial Control Technology and the College of Control Science and Engineering, Zhejiang University（浙江大学工业控制技术国家重点实验室与控制科学与工程学院）；School of Electrical Engineering and Computer Science, Pennsylvania State University（宾夕法尼亚州立大学电气与计算机工程学院）
University of Pittsburgh（匹兹堡大学）；Nanjing University of Aeronautics & Astronautics（南京航空航天大学）
Bilevel optimization has been recently used in many machine learning problems such as hyperparameter optimization, policy optimization, and meta learning. Although many bilevel optimization methods have been proposed, they still suffer from the high computational complexities and do not consider the more general bilevel problems with nonsmooth regularization. In the paper, thus, we propose a class of enhanced bilevel optimization methods with using Bregman distance to solve bilevel optimization problems, where the outer subproblem is nonconvex and possibly nonsmooth, and the inner subproblem i…
Georgia Institute of Technology（佐治亚理工学院）；Syracuse University（雪城大学）
This paper studies how to apply differential privacy to constrained optimization problems whose inputs are sensitive. This task raises significant challenges since random perturbations of the input data often render the constrained optimization problem infeasible or change significantly the nature of its optimal solutions. To address this difficulty, this paper proposes a bilevel optimization model that can be used as a post-processing step: It redistributes the noise introduced by a differentially private mechanism optimally while restoring feasibility and near-optimality. The paper shows tha…
The optimisation of nonsmooth, nonconvex functions without access to gradients is a particularly challenging problem that is frequently encountered, for example in model parameter optimisation problems. Bilevel optimisation of parameters is a standard setting in areas such as variational regularisation problems and supervised machine learning. We present efficient and robust derivative-free methods called randomised Itoh--Abe methods. These are generalisations of the Itoh--Abe discrete gradient method, a well-known scheme from geometric integration, which has previously only been considered in…
Bilevel optimization have gained growing interests, with numerous applications found in meta learning, minimax games, reinforcement learning, and nested composition optimization. This paper studies the problem of distributed bilevel optimization over a network where agents can only communicate with neighbors, including examples from multi-task, multi-agent learning and federated learning. In this paper, we propose a gossip-based distributed bilevel learning algorithm that allows networked agents to solve both the inner and outer optimization problems in a single timescale and share information…
Nanjing University of Aeronautics and Astronautics（南京航空航天大学）
Bilevel optimization is a popular hierarchical model in machine learning, and has been widely applied to many machine learning tasks such as meta learning, hyperparameter learning and policy optimization. Although many bilevel optimization algorithms recently have been developed, few adaptive algorithm focuses on the bilevel optimization under the distributed setting. It is well known that the adaptive gradient methods show superior performances on both distributed and non-distributed optimization. In the paper, thus, we propose a novel adaptive federated bilevel optimization algorithm (i.e.,A…
Nanjing University of Aeronautics and Astronautics（南京航空航天大学）；University of Pittsburgh（匹兹堡大学）
Bilevel optimization recently has attracted increased interest in machine learning due to its many applications such as hyper-parameter optimization and meta learning. Although many bilevel methods recently have been proposed, these methods do not consider using adaptive learning rates. It is well known that adaptive learning rates can accelerate optimization algorithms. To fill this gap, in the paper, we propose a novel fast adaptive bilevel framework to solve stochastic bilevel optimization problems that the outer problem is possibly nonconvex and the inner problem is strongly convex. Our fr…
