Bilevel optimization minimizes an objective function, defined by an upper-level problem whose feasible region is the solution of a lower-level problem. We study the oracle complexity of finding an $ε$-stationary point with first-order methods when the upper-level problem is nonconvex, and the lower-level problem is strongly convex. Recent works achieve a $\tilde{\mathcal{O}}(\bar κ_y^{7/2} ε^{-2})$ upper bound that is near-optimal in $ε$. In this work, we establish a new $Ω(κ_y^{5/2} ε^{-2})$ lower bound, where $κ_y \le \bar κ_y$ is the lower-level condition number. Our lower bound establishes the first provable gap {in terms of condition number dependency} between bilevel problems and minimax problems in this setup, and \textit{is tight up to logarithmic factors when the lower-level function is quadratic.} Our lower bounds can be extended to various settings. (1) For second-order and arbitrarily smooth problems, we show lower bounds of $Ω(κ_y^{9/4} ε^{-7/4})$ and $Ω(κ_y^{13/6} ε^{-5/3})$, respectively. (2) For convex--strongly-convex problems, we improve the previously best lower bound (Ji and Liang, JMLR 2022) from $Ω(κ_y /\sqrtε)$ to $Ω(κ_y^{3/2} / \sqrtε)$. (3) For stochastic nonconvex--strongly-convex problems, we also show the lower bounds of $Ω(κ_y^4 ε^{-4})$ and $Ω(κ_y^{9/2} ε^{-4})$ for stochastic Hessian-vector-product and stochastic first-order methods, respectively.
Tongji University（同济大学）；University of Connecticut（康涅狄格大学）
Bilevel optimization plays an essential role in many machine learning tasks, ranging from hyperparameter optimization to meta-learning. Existing studies on bilevel optimization, however, focus on either centralized or synchronous distributed setting. The centralized bilevel optimization approaches require collecting massive amount of data to a single server, which inevitably incur significant communication expenses and may give rise to data privacy risks. Synchronous distributed bilevel optimization algorithms, on the other hand, often face the straggler problem and will immediately stop worki…
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…
Bilevel Optimization Programming is used to model complex and conflicting interactions between agents, for example in Robust AI or Privacy-preserving AI. Integrating bilevel mathematical programming within deep learning is thus an essential objective for the Machine Learning community. Previously proposed approaches only consider single-level programming. In this paper, we extend existing single-level optimization programming approaches and thus propose Differentiating through Bilevel Optimization Programming (BiGrad) for end-to-end learning of models that use Bilevel Programming as a layer. B…
Wisconsin Institute for Discovery（威斯康星发现研究所）；UW-Madison（威斯康星大学麦迪逊分校）
We consider stochastic unconstrained bilevel optimization problems when only the first-order gradient oracles are available. While numerous optimization methods have been proposed for tackling bilevel problems, existing methods either tend to require possibly expensive calculations regarding Hessians of lower-level objectives, or lack rigorous finite-time performance guarantees. In this work, we propose a Fully First-order Stochastic Approximation (F2SA) method, and study its non-asymptotic convergence properties. Specifically, we show that F2SA converges to an $ε$-stationary solution of the b…
Bilevel optimization has become a powerful framework in various machine learning applications including meta-learning, hyperparameter optimization, and network architecture search. There are generally two classes of bilevel optimization formulations for machine learning: 1) problem-based bilevel optimization, whose inner-level problem is formulated as finding a minimizer of a given loss function; and 2) algorithm-based bilevel optimization, whose inner-level solution is an output of a fixed algorithm. For the first class, two popular types of gradient-based algorithms have been proposed for hy…
Rensselaer Polytechnic Institute（伦斯勒理工学院）；University of California, Los Angeles（加利福尼亚大学洛杉矶分校）
Stochastic bilevel optimization generalizes the classic stochastic optimization from the minimization of a single objective to the minimization of an objective function that depends the solution of another optimization problem. Recently, stochastic bilevel optimization is regaining popularity in emerging machine learning applications such as hyper-parameter optimization and model-agnostic meta learning. To solve this class of stochastic optimization problems, existing methods require either double-loop or two-timescale updates, which are sometimes less efficient. This paper develops a new opti…
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…
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…
Bilevel Optimization has witnessed notable progress recently with new emerging efficient algorithms and has been applied to many machine learning tasks such as data cleaning, few-shot learning, and neural architecture search. However, little attention has been paid to solve the bilevel problems under distributed setting. Federated learning (FL) is an emerging paradigm which solves machine learning tasks over distributed-located data. FL problems are challenging to solve due to the heterogeneity and communication bottleneck. However, it is unclear how these challenges will affect the convergenc…
The Ohio State University（俄亥俄州立大学）；University of Michigan, Ann Arbor（密歇根大学安娜堡分校）
Bilevel optimization has found extensive applications in modern machine learning problems such as hyperparameter optimization, neural architecture search, meta-learning, etc. While bilevel problems with a unique inner minimal point (e.g., where the inner function is strongly convex) are well understood, such a problem with multiple inner minimal points remains to be challenging and open. Existing algorithms designed for such a problem were applicable to restricted situations and do not come with a full guarantee of convergence. In this paper, we adopt a reformulation of bilevel optimization to…
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…
We study a class of algorithms for solving bilevel optimization problems in both stochastic and deterministic settings when the inner-level objective is strongly convex. Specifically, we consider algorithms based on inexact implicit differentiation and we exploit a warm-start strategy to amortize the estimation of the exact gradient. We then introduce a unified theoretical framework inspired by the study of singularly perturbed systems (Habets, 1974) to analyze such amortized algorithms. By using this framework, our analysis shows these algorithms to match the computational complexity of oracl…
Bilevel optimization has arisen as a powerful tool for many machine learning problems such as meta-learning, hyperparameter optimization, and reinforcement learning. In this paper, we investigate the nonconvex-strongly-convex bilevel optimization problem. For deterministic bilevel optimization, we provide a comprehensive convergence rate analysis for two popular algorithms respectively based on approximate implicit differentiation (AID) and iterative differentiation (ITD). For the AID-based method, we orderwisely improve the previous convergence rate analysis due to a more practical parameter …
This chapter presents the Bilevel Optimization LIBrary of the test problems (BOLIB for short), which contains a collection of test problems, with continuous variables, to help support the development of numerical solvers for bilevel optimization. The library contains 173 examples with 138 nonlinear, 24 linear, and 11 simple bilevel optimization problems. This BOLIB collection is probably the largest bilevel optimization library of test problems. Moreover, as the library is computationenabled with the MATLAB m-files of all the examples, it provides a uniform basis for testing and comparing algo…
This chapter presents a self-contained approach of variational analysis and generalized differentiation to deriving necessary optimality in problems of bilevel optimization with Lipschitzian data. We mainly concentrate on optimistic models, although the developed machinery also applies to pessimistic versions. Some open problems are posed and discussed.
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…
The Ohio State University（俄亥俄州立大学）；University of Michigan（密歇根大学）
Bilevel optimization has been widely applied in many important machine learning applications such as hyperparameter optimization and meta-learning. Recently, several momentum-based algorithms have been proposed to solve bilevel optimization problems faster. However, those momentum-based algorithms do not achieve provably better computational complexity than $\mathcal{\widetilde O}(ε^{-2})$ of the SGD-based algorithm. In this paper, we propose two new algorithms for bilevel optimization, where the first algorithm adopts momentum-based recursive iterations, and the second algorithm adopts recurs…
University of Michigan（密歇根大学）；University of California, Riverside（加州大学河滨分校）；University of British Columbia（不列颠哥伦比亚大学）
Standard federated optimization methods successfully apply to stochastic problems with single-level structure. However, many contemporary ML problems -- including adversarial robustness, hyperparameter tuning, and actor-critic -- fall under nested bilevel programming that subsumes minimax and compositional optimization. In this work, we propose \fedblo: A federated alternating stochastic gradient method to address general nested problems. We establish provable convergence rates for \fedblo in the presence of heterogeneous data and introduce variations for bilevel, minimax, and compositional op…
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…
Stochastic bilevel optimization, which captures the inherent nested structure of machine learning problems, is gaining popularity in many recent applications. Existing works on bilevel optimization mostly consider either unconstrained problems or constrained upper-level problems. This paper considers the stochastic bilevel optimization problems with equality constraints both in the upper and lower levels. By leveraging the special structure of the equality constraints problem, the paper first presents an alternating implicit projected SGD approach and establishes the $\tilde{\cal O}(ε^{-2})$ s…
The Ohio State University（俄亥俄州立大学）；Wayne State University（韦恩州立大学）；Air Force Research Laboratory（美国空军研究实验室）
Decentralized bilevel optimization has received increasing attention recently due to its foundational role in many emerging multi-agent learning paradigms (e.g., multi-agent meta-learning and multi-agent reinforcement learning) over peer-to-peer edge networks. However, to work with the limited computation and communication capabilities of edge networks, a major challenge in developing decentralized bilevel optimization techniques is to lower sample and communication complexities. This motivates us to develop a new decentralized bilevel optimization called DIAMOND (decentralized single-timescal…
Solving a bilevel optimization problem is at the core of several machine learning problems such as hyperparameter tuning, data denoising, meta- and few-shot learning, and training-data poisoning. Different from simultaneous or multi-objective optimization, the steepest descent direction for minimizing the upper-level cost in a bilevel problem requires the inverse of the Hessian of the lower-level cost. In this work, we propose a novel algorithm for solving bilevel optimization problems based on the classical penalty function approach. Our method avoids computing the Hessian inverse and can han…
In this paper we present BilevelJuMP, a new Julia package to support bilevel optimization within the JuMP framework. The package is a Julia library that enables the user to describe both upper and lower-level optimization problems using the JuMP algebraic syntax. Due to the generality and flexibility our library inherits from JuMP's syntax, our package allows users to model bilevel optimization problems with conic constraints in the lower level and all JuMP supported constraints in the upper level (Conic, Quadratic, Non-Linear, Integer, etc.). Moreover, the user-defined problem can be subseque…
We consider the tropical analogues of a particular bilevel optimization problem studied by Dempe and Franke and suggest some methods of solving these new tropical bilevel optimization problems. In particular, it is found that the algorithm developed by Dempe and Franke can be formulated and its validity can be proved in a more general setting, which includes the tropical bilevel optimization problems in question. We also show how the feasible set can be decomposed into a finite number of tropical polyhedra, to which the tropical linear programming solvers can be applied.
A Provably Convergent Plug-and-Play Framework for Stochastic Bilevel Optimization
Institute of Operations Research and Information Engineering, Beijing University of Technology（北京理工大学运筹学与信息工程学院）；National Center for Applied Mathematics Shenzhen（深圳应用数学中心）；Department of Mathematics, Southern University of Science and Technology（南方科技大学数学系）；School of Science, Beijing University of Posts and Telecommunications（北京邮电大学理学院）；Department of Mathematics, and National Center for Applied Mathematics Shenzhen, Southern University of Science and Technology（南方科技大学数学系和深圳应用数学中心）
We investigate the complexity of bilevel combinatorial optimization with uncertainty in the follower's objective, in a robust optimization approach. We show that the robust counterpart of the bilevel problem under interval uncertainty can be $Σ^{\text P}_2$-hard, even when the certain bilevel problem is NP-equivalent and the follower's problem is tractable. On the contrary, in the discrete uncertainty case, the robust bilevel problem is at most one level harder than the follower's problem.
This paper proposes a Real-Time Market (RTM) platform for an aggregator and its corresponding prosumers to participate in the electricity wholesale market. The proposed energy market platform is modeled as a bilevel optimization problem where the aggregator and the prosumers are considered as self-interest agents. The current state-of-the-art Mathematical Programming with Equilibrium Constraints (MPEC) and Mixed-Integer Programming (MIP) based approaches to solve bilevel optimization problems are not satisfactory for real-time applications. The computation time for such approaches grows expone…
This article studies Gauss-Newton-type methods for over-determined systems to find solutions to bilevel programming problems. To proceed, we use the lower-level value function reformulation of bilevel programs and consider necessary optimality conditions under appropriate assumptions. First under strict complementarity for upper- and lower-level feasibility constraints, we prove the convergence of a Gauss-Newton-type method in computing points satisfying these optimality conditions under additional tractable qualification conditions. Potential approaches to address the shortcomings of the meth…
Bilevel optimization is a powerful tool for modeling hierarchical decision making processes. However, the resulting problems are challenging to solve - both in theory and practice. Fortunately, there have been significant algorithmic advances in the field so that we can solve much larger and also more complicated problems today compared to what was possible to solve two decades ago. This results in more and more challenging bilevel problems that researchers try to solve today. In this article, we give a brief introduction to one of these more challenging classes of bilevel problems: bilevel op…
Massachusetts Institute of Technology（麻省理工学院）；IBM Thomas J. Watson Research Center（IBM托马斯·J·沃森研究中心）；University of California San Diego（加州大学圣地亚哥分校）
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 descent-ascent bilevel optimization algorithm, to solve the generic problem and present a novel analysis showing that MORBiT converges to the first-order stationary point at a rate of $\widetilde{\mathcal{O}}(n^{1/2} K^{-2/5})$ for a class of weakly convex problems with $n$ objectives upon $K$ iterations of the algorithm. Our analysis utilizes novel results to handl…
Shandong University（山东大学）；Renmin University of China（中国人民大学）；Tsinghua University（清华大学）；Huawei（华为）
The (gradient-based) bilevel programming framework is widely used in hyperparameter optimization and has achieved excellent performance empirically. Previous theoretical work mainly focuses on its optimization properties, while leaving the analysis on generalization largely open. This paper attempts to address the issue by presenting an expectation bound w.r.t. the validation set based on uniform stability. Our results can explain some mysterious behaviours of the bilevel programming in practice, for instance, overfitting to the validation set. We also present an expectation bound for the clas…
This paper studies bilevel polynomial optimization problems. To solve them, we give a method based on polynomial optimization relaxations. Each relaxation is obtained from the Kurash-Kuhn-Tucker (KKT) conditions for the lower level optimization and the exchange technique for semi-infinite programming. For KKT conditions, Lagrange multipliers are represented as polynomial or rational functions. The Moment-SOS relaxations are used to solve the polynomial optimizattion relaxations. Under some general assumptions, we prove the convergence of the algorithm for solving bilevel polynomial optimizatio…
Bilevel optimization is defined as a mathematical program, where an optimization problem contains another optimization problem as a constraint. These problems have received significant attention from the mathematical programming community. Only limited work exists on bilevel problems using evolutionary computation techniques; however, recently there has been an increasing interest due to the proliferation of practical applications and the potential of evolutionary algorithms in tackling these problems. This paper provides a comprehensive review on bilevel optimization from the basic principles…
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…
A large number of application problems involve two levels of optimization, where one optimization task is nested inside the other. These problems are known as bilevel optimization problems and have been studied by both classical optimization community and evolutionary optimization community. Most of the solution procedures proposed until now are either computationally very expensive or applicable to only small classes of bilevel optimization problems adhering to mathematically simplifying assumptions. In this paper, we propose an evolutionary optimization method that tries to reduce the comput…
