Estimating hyperparameters has been a long-standing problem in machine learning. We consider the case where the task at hand is modeled as the solution to an optimization problem. Here the exact gradient with respect to the hyperparameters cannot be feasibly computed and approximate strategies are required. We introduce a unified framework for computing hypergradients that generalizes existing methods based on the implicit function theorem and automatic differentiation/backpropagation, showing that these two seemingly disparate approaches are actually tightly connected. Our framework is extremely flexible, allowing its subproblems to be solved with any suitable method, to any degree of accuracy. We derive a priori and computable a posteriori error bounds for all our methods, and numerically show that our a posteriori bounds are usually more accurate. Our numerical results also show that, surprisingly, for efficient bilevel optimization, the choice of hypergradient algorithm is at least as important as the choice of lower-level solver.
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）
An adaptively inexact first-order method for bilevel optimization with application to hyperparameter learning
EPSRC Centre for Doctoral Training in Statistical Applied Mathematics at Bath (SAMBa)（EPSRC统计应用数学博士培训中心（SAMBa））；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）
Alternating Gradient-Type Algorithm for Bilevel Optimization with Inexact Lower-Level Solutions via Moreau Envelope-based Reformulation
Southern University of Science and Technology（南方科技大学）；National Center for Applied Mathematics Shenzhen（深圳应用数学国家研究中心）；City University of Hong Kong（香港城市大学）
Bilevel Learning via Strategically Inexact Stochastic Gradient Descent
EPSRC Centre for Doctoral Training in Statistical Applied Mathematics at Bath（巴斯大学统计应用数学博士培训中心）；Australian Research Council（澳大利亚研究理事会）；Indian Institute of Technology (IIT), Kharagpur（印度理工学院（Kharagpur分校））；University of Melbourne（墨尔本大学）；Department of Electronics & Electrical Communication Engineering（电子与电气通信工程系）；School of Mathematics and Statistics（数学与统计学学院）
Derivative-Free Bilevel Optimization with Inexact Lower-Level Solutions
Dipartimento di Ingegneria Informatica, Automatica e Gestionale “A. Ruberti”, “Sapienza” Università di Roma（罗马大学信息与自动化与管理工程学院）；Istituto di Analisi dei Sistemi ed Informatica “A. Ruberti”, Consiglio Nazionale delle Ricerche（意大利国家研究委员会系统与信息技术分析研究所）
Value Function Based Difference-of-Convex Algorithm for Bilevel Hyperparameter Selection Problems
University of Waterloo（滑铁卢大学）；University of Victoria（维多利亚大学）；Southern University of Science and Technology（南方科技大学）；National Center for Applied Mathematics Shenzhen（深圳国家应用数学中心）
DUET: Decentralized Bilevel Optimization without Lower-Level Strong Convexity
Department of Electrical and Computer Engineering, The Ohio State University（俄亥俄州立大学电气与计算机工程系）；Department of Computer Science and Engineering, University of North Texas（北卡罗来纳州立大学计算机科学与工程系）；Department of Computer Science and Engineering, The Chinese University of Hong Kong（香港中文大学计算机科学与工程系）
An Adaptive Algorithm for Bilevel Optimization on Riemannian Manifolds
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（复旦大学现代应用数学上海市重点实验室）
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）
Stochastic Regret Guarantees for Online Zeroth- and First-Order Bilevel Optimization
Amirkabir University of Technology（阿姆斯泰尔大学）；University of Pennsylvania（宾夕法尼亚大学）；Samsung SDS Research America（三星SDS美国研究部）；University of California, Los Angeles（加州大学洛杉矶分校）
Moreau Envelope Based Difference-of-weakly-Convex Reformulation and Algorithm for Bilevel Programs
University of British Columbia（不列颠哥伦比亚大学）；University of Victoria（维多利亚大学）；Southern University of Science and Technology（南方科技大学）；National Center for Applied Mathematics Shenzhen（深圳国家应用数学中心）；Peng Cheng Laboratory（鹏城实验室）
Bilevel Optimization over Saddle Points of Zero-Sum Markov Games
Shun Hing Institute of Advanced Engineering, The Chinese University of Hong Kong（香港中文大学先进工程学院）；Department of Computer Science and Engineering, The Chinese University of Hong Kong（香港中文大学计算机科学与工程系）
An Augmented Lagrangian Value Function Method for Lower-level Constrained Stochastic Bilevel Optimization
School of Mathematical Science, Peking University（北京大学数学科学学院）；Department of Electrical and Computer Engineering, University of Minnesota（明尼苏达大学电气与计算机工程系）
