Constrained Bi-Level Optimization: Proximal Lagrangian Value Function Approach and Hessian-free Algorithm

Wei Yao, Chengming Yu, Shangzhi Zeng, Jin Zhang

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Keywords: Bi-level Optimization, Constrained Optimization, Hessian-free, Single-loop, Value Function, Convergence Analysis

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TL;DR: This paper presents a new approach and algorithm for solving a class of constrained Bi-Level Optimization problems in which the lower-level problem involves constraints coupling both upper-level and lower-level variables.

Abstract: This paper presents a new approach and algorithm for solving a class of constrained Bi-Level Optimization (BLO) problems in which the lower-level problem involves constraints coupling both upper-level and lower-level variables. Such problems have recently gained significant attention due to their broad applicability in machine learning. However, conventional gradient-based methods unavoidably rely on computationally intensive calculations related to the Hessian matrix. To address this challenge, we devise a smooth proximal Lagrangian value function to handle the constrained lower-level problem. Utilizing this construct, we introduce a single-level reformulation for constrained BLOs that transforms the original BLO problem into an equivalent optimization problem with smooth constraints. Enabled by this reformulation, we develop a Hessian-free gradient-based algorithm—termed proximal Lagrangian Value function-based Hessian-free Bi-level Algorithm (LV-HBA)—that is straightforward to implement in a single loop manner. Consequently, LV-HBA is especially well-suited for machine learning applications. Furthermore, we offer non-asymptotic convergence analysis for LV-HBA, eliminating the need for traditional strong convexity assumptions for the lower-level problem while also being capable of accommodating non-singleton scenarios. Empirical results substantiate the algorithm's superior practical performance.

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Primary Area: optimization

Submission Number: 3552