Overcoming Lower-Level Constraints in Bilevel Optimization: A Novel Approach with Regularized Gap Functions

TL;DR

This paper introduces a novel single-loop, Hessian-free constrained bilevel optimization algorithm that handles general lower-level constraints using a regularized gap function, with proven convergence and broad applicability.

Contribution

The work develops a new algorithm for constrained bilevel optimization that overcomes limitations of existing methods, applicable to more general constraints and without strong convexity assumptions.

Findings

Effective on synthetic problems, hyperparameter learning, and GANs.

Converges non-asymptotically under convexity assumptions.

Handles more general lower-level constraints than previous methods.

Abstract

Constrained bilevel optimization tackles nested structures present in constrained learning tasks like constrained meta-learning, adversarial learning, and distributed bilevel optimization. However, existing bilevel optimization methods mostly are typically restricted to specific constraint settings, such as linear lower-level constraints. In this work, we overcome this limitation and develop a new single-loop, Hessian-free constrained bilevel algorithm capable of handling more general lower-level constraints. We achieve this by employing a doubly regularized gap function tailored to the constrained lower-level problem, transforming constrained bilevel optimization into an equivalent single-level optimization problem with a single smooth constraint. We rigorously establish the non-asymptotic convergence analysis of the proposed algorithm under the convexity of lower-level problem, avoiding the need for strong convexity assumptions on the lower-level objective or coupling convexity assumptions on lower-level constraints found in existing literature. Additionally, the generality of our method allows for its extension to bilevel optimization with minimax lower-level problem. We evaluate the effectiveness and efficiency of our algorithm on various synthetic problems, typical hyperparameter learning tasks, and generative adversarial network.