Overcoming Lower-Level Constraints in Bilevel Optimization: A Novel Approach with Regularized Gap Functions
Wei Yao Haian Yin Shangzhi Zeng Jin Zhang
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.
bilevel optimization, constrained optimization, gap function, single-loop, Hessian-free, convergence analysis
We introduce a novel single-loop, Hessian-free constrained bilevel algorithm capable of handling broader lower-level constraints by using a doubly regularized gap function.