First-order penalty methods for bilevel optimization

TL;DR

This paper introduces first-order penalty methods for bilevel optimization problems with nonsmooth lower levels and nonconvex upper levels, providing complexity guarantees and demonstrating the approach as a novel minimax reformulation.

Contribution

It is the first to show bilevel problems can be approximately solved via minimax reformulation and offers the first implementable method with complexity bounds.

Findings

Proposed penalty methods achieve $O( ext{operation complexity})$ for $oldsymbol{ ext{ε-KKT}}$ solutions.

Numerical results demonstrate the effectiveness of the methods.

Bilevel optimization can be approximately solved as a structured minimax problem.

Abstract

In this paper we study a class of unconstrained and constrained bilevel optimization problems in which the lower level is a possibly nonsmooth convex optimization problem, while the upper level is a possibly nonconvex optimization problem. We introduce a notion of $\varepsilon$-KKT solution for them and show that an $\varepsilon$-KKT solution leads to an $O(\sqrt{\varepsilon})$- or $O(\varepsilon)$-hypergradient based stionary point under suitable assumptions. We also propose first-order penalty methods for finding an $\varepsilon$-KKT solution of them, whose subproblems turn out to be a structured minimax problem and can be suitably solved by a first-order method recently developed by the authors. Under suitable assumptions, an \emph{operation complexity} of $O(\varepsilon^{-4}\log\varepsilon^{-1})$ and $O(\varepsilon^{-7}\log\varepsilon^{-1})$, measured by their fundamental operations, is established for the proposed penalty methods for finding an $\varepsilon$-KKT solution of the unconstrained and constrained bilevel optimization problems, respectively. Preliminary numerical results are presented to illustrate the performance of our proposed methods. To the best of our knowledge, this paper is the first work to demonstrate that bilevel optimization can be approximately solved as minimax optimization, and moreover, it provides the first implementable method with complexity guarantees for such sophisticated bilevel optimization.