Functionally Constrained Algorithm Solves Convex Simple Bilevel Problem

Huaqing Zhang, Lesi Chen, Jing Xu, Jingzhao Zhang

Keywords: bilevel optimization, first-order methods, optimal complexity

Abstract:

This paper studies simple bilevel problems, where a convex upper-level function is minimized over the optimal solutions of a convex lower-level problem. We first show the fundamental difficulty of simple bilevel problems, that the approximate optimal value of such problems is not obtainable by first-order zero-respecting algorithms. Then we follow recent works to pursue the weak approximate solutions. For this goal, we propose a novel method by reformulating them into functionally constrained problems. Our method achieves near-optimal rates for both
smooth and nonsmooth problems. To the best of our knowledge, this is the first near-optimal algorithm that works under standard assumptions of smoothness or Lipschitz continuity for the objective functions.

Primary Area: Optimization (convex and non-convex, discrete, stochastic, robust)

Submission Number: 8160