Accelerating Black-Box Bilevel Optimization with Rank-Based Upper-Level Value Function Approximation

TL;DR

This paper introduces a rank-based approximation framework for black-box bilevel optimization that significantly reduces computational costs and improves performance on complex, multimodal problems.

Contribution

It proposes a novel method that directly approximates upper-level value function rankings, bypassing costly lower-level optimizations, especially effective for challenging landscapes.

Findings

Achieves competitive results on standard benchmarks.

Solves problems intractable for previous methods.

Reduces computational burden of bilevel optimization.

Abstract

Bilevel optimization is a field of significant theoretical and practical interest, yet solving such optimization problems remains challenging. Evolutionary methods have been employed to address these problems in the black-box setting; however, they incur high computational cost due to the nested nature of bilevel optimization. Although previous methods have attempted to reduce this cost through various heuristic techniques, such approaches limit versatility on challenging optimization landscapes, such as those with multimodality and significant interaction between upper- and lower-level decision variables. In this study, we propose an efficient framework that exploits the invariance of rank-based evolutionary algorithms to monotonic transformations, thereby reducing the computational burden of the lower-level optimization loop. Specifically, our method directly approximates the rankings of the upper-level value function, bypassing the need to run the lower-level optimizer until convergence for each upper-level iteration. We apply this framework to the setting where both levels are continuous, adopting CMA-ES as the optimizer. We demonstrate that our method achieves competitive performance on standard bilevel optimization benchmarks and can solve problems that are intractable with previously proposed methods, particularly those with multimodality and strong inter-variable interactions.