No-regret optimization of time-varying bilevel problems

TL;DR

This paper introduces W-SparQ-BL, a Bayesian optimization method for time-varying bilevel problems with noisy observations, achieving sublinear regret and demonstrating effectiveness in game-theoretic scenarios.

Contribution

It develops a novel Gaussian process-based framework for dynamic bilevel optimization under uncertainty, with theoretical regret guarantees and practical validation.

Findings

Achieves sublinear dynamic regret in time-varying settings.

Effectively models lower-level responses with Gaussian processes.

Demonstrates success on game-theoretic problems.

Abstract

Bilevel optimization problems arise in many applications where decisions must account for the optimal response of another system, such as in game-theoretic settings. However, these problems are notoriously challenging, as even linear bilevel programs are strongly NP-hard. In this work, we consider bilevel optimization with a known upper-level objective and an unknown, potentially time-varying lower-level response, accessible only through noisy zeroth-order observations. We propose W-SparQ-BL, a Bayesian optimization framework that models the lower-level mapping using multi-output Gaussian processes and enables efficient optimization under uncertainty. Our approach leverages a sparse, observation-based approximation to control the effect of noise and temporal variability, while requiring only limited access to additional information over time. We establish regularity results linking the lower-level response to standard RKHS assumptions for common kernels, including Mat\'ern and squared exponential. We prove that W-SparQ-BL achieves sublinear dynamic regret in both stationary and time-varying settings. Experiments on representative time-varying game-theoretic problems demonstrate the effectiveness of our approach.