BILBO: BILevel Bayesian Optimization

TL;DR

BILBO addresses the challenge of optimizing both upper- and lower-level objectives in bilevel problems without repeated lower-level solves or gradient information. By sampling from confidence-bound trusted sets and employing a conditional reassignment strategy, it bounds lower-level suboptimality and promotes targeted exploration of the lower-level landscape, achieving sublinear regret under common kernels. The method demonstrates strong empirical performance on synthetic multimodal benchmarks and real-world problems (energy and chemical processes), outperforming nested and baseline trusted-set approaches. This work offers a scalable, derivative-free, and theoretically grounded framework for general bilevel optimization in noisy, constrained settings, with practical impact for complex hierarchical decision problems.

Abstract

Bilevel optimization is characterized by a two-level optimization structure, where the upper-level problem is constrained by optimal lower-level solutions, and such structures are prevalent in real-world problems. The constraint by optimal lower-level solutions poses significant challenges, especially in noisy, constrained, and derivative-free settings, as repeating lower-level optimizations is sample inefficient and predicted lower-level solutions may be suboptimal. We present BILevel Bayesian Optimization (BILBO), a novel Bayesian optimization algorithm for general bilevel problems with blackbox functions, which optimizes both upper- and lower-level problems simultaneously, without the repeated lower-level optimization required by existing methods. BILBO samples from confidence-bounds based trusted sets, which bounds the suboptimality on the lower level. Moreover, BILBO selects only one function query per iteration, where the function query selection strategy incorporates the uncertainty of estimated lower-level solutions and includes a conditional reassignment of the query to encourage exploration of the lower-level objective. The performance of BILBO is theoretically guaranteed with a sublinear regret bound for commonly used kernels and is empirically evaluated on several synthetic and real-world problems.

Figures (7)

• Figure 1: Example of bilevel optimization with upper-level variable $\mathbf{x}$ and lower-level variable $\mathbf{z}$. The bilevel solution (blue cross) is constrained by lower-level (LL) solutions (yellow line) and differs from the non-bilevel solution (red cross).
• Figure 2: Key components of BILBO, where pink shaded areas represent trusted sets $\mathcal{S}^+_t \cap \mathcal{P}^+_t$. (a) Query point selection within trusted sets following \ref{['eqn:query']}. (b) Conditional reassignment following \ref{['eq:new_zt']}, where lower-level query $\mathbf{z}_t$ may be reassigned to estimated lower-level solution $\bar{\mathbf{z}}_t(\mathbf{x}_t)$ if lower-level objective is selected for query.
• Figure 3: BraninHoo+GoldsteinPrice experiment details. LL refers to lower-level. (a) Upper-level objective, Branin-Hoo. (b) Lower-level objective, Goldstein-Price. (c) BILBO's upper-level estimate. (d) BILBO's lower-level estimate. (e-f) BILBO's iterative queries.
• Figure 4: Instantaneous regrets (log-scale) over number of queries, averaged over 5 runs, for synthetic experiments.
• Figure 5: Real-world experiments. (a-b) Functions from energy experiment and (c-d) BILBO outputs, with optimal solution (red dot) and predicted solution (blue cross). (e-f) Regret plots in log-scale for energy and chemical experiments respectively.

Theorems & Definitions (23)

• Definition 4.1: Confidence bounds
• Corollary 4.2
• Definition 4.3: Trusted set of feasible solutions
• Lemma 4.4
• Definition 4.5: Trusted set of optimal lower-level solutions
• Lemma 4.6
• Definition 4.7: Function query
• Lemma 4.8
• Theorem 4.9
• Lemma 4.10