BILBO: BILevel Bayesian Optimization

2025-02-04

AI Summary

In black-box bilevel optimization, repeated lower-level optimization is computationally inefficient, and solutions become unreliable under observational noise. Method: This paper introduces the first joint Bayesian optimization framework, departing from conventional nested optimization. It establishes a cooperative modeling mechanism between upper- and lower-level problems: leveraging Gaussian process regression and a variant of the upper confidence bound (UCB), it explicitly models lower-level solution uncertainty and incorporates a conditional reweighting strategy, enabling a single-query confidence-bound credible set that simultaneously enforces suboptimality constraints and enhances global exploration. Contribution/Results: We prove a sublinear regret bound. Experiments on synthetic and real-world tasks demonstrate consistent superiority over state-of-the-art methods, achieving up to a 3.2× improvement in query efficiency.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Learning to Search
• Constraint Satisfaction and Optimization: Constraint Optimization

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Uses of LLMs and GenAI for marketplace design, bidding, and strategic interactions
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

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.

Problem

• Optimizes bilevel problems efficiently
• Reduces repeated lower-level optimizations
• Handles noisy and derivative-free settings

Innovation

• Bayesian optimization algorithm
• simultaneous upper-lower optimization
• confidence-bounds trusted sets