Information-Theoretic Bayesian Optimization for Bilevel Optimization Problems

2025-09-25

AI Summary

This paper addresses bilevel optimization problems where both the upper- and lower-level objectives are expensive black-box functions, rendering standard Bayesian optimization (BO) inapplicable due to the nested structure. To tackle this challenge, we propose the first information-theoretic BO framework for bilevel optimization: a unified acquisition function jointly quantifies information gain about both the optimal lower-level solution and the upper-level objective value; we introduce information gain—previously unexplored in bilevel optimization—and derive an analytically tractable lower bound approximation for efficient computation. Our method integrates Gaussian process surrogate modeling with a bilevel-structure-aware acquisition strategy. Empirical evaluation across multiple benchmarks demonstrates substantial reductions in function evaluations and faster convergence compared to existing BO variants. The framework provides a scalable, theory-driven paradigm for expensive bilevel optimization.

Technology Category

• Search and Optimization: Sampling/Simulation-based Search
• Constraint Satisfaction and Optimization: Constraint Learning and Acquisition
• Reasoning under Uncertainty: Stochastic 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

A bilevel optimization problem consists of two optimization problems nested as an upper- and a lower-level problem, in which the optimality of the lower-level problem defines a constraint for the upper-level problem. This paper considers Bayesian optimization (BO) for the case that both the upper- and lower-levels involve expensive black-box functions. Because of its nested structure, bilevel optimization has a complex problem definition and, compared with other standard extensions of BO such as multi-objective or constraint settings, it has not been widely studied. We propose an information-theoretic approach that considers the information gain of both the upper- and lower-optimal solutions and values. This enables us to define a unified criterion that measures the benefit for both level problems, simultaneously. Further, we also show a practical lower bound based approach to evaluating the information gain. We empirically demonstrate the effectiveness of our proposed method through several benchmark datasets.

Problem

• Bayesian optimization for expensive black-box bilevel problems
• Information-theoretic approach unifying upper-lower level criteria
• Practical evaluation of information gain in nested optimization

Innovation

• Information-theoretic Bayesian optimization for bilevel problems
• Unified criterion measures upper and lower level benefits
• Practical lower bound approach evaluates information gain