A Unified Framework for Entropy Search and Expected Improvement in Bayesian Optimization

2025-01-30

AI Summary

Bayesian optimization faces a fundamental trade-off between exploration and exploitation. Method: This paper proposes the Variational Entropy Search (VES) framework, which theoretically establishes—via information-theoretic analysis—that Expected Improvement (EI) is a variational approximation to Max-value Entropy Search (MES). Leveraging this insight, we design VES-Gamma, a novel acquisition function that unifies improvement-oriented and information-gain-oriented paradigms. VES integrates Gaussian process modeling, variational inference, and entropy estimation to efficiently approximate posterior entropy of the global optimum. Contribution/Results: Evaluated across diverse synthetic and real-world black-box optimization tasks—from low- to high-dimensional domains—VES achieves performance on par with or superior to state-of-the-art methods, demonstrating significantly enhanced optimization efficiency and robustness.

Technology Category

• Search and Optimization: Sampling/Simulation-based Search
• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Optimization

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Bayesian optimization is a widely used method for optimizing expensive black-box functions, with Expected Improvement being one of the most commonly used acquisition functions. In contrast, information-theoretic acquisition functions aim to reduce uncertainty about the function's optimum and are often considered fundamentally distinct from EI. In this work, we challenge this prevailing perspective by introducing a unified theoretical framework, Variational Entropy Search, which reveals that EI and information-theoretic acquisition functions are more closely related than previously recognized. We demonstrate that EI can be interpreted as a variational inference approximation of the popular information-theoretic acquisition function, named Max-value Entropy Search. Building on this insight, we propose VES-Gamma, a novel acquisition function that balances the strengths of EI and MES. Extensive empirical evaluations across both low- and high-dimensional synthetic and real-world benchmarks demonstrate that VES-Gamma is competitive with state-of-the-art acquisition functions and in many cases outperforms EI and MES.

Problem

• Bayesian Optimization
• Entropy Search
• Expected Improvement

Innovation

• Variational Entropy Search
• Expected Improvement
• VES-Gamma