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

TL;DR

This paper introduces a unified theoretical framework linking Expected Improvement and information-theoretic acquisition functions in Bayesian optimization, leading to a new method that outperforms existing approaches.

Contribution

It reveals the close relationship between EI and information-theoretic functions, proposing a new acquisition function VES-Gamma that combines their strengths.

Findings

VES-Gamma outperforms EI and MES on various benchmarks.

EI can be viewed as a variational approximation of MES.

The unified framework bridges two major classes of acquisition functions.

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.