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

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.

Imagine trying to find the best recipe for a cake by baking as few cakes as possible, since each one takes a lot of time and ingredients. This is similar to a problem in optimization where we want to find the best settings for a complex system without too many expensive experiments. Researchers have developed different strategies for this, but two popular approaches were thought to be fundamentally distinct, like choosing the next recipe based on either what is most likely to be the best (Expected Improvement) or what will teach us the most about all possible good recipes (Max-value Entropy Search).Our research shows that these two strategies are surprisingly connected. We developed a new framework that reveals the "most likely to be best" strategy is actually a simplified version of the "learn the most" strategy. This new understanding allowed us to create a hybrid strategy, called VES-Gamma, that combines the advantages of both.This matters because our new hybrid approach outperforms the individual strategies in many scenarios, from complex simulations to real-world problems. This enables scientists and engineers to find better solutions to their problems faster and with fewer costly experiments, accelerating discovery and innovation.