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

Nuojin Cheng, Leonard Papenmeier, Stephen Becker, Luigi Nardi
Proceedings of the 42nd International Conference on Machine Learning, PMLR 267:10106-10120, 2025.

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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v267-cheng25g, title = {A Unified Framework for Entropy Search and Expected Improvement in {B}ayesian Optimization}, author = {Cheng, Nuojin and Papenmeier, Leonard and Becker, Stephen and Nardi, Luigi}, booktitle = {Proceedings of the 42nd International Conference on Machine Learning}, pages = {10106--10120}, year = {2025}, editor = {Singh, Aarti and Fazel, Maryam and Hsu, Daniel and Lacoste-Julien, Simon and Berkenkamp, Felix and Maharaj, Tegan and Wagstaff, Kiri and Zhu, Jerry}, volume = {267}, series = {Proceedings of Machine Learning Research}, month = {13--19 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v267/main/assets/cheng25g/cheng25g.pdf}, url = {https://proceedings.mlr.press/v267/cheng25g.html}, 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.} }
Endnote
%0 Conference Paper %T A Unified Framework for Entropy Search and Expected Improvement in Bayesian Optimization %A Nuojin Cheng %A Leonard Papenmeier %A Stephen Becker %A Luigi Nardi %B Proceedings of the 42nd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2025 %E Aarti Singh %E Maryam Fazel %E Daniel Hsu %E Simon Lacoste-Julien %E Felix Berkenkamp %E Tegan Maharaj %E Kiri Wagstaff %E Jerry Zhu %F pmlr-v267-cheng25g %I PMLR %P 10106--10120 %U https://proceedings.mlr.press/v267/cheng25g.html %V 267 %X 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.
APA
Cheng, N., Papenmeier, L., Becker, S. & Nardi, L.. (2025). A Unified Framework for Entropy Search and Expected Improvement in Bayesian Optimization. Proceedings of the 42nd International Conference on Machine Learning, in Proceedings of Machine Learning Research 267:10106-10120 Available from https://proceedings.mlr.press/v267/cheng25g.html.

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