The Acquisition Function

  • Francesco ArchettiEmail author
  • Antonio Candelieri
Part of the SpringerBriefs in Optimization book series (BRIEFSOPTI)


The acquisition function is the mechanism to implement the trade-off between exploration and exploitation in BO. More precisely, any acquisition function aims to guide the search of the optimum towards points with potential low values of objective function either because the prediction of \(f\left( x \right)\), based on the probabilistic surrogate model, is low or the uncertainty, also based on the same model, is high (or both).


  1. Astudillo, R., Frazier, P.: Bayesian optimization of composite functions. In: International Conference on Machine Learning, pp. 354–363 (2019, May)Google Scholar
  2. Auer, P.: Using confidence bounds for exploitation-exploration trade-offs. J. Mach. Learn. Res. 3(3), 397–422 (2002)MathSciNetzbMATHGoogle Scholar
  3. Basu, K., Ghosh, S.: Analysis of Thompson sampling for Gaussian process optimization in the bandit setting (2017). arXiv preprint arXiv:1705.06808
  4. Berger, J.O.: Statistical Decision Theory and Bayesian Analysis. Springer Science & Business Media (2013)Google Scholar
  5. Brochu, E., Cora, V.M., de Freitas, N.: A tutorial on bayesian optimization of expensive cost functions, with application to active user modeling and hierarchical reinforcement learning (2010). arXiv preprint arXiv:1012.2599
  6. Frazier, P., Powell, W., Dayanik, S.: The knowledge-gradient policy for correlated normal beliefs. Informs J. Comput. 21(4), 599–613 (2009)MathSciNetCrossRefGoogle Scholar
  7. Glasserman, P.: Performance continuity and differentiability in Monte Carlo optimization. In: 1988 Winter Simulation Conference Proceedings, pp. 518–524. IEEE (1988)Google Scholar
  8. González, J., Osborne, M., Lawrence, N.D.: GLASSES: Relieving the Myopia of Bayesian Optimisation (2016)Google Scholar
  9. Hennig, P., Schuler, C.J.: Entropy search for information-efficient global optimization. J. Mach. Learn. Res. 13, 1809–1837 (2012)Google Scholar
  10. Hernández-Lobato, J.M., Hoffman, M.W., Ghahramani, Z.: Predictive entropy search for efficient global optimization of black-box functions. Adv. Neural. Inf. Process. Syst. 25, 144–149 (2014). Scholar
  11. Jones, D.R., Schonlau, M., Welch, W.J.: Efficient global optimization of expensive black-box functions. J. Glob. Optim. 13(4), 455–492 (1998)Google Scholar
  12. Kingma, D.P., Welling, M.: Auto-encoding variational bayes (2013). arXiv preprint arXiv:1312.6114
  13. Krause, A., Golovin, D.:. Submodular function maximization. In: Tractability: practical Approaches to Hard Problems, pp. 71–104. Cambridge University Press (2014)Google Scholar
  14. Kushner, H.J.: A new method of locating the maximum point of an arbitrary multi-peak curve in the presence of noise. J. Basic Eng. 86, 97–106 (1964)CrossRefGoogle Scholar
  15. Lam, R., Willcox, K., Wolpert, D.H.: Bayesian optimization with a finite budget: an approximate dynamic programming approach. Adv. Neural. Inf. Process. Syst. 30, 883–891 (2016). Scholar
  16. Marchant, R., Ramos, F., Sanner, S.: Sequential Bayesian optimisation for spatial-temporal monitoring. In: International Conference on Uncertainty in Artificial Intelligence, pp. 553–562 (2014)Google Scholar
  17. Mockus, J., Tiesis, V., and Zilinskas, A.: The application of Bayesian methods for seeking the extremum. In: Dixon, L., Szego, G. (eds.) Towards Global Optimisation, 2, pp. 117–130. Elsevier (1978)Google Scholar
  18. Nguyen, V., Osborne, M.A.: Knowing the what but not the where in Bayesian optimization (2019). arXiv preprint arXiv:1905.02685
  19. Noè, U., Husmeier, D.: On a new improvement-based acquisition function for Bayesian optimization (2018). arXiv preprint arXiv:1808.06918
  20. Osborne, M.A., Garnett, R., Roberts, S.J.: Gaussian processes for global optimization. In: 3rd International Conference on Learning and Intelligent Optimization (LION3), vol. 2009 (2009)Google Scholar
  21. Poloczek, M., Wang, J., Frazier, P.: Multi-information source optimization.In: Guyon, I., Luxburg, U.V., Bengio, S., Wallach, H., Fergus, R., Vishwanathan, S., Garnett, R. (eds.) Advances in Neural Information Processing Systems, 30, pp. 4291–4301. Curran Associates, Red Hook, NY (2017)Google Scholar
  22. Russo, D., Van Roy, B., Kazerouni, A., Osband, I., Wen, Z.: A tutorial on Thompson sam-pling. Found. Trends Mach. Learn. 11, 1–96 (2018). Scholar
  23. Srinivas, N., Krause, A., Kakade, S.M., Seeger, M.W.: Information-theoretic regret bounds for Gaussian process optimization in the bandit setting. In: IEEE Transactions on Information Theory, pp. 3250–3265 (2012)MathSciNetCrossRefGoogle Scholar
  24. Toscano-Palmerin, S., Frazier, P.I.: Bayesian optimization with expensive integrands (2018). arXiv preprint arXiv:1803.08661
  25. Villemonteix, J., Vazquez, E., Walter, E.: An informational approach to the global optimization of expensive-to-evaluate functions. J. Global Optim. 44(4), 509 (2009)MathSciNetCrossRefGoogle Scholar
  26. Volpp, M., Fröhlich, L., Doerr, A., Hutter, F., Daniel, C.: Meta-learning acquisition functions for Bayesian optimization (2019). arXiv preprint arXiv:1904.02642
  27. Wang, Z., Jegelka, S.: Max-value entropy search for efficient Bayesian optimization. In: Proceedings of the 34th International Conference on Machine Learning. Sydney, Australia (2017)Google Scholar
  28. Wilson, J., Hutter, F., Deisenroth, M.: Maximizing acquisition functions for Bayesian optimization. In: Advances in Neural Information Processing Systems, pp. 9906–9917 (2018)Google Scholar
  29. Wu, J., Frazier, P.: The parallel knowledge gradient method for batch bayesian optimization. In: Advances in Neural Information Processing Systems, pp. 3126–3134 (2016)Google Scholar
  30. Wu, J., Frazier, P.I.: Discretization-free knowledge gradient methods for bayesian optimization (2017). arXiv preprint arXiv:1707.06541
  31. Wu, J., Poloczek, M., Wilson, A.G., Frazier, P.: Bayesian optimization with gradients. In: Advances in Neural Information Processing Systems, pp. 5273–5284, (4.2.5) (2017)Google Scholar
  32. Yan, L., Duan, X., Liu, B., Xu, J.: Bayesian optimization based on K-optimality. Entropy 20(8), 594 (2018)CrossRefGoogle Scholar

Copyright information

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Computer Science, Systems and CommunicationsUniversity of Milano-BicoccaMilanItaly

Personalised recommendations