Robust Gaussian Process-Based Global Optimization Using a Fully Bayesian Expected Improvement Criterion

  • Romain Benassi
  • Julien Bect
  • Emmanuel Vazquez
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6683)


We consider the problem of optimizing a real-valued continuous function f, which is supposed to be expensive to evaluate and, consequently, can only be evaluated a limited number of times. This article focuses on the Bayesian approach to this problem, which consists in combining evaluation results and prior information about f in order to efficiently select new evaluation points, as long as the budget for evaluations is not exhausted.

The algorithm called efficient global optimization (EGO), proposed by Jones, Schonlau and Welch (J. Global Optim., 13(4):455–492, 1998), is one of the most popular Bayesian optimization algorithms. It is based on a sampling criterion called the expected improvement (EI), which assumes a Gaussian process prior about f. In the EGO algorithm, the parameters of the covariance of the Gaussian process are estimated from the evaluation results by maximum likelihood, and these parameters are then plugged in the EI sampling criterion. However, it is well-known that this plug-in strategy can lead to very disappointing results when the evaluation results do not carry enough information about f to estimate the parameters in a satisfactory manner.

We advocate a fully Bayesian approach to this problem, and derive an analytical expression for the EI criterion in the case of Student predictive distributions. Numerical experiments show that the fully Bayesian approach makes EI-based optimization more robust while maintaining an average loss similar to that of the EGO algorithm.


Global Optimization Covariance Function Bayesian Approach Gaussian Process Evaluation Point 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Törn, A., Zilinskas, A.: Global Optimization. Springer, Berlin (1989)CrossRefzbMATHGoogle Scholar
  2. 2.
    Pintér, J.D.: Global optimization. Continuous and Lipschitz optimization: algorithms, implementations and applications. Springer, Heidelberg (1996)zbMATHGoogle Scholar
  3. 3.
    Zhigljavsky, A., Zilinskas, A.: Stochastic global optimization. Springer, Heidelberg (2007)zbMATHGoogle Scholar
  4. 4.
    Conn, A.R., Scheinberg, K., Vicente, L.N.: Introduction to derivative-free optimization. SIAM, Philadelphia (2009)CrossRefzbMATHGoogle Scholar
  5. 5.
    Tenne, Y., Goh, C.K.: Computational intelligence in optimization: applications and implementations. Springer, Heidelberg (2010)CrossRefzbMATHGoogle Scholar
  6. 6.
    Mockus, J., Tiesis, V., Zilinskas, A.: The application of Bayesian methods for seeking the extremum. In: Dixon, L., Szego, G. (eds.) Towards Global Optimization, vol. 2, pp. 117–129. Elsevier, Amsterdam (1978)Google Scholar
  7. 7.
    Mockus, J.: Bayesian approach to Global Optimization: Theory and Applications. Kluwer Acad. Publ., Dordrecht (1989)CrossRefzbMATHGoogle Scholar
  8. 8.
    Betrò, B.: Bayesian methods in global optimization. Journal of Global Optimization 1, 1–14 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Locatelli, M., Schoen, F.: An adaptive stochastic global optimization algorithm for one-dimensional functions. Annals of Operations Research 58(4), 261–278 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Auger, A., Teytaud, O.: Continuous lunches are free plus the design of optimal optimization algorithms. Algorithmica 57(1), 121–146 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Ginsbourger, D., Le Riche, R.: Towards Gaussian process-based optimization with finite time horizon. In: mODa 9 Advances in Model-Oriented Design and Analysis. Contribution to Statistics, pp. 89–96. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  12. 12.
    Grünewälder, S., Audibert, J.-Y., Opper, M., Shawe-Taylor, J.: Regret bounds for Gaussian process bandit problems. In: Proceedings of the 13th International Conference on Artificial Intelligence and Statistics (AISTATS 2010). JMLR W&CP, vol. 9, pp. 273–280 (2010)Google Scholar
  13. 13.
    Bertsekas, D.P.: Dynamic programming and optimal control. Athena Scientific, Belmont (1995)zbMATHGoogle Scholar
  14. 14.
    Jones, D.R., Schonlau, M., Welch, W.J.: Efficient global optimization of expensive black-box functions. Journal of Global Optimization 13(4), 455–492 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Forrester, A.I.J., Jones, D.R.: Global optimization of deceptive functions with sparse sampling. In: 12th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, September 10-12 (2008)Google Scholar
  16. 16.
    Locatelli, M.: Bayesian algorithms for one-dimensional global optimization. Journal of Global Optimization 10(1), 57–76 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Osborne, M.A.: Bayesian Gaussian Processes for Sequential Prediction Optimisation and Quadrature. PhD thesis, University of Oxford (2010)Google Scholar
  18. 18.
    Osborne, M.A., Garnett, R., Roberts, S.J.: Gaussian processes for global optimization. In: 3rd International Conference on Learning and Intelligent Optimization (LION3), Online Proceedings, Trento, Italy (2009)Google Scholar
  19. 19.
    Osborne, M.A., Roberts, S.J., Rogers, A., Ramchurn, S.D., Jennings, N.R.: Towards real-time information processing of sensor network data using computationally efficient multi-output Gaussian processes. In: Proceedings of the 7th International Conference on Information Processing in Sensor Networks, pp. 109–120. IEEE Computer Society, Los Alamitos (2008)Google Scholar
  20. 20.
    Williams, B., Santner, T., Notz, W.: Sequential Design of Computer Experiments to Minimize Integrated Response Functions. Statistica Sinica 10(4), 1133–1152 (2000)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Schonlau, M.: Computer experiments and global optimization. PhD thesis, University of Waterloo, Waterloo, Ontario, Canada (1997)Google Scholar
  22. 22.
    Schonlau, M., Welch, W.J.: Global optimization with nonparametric function fitting. In: Proceedings of the ASA, Section on Physical and Engineering Sciences, pp. 183–186. Amer. Statist. Assoc. (1996)Google Scholar
  23. 23.
    Schonlau, M., Welch, W.J., Jones, D.R.: A data analytic approach to Bayesian global optimization. In: Proceedings of the ASA, Section on Physical and Engineering Sciences, pp. 186–191. Amer. Statist. Assoc. (1997)Google Scholar
  24. 24.
    Forrester, A.I.J., Keane, A.J.: Recent advances in surrogate-based optimization. Progress in Aerospace Sciences 45(1-3), 50–79 (2009)CrossRefGoogle Scholar
  25. 25.
    Jones, D.R.: A taxonomy of global optimization methods based on response surfaces. Journal of Global Optimization 21(4), 345–383 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Robert, C.P., Casella, G.: Monte Carlo statistical methods. Springer, Heidelberg (2004)CrossRefzbMATHGoogle Scholar
  27. 27.
    Del Moral, P., Doucet, A., Jasra, A.: Sequential Monte Carlo samplers. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 68(3), 411–436 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Liu, J.S.: Monte Carlo strategies in scientific computing. Springer, Heidelberg (2008)zbMATHGoogle Scholar
  29. 29.
    O’Hagan, A.: Bayes-Hermite quadrature. Journal of Statistical Planning and Inference 29(3), 245–260 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    O’Hagan, A.: Curve Fitting and Optimal Design for Prediction. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 40(1), 1–42 (1978)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Handcock, M.S., Stein, M.L.: A Bayesian analysis of Kriging. Technometrics 35(4), 403–410 (1993)CrossRefGoogle Scholar
  32. 32.
    Ginsbourger, D., Helbert, C., Carraro, L.: Discrete mixtures of kernels for kriging-based optimization. Quality and Reliability Engineering International 24, 681–691 (2008)CrossRefGoogle Scholar
  33. 33.
    O’Hagan, A.: Some Bayesian numerical analysis. In: Bayesian Statistics 4: Proceedings of the Fourth Valencia International Meeting, April 15-20, 1991. Oxford University Press, Oxford (1992)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Romain Benassi
    • 1
  • Julien Bect
    • 1
  • Emmanuel Vazquez
    • 1
  1. 1.SUPELECGif-sur-YvetteFrance

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