Convergence properties of the expected improvement algorithm with fixed mean and covariance functions

• Published in: Journal of statistical planning and inference Vol. 140; no. 11; pp. 3088 - 3095
• Main Authors: Vazquez, Emmanuel, Bect, Julien
• Format: Journal Article
• Language: English
• Published: Kidlington Elsevier B.V 01.11.2010; Elsevier
• Subjects: Bayesian optimization; Calculus of variations and optimal control; Computer experiments; Exact sciences and technology; Gaussian process; General topics; Global optimization; Machine Learning; Mathematical analysis; Mathematics; Numerical analysis; Numerical analysis. Scientific computation; Numerical methods in mathematical programming, optimization and calculus of variations; Numerical methods in optimization and calculus of variations; Optimization and Control; Probability; Probability and statistics; Probability theory and stochastic processes; RKHS; Sciences and techniques of general use; Sequential design; Statistics; Statistics Theory; Stochastic processes; Computer experiments; Global optimization; RKHS; Gaussian process; Sequential design; Bayesian optimization; Density estimation; Covariance function; Optimization method; Gaussian distribution; Probability distribution; Optimization; Convergence; Sequential method; Variational calculus; Hilbert space; Mathematical programming; Bayes estimation; Prior distribution; Continuous function; Statistical decision; Prior probability; Algorithm; Mean estimation; Statistical method; Numerical analysis; Gaussian processes; sequential design; computer experiments; global optimization
• ISSN: 0378-3758; 1873-1171; 1873-1171
• DOI: 10.1016/j.jspi.2010.04.018

This paper deals with the convergence of the expected improvement algorithm, a popular global optimization algorithm based on a Gaussian process model of the function to be optimized. The first result is that under some mild hypotheses on the covariance function k of the Gaussian process, the expected improvement algorithm produces a dense sequence of evaluation points in the search domain, when the function to be optimized is in the reproducing kernel Hilbert space generated by k. The second result states that the density property also holds for P -almost all continuous functions, where P is the (prior) probability distribution induced by the Gaussian process.