Open Access
November 2019 A supermartingale approach to Gaussian process based sequential design of experiments
Julien Bect, François Bachoc, David Ginsbourger
Bernoulli 25(4A): 2883-2919 (November 2019). DOI: 10.3150/18-BEJ1074


Gaussian process (GP) models have become a well-established framework for the adaptive design of costly experiments, and notably of computer experiments. GP-based sequential designs have been found practically efficient for various objectives, such as global optimization (estimating the global maximum or maximizer(s) of a function), reliability analysis (estimating a probability of failure) or the estimation of level sets and excursion sets. In this paper, we study the consistency of an important class of sequential designs, known as stepwise uncertainty reduction (SUR) strategies. Our approach relies on the key observation that the sequence of residual uncertainty measures, in SUR strategies, is generally a supermartingale with respect to the filtration generated by the observations. This observation enables us to establish generic consistency results for a broad class of SUR strategies. The consistency of several popular sequential design strategies is then obtained by means of this general result. Notably, we establish the consistency of two SUR strategies proposed by Bect, Ginsbourger, Li, Picheny and Vazquez (Stat. Comput. 22 (2012) 773–793) – to the best of our knowledge, these are the first proofs of consistency for GP-based sequential design algorithms dedicated to the estimation of excursion sets and their measure. We also establish a new, more general proof of consistency for the expected improvement algorithm for global optimization which, unlike previous results in the literature, applies to any GP with continuous sample paths.


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Julien Bect. François Bachoc. David Ginsbourger. "A supermartingale approach to Gaussian process based sequential design of experiments." Bernoulli 25 (4A) 2883 - 2919, November 2019.


Received: 1 August 2017; Revised: 1 September 2018; Published: November 2019
First available in Project Euclid: 13 September 2019

zbMATH: 07110115
MathSciNet: MR4003568
Digital Object Identifier: 10.3150/18-BEJ1074

Keywords: Active learning , convergence , sequential design of experiments , stepwise uncertainty reduction , supermartingale , uncertainty functional

Rights: Copyright © 2019 Bernoulli Society for Mathematical Statistics and Probability

Vol.25 • No. 4A • November 2019
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