A supermartingale approach to Gaussian process based sequential design of experiments

Abstract

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. Comp., 2012)-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.MSC 2010 subject classifications: Primary 60G15, 62L05; secondary 68T05.

Adaptive Design of Experiments for Conservative Estimation of Excursion Sets

“…Reduction approach (SUR, see, e.g., Fleuret and Geman, 1999;Bect et al, 2012;Chevalier et al, 2014a;Bect et al, 2018) and we select a sequence of points in order to reduce the uncertainty on selected quantities of interest. In the remainder of the paper we consider that the first n points x 1 , .…”

“…a minimizer of the future uncertainty in expectation. For a more complete and theoretical perspective on SUR strategies see, e.g., Bect et al (2018) and references therein. There are many ways to proceed with the minimization introduced above, see, e.g., Osborne et al (5) is called batch sequential one-step lookahead sampling criterion and is denoted by J n :…”

A Bayesian optimization approach to find Nash equilibria

“…Finally, this work could also be extended to continuous domains, for which related convergence properties may be investigated in light of recent advances on SUR approaches (Bect et al, 2016).…”

Gaussian process optimization with failures: classification and convergence proof

“…crash likeliness) of a new design x. Second, leveraging recent results on sequential strategies [2], we propose an algorithm in the form of EGO with guaranteed convergence.…”