A supermartingale approach to Gaussian process based sequential design of experiments
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Gaussian process (GP) models have become a well-established frameworkfor the adaptive design of costly experiments, and notably of computerexperiments. GP-based sequential designs have been found practicallyefficient 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 theestimation of level sets and excursion sets. In this paper, we dealwith convergence properties of an important class of sequential designapproaches, known as stepwise uncertainty reduction (SUR) strategies.Our approach relies on the key observation that the sequence ofresidual uncertainty measures, in SUR strategies, is generally asupermartingale with respect to the filtration generated by theobservations. We study the existence of SUR strategies and establishgeneric convergence results for a broad class thereof. We alsointroduce a special class of uncertainty measures defined in terms ofregular loss functions, which makes it easier to check that ourconvergence results apply in particular cases. Applications of thelatter include proofs of convergence for the two main SUR strategiesproposed by Bect, Ginsbourger, Li, Picheny and Vazquez (Stat. Comp.,2012). To the best of our knowledge, these are the first convergenceproofs for GP-based sequential design algorithms dedicated to theestimation of excursions sets and their measure. Coming to globaloptimization algorithms, we also show that the knowledge gradientstrategy can be cast in the SUR framework with an uncertaintyfunctional stemming from a regular loss, resulting in furtherconvergence results. We finally establish a new proof of convergencefor the expected improvement algorithm, which is the first proof forthis algorithm that applies to any GP with continuous sample paths.
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