A supermartingale approach to Gaussian process based sequential design of experiments

Julien Bect (1, 2) , François Bachoc (3) , David Ginsbourger (4, 5)

1 L2S - Laboratoire des signaux et systèmes
2 GdR MASCOT-NUM - Méthodes d'Analyse Stochastique des Codes et Traitements Numériques
3 IMT - Institut de Mathématiques de Toulouse UMR5219
4 IDIAP Research Institute
5 IMSV - Institute of Mathematical Statistics and Actuarial Science [Bern]

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.

Mots clés: Stepwise Uncertainty Reduction; Sequential Design of Experiments; Active Learning; Supermartingale; Uncertainty functional; Convergence

Domaines: Statistiques [math.ST]; Machine Learning [stat.ML]; Probabilités [math.PR]

Julien Bect, François Bachoc, David Ginsbourger. A supermartingale approach to Gaussian process based sequential design of experiments. Bernoulli, 2019, 25 (4A), pp.2883-2919. ⟨10.3150/18-bej1074⟩. ⟨hal-01351088v4⟩