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 deal with convergence properties of an important class of sequential design approaches, 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. We study the existence of SUR strategies and establish generic convergence results for a broad class thereof. We also introduce a special class of uncertainty measures defined in terms of regular loss functions, which makes it easier to check that our convergence results apply in particular cases. Applications of the latter include proofs of convergence for the two main SUR strategies proposed by Bect, Ginsbourger, Li, Picheny and Vazquez (Stat. Comp., 2012). To the best of our knowledge, these are the first convergence proofs for GP-based sequential design algorithms dedicated to the estimation of excursions sets and their measure. Coming to global optimization algorithms, we also show that the knowledge gradient strategy can be cast in the SUR framework with an uncertainty functional stemming from a regular loss, resulting in further convergence results. We finally establish a new proof of convergence for the expected improvement algorithm, which is the first proof for this algorithm that applies to any GP with continuous sample paths.

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

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

Julien Bect, François Bachoc, David Ginsbourger. A supermartingale approach to Gaussian process based sequential design of experiments. 2017. ⟨hal-01351088v2⟩