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**A supermartingale approach to Gaussian process based sequential design of experiments.**
*(English)*
Zbl 1428.62369

Summary: 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 J. Bect et al. [Stat. Comput. 22, No. 3, 773–793 (2012; Zbl 1252.62081)] – 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:

62L05 | Sequential statistical design |

68T05 | Learning and adaptive systems in artificial intelligence |

60G48 | Generalizations of martingales |

60G15 | Gaussian processes |

### Keywords:

active learning; convergence; sequential design of experiments; stepwise uncertainty reduction; supermartingale; uncertainty functional### Citations:

Zbl 1252.62081### Software:

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\textit{J. Bect} et al., Bernoulli 25, No. 4A, 2883--2919 (2019; Zbl 1428.62369)

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