Towards Gaussian Process-based Optimization with Finite Time Horizon

During the last decade, Kriging-based sequential optimization algorithms have become standard methods in computer experiments. These algorithms rely on the iterative maximization of sampling criteria such as the

Expected Improvement

(

EI

), which takes advantage of Kriging conditional distributions to make an explicit trade-off between promising and uncertain points in the search space. We have recently worked on a multipoint

EI

criterion meant to choose simultaneously several points for synchronous parallel computation. The results presented in this article concern sequential procedures with a fixed number of iterations. We show that maximizing the usual

EI

at each iteration is suboptimal. In essence, the latter amounts to considering the current iteration as the last one. This work formulates the problem of optimal strategy for finite horizon sequential optimization, provides the solution to this problem in terms of a new multipoint

EI

, and illustrates the suboptimality of maximizing the 1-point

EI

at each iteration on the basis of a first counter-example.