User preferences in Bayesian multi-objective optimization: the expected weighted hypervolume improvement criterion

In this article, we present a framework for taking into account user preferences in multi-objective Bayesian optimization in the case where the objectives are expensive-to-evaluate black-box functions. A novel expected improvement criterion to be used within Bayesian optimization algorithms is introduced. This criterion, which we call the expected weighted hypervolume improvement (EWHI) criterion, is a generalization of the popular expected hypervolume improvement to the case where the hypervolume of the dominated region is defined using an absolutely continuous measure instead of the Lebesgue measure. The EWHI criterion takes the form of an integral for which no closed form expression exists in the general case. To deal with its computation, we propose an importance sampling approximation method. A sampling density that is optimal for the computation of the EWHI for a predefined set of points is crafted and a sequential Monte-Carlo (SMC) approach is used to obtain a sample approximately distributed from this density. The ability of the criterion to produce optimization strategies oriented by user preferences is demonstrated on a simple bi-objective test problem in the cases of a preference for one objective and of a preference for certain regions of the Pareto front.