Mean-Variance Analysis in Bayesian Optimization under Uncertainty

Decision making in an uncertain environment has been studied in various domains. For example, in financial engineering, the mean-variance analysis [1, 2, 3] has been introduced as a framework for making investment decisions, taking into account the tradeoff between the return (mean) and the risk (variance) of the investment. In this paper we study active learning (AL) in an uncertain environment. In many practical AL problems, there are two types of parameters called design parameters and environmental parameters. For example, in a product design, while the design parameters are fully controllable, the environmental parameters vary depending on the environment in which the product is used. In this paper, we examine AL problems under such an uncertain environment, where the goal is to efficiently find the optimal design parameters by properly taking into account the uncertainty of the environmental parameters. Concretely, let f(x, w) be a blackbox function indicating the performance of a product, where x X is the set of controllable design parameters and w Ω is the set of uncontrollable environmental parameters whose uncertainty is characterized by a probability distribution p(w).