  We consider active learning (AL) in an uncertain environment in which
trade-off between multiple risk measures need to be considered. As an AL
problem in such an uncertain environment, we study Mean-Variance Analysis in
Bayesian Optimization (MVA-BO) setting. Mean-variance analysis was developed in
the field of financial engineering and has been used to make decisions that
take into account the trade-off between the average and variance of investment
uncertainty. In this paper, we specifically focus on BO setting with an
uncertain component and consider multi-task, multi-objective, and constrained
optimization scenarios for the mean-variance trade-off of the uncertain
component. When the target blackbox function is modeled by Gaussian Process
(GP), we derive the bounds of the two risk measures and propose AL algorithm
for each of the above three problems based on the risk measure bounds. We show
the effectiveness of the proposed AL algorithms through theoretical analysis
and numerical experiments.
