Bayesian Optimization with Lower Confidence Bounds for Minimization Problems with Known Outer Structure

Abstract

This paper considers Bayesian optimization (BO) for problems with known outer problem structure. In contrast to the classic BO setting, where the objective function itself is unknown and needs to be iteratively estimated from noisy observations, we analyze the case where the objective has a known outer structure - given in terms of a loss function - while the inner structure - an unknown input-output model - is again iteratively estimated from noisy observations of the model outputs. We introduce a novel lower confidence bound algorithm for this particular problem class which exploits the known outer problem structure. The proposed method is analyzed in terms of regret for the special case of convex loss functions and probabilistic parametric models which are linear in the uncertain parameters. Numerical examples illustrate the superior performance of structure-exploiting methods compared to structure-agnostic approaches.