Sequential- and Parallel- Constrained Max-value Entropy Search via Information Lower Bound

Bayesian optimization (BO) is known as a powerful tool for optimizing an unknown, expensive function through querying the function values sequentially. On the other hand, in many practical problems, additional unknown constraints also need to be considered. In this paper, we propose an information-theoretic approach called Constrained Max-value Entropy Search via Information lower BOund (CMES-IBO) for the constrained BO (CBO). Although information-theoretic methods have been studied in CBO literature, they have not revealed any relation between their acquisition functions and the original mutual information. In contrast, our acquisition function is an unbiased consistent estimator of a lower bound of mutual information. We show that our CMES-IBO has several advantageous properties such as non-negativity, estimation error bounds of the acquisition function, and well-definedness of the criterion, none of which have been shown for the existing information-theoretic CBO. Furthermore, by using conditional mutual information, we extend CMES-IBO to the parallel setting in which multiple queries can be issued simultaneously. We demonstrate the effectiveness of CMES-IBO by several benchmark functions.