Improved Max-value Entropy Search for Multi-objective Bayesian Optimization with Constraints

We present MESMOC+, an improved version of Max-value Entropy search for Multi-Objective Bayesian optimization with Constraints (MESMOC). MESMOC+ can be used to solve constrained multi-objective problems when the objectives and the constraints are expensive to evaluate. MESMOC+ works by minimizing the entropy of the solution of the optimization problem in function space, i.e., the Pareto frontier, to guide the search for the optimum. The cost of MESMOC+ is linear in the number of objectives and constraints. Furthermore, it is often significantly smaller than the cost of alternative methods based on minimizing the entropy of the Pareto set. The reason for this is that it is easier to approximate the required computations in MESMOC+. Moreover, MESMOC+'s acquisition function is expressed as the sum of one acquisition per each black-box (objective or constraint). Thus, it can be used in a decoupled evaluation setting in which one chooses not only the next input location to evaluate, but also which black-box to evaluate there. We compare MESMOC+ with related methods in synthetic and real optimization problems. These experiments show that the entropy estimation provided by MESMOC+ is more accurate than that of previous methods. This leads to better optimization results. MESMOC+ is also competitive with other information-based methods for constrained multi-objective Bayesian optimization, but it is significantly faster.