Predictive Entropy Search for Multi-objective Bayesian Optimization with Constraints

Eduardo C. Garrido Merchán, Daniel Hernandez Lobato

Abstract: This work presents PESMOC, Predictive Entropy Search for Multi-objective Bayesian Optimization with Constraints, an information-based strategy for the simultaneous optimization of multiple expensive-to-evaluate black-box functions under the presence of several constraints. Iteratively, PESMOC chooses an input location on which to evaluate the objective functions and the constraints so as to maximally reduce the entropy of the Pareto set of the corresponding optimization problem. The constraints considered in PESMOC are assumed to have similar properties to those of the objectives in typical Bayesian optimization problems. That is, they do not have a known expression (which prevents any gradient computation), their evaluation is considered to be very expensive, and the resulting observations may be corrupted by noise. Importantly, in PESMOC the acquisition function is decomposed as a sum of objective and constraint specific acquisition functions. This enables the use of the algorithm in decoupled evaluation scenarios in which objectives and constraints can be evaluated separately and perhaps with different costs. Therefore, PESMOC not only makes intelligent decisions about where to evaluate next the problem objectives and constraints, but also about which objective or constraint to evaluate next. We present strong empirical evidence in the form of synthetic, benchmark and real-world experiments that illustrate the effectiveness of PESMOC. In these experiments PESMOC outperforms other state-of-the-art methods for constrained multi-objective Bayesian optimization based on a generalization of the expected improvement. The results obtained also show that a decoupled evaluation scenario can lead to significant improvements over a coupled one in which objectives and constraints are evaluated at the same input.