Joint entropy search for multi-objective Bayesian optimization with constraints and multiple fidelities

Bayesian optimization (BO) methods can be used to solve efficiently problems with several objectives and constraints. Each objective and constraint is considered a black-box function that is expensive to evaluate, lacking a closed-form expression. BO methods use a model of each black-box to guide the search for the problem’s solution. Specifically, they make intelligent decisions about where each black-box function should be evaluated next with the goal of finding the solution using a few evaluations only. Sometimes, however, the black-boxes may be evaluated at different fidelity levels. A lower fidelity is simply a cheap proxy for the corresponding black-box. These lower fidelities correlate with the actual black-boxes to optimize and can, therefore, be used to reduce the overall cost of solving the optimization problem. Here, we propose Multi-fidelity Joint Entropy Search for Multi-objective Bayesian Optimization with Constraints (MF-JESMOC), a BO method for solving the aforementioned problems. MF-JESMOC chooses the next point, and fidelity level at which to evaluate the black-boxes, as the combination that is expected to reduce the most the joint entropy of the Pareto set and the Pareto front, normalized by the fidelity’s evaluation cost. We use Deep Gaussian processes to model each black-box and the dependencies between fidelities. These are powerful probabilistic models that can learn the dependency structure among fidelity levels of each black-box. Several experiments show that MF-JESMOC outperforms other state-of-the-art methods for multi-objective BO with constraints and different fidelity levels in both synthetic and real-world problems.