However, please note that we clearly mentioned that this work is an extension of MES to MO setting.

We propose Pareto-frontier entropy search (PFES) for multi-objective Bayesian optimization (MBO). Unlike the existing entropy search for MBO which considers the entropy of the input space, we define the entropy of Pareto-frontier in the output space. By using a sampled Pareto-frontier from the current model, PFES provides a simple formula for directly evaluating the entropy. Besides the usual MBO setting, in which all the objectives are simultaneously observed, we also consider the "decoupled" setting, in which the objective functions can be observed separately. PFES can easily derive an acquisition function for the decoupled setting through the entropy of the marginal density for each output variable. For the both settings, by conditioning on the sampled Pareto-frontier, dependence among different objectives arises in the entropy evaluation. PFES can incorporate this dependency into the acquisition function, while the existing information-based MBO employs an independent Gaussian approximation. Our numerical experiments show effectiveness of PFES through synthetic functions and real-world datasets from materials science.

We present PESMO, a Bayesian method for identifying the Pareto set of multi-objective optimization problems, when the functions are expensive to evaluate. The central idea of PESMO is to choose evaluation points so as to maximally reduce the entropy of the posterior distribution over the Pareto set. Critically, the PESMO multi-objective acquisition function can be decomposed as a sum of objective-specific acquisition functions, which enables the algorithm to be used in \emph{decoupled} scenarios in which the objectives can be evaluated separately and perhaps with different costs. This decoupling capability also makes it possible to identify difficult objectives that require more evaluations. PESMO also offers gains in efficiency, as its cost scales linearly with the number of objectives, in comparison to the exponential cost of other methods. We compare PESMO with other related methods for multi-objective Bayesian optimization on synthetic and real-world problems. The results show that PESMO produces better recommendations with a smaller number of evaluations of the objectives, and that a decoupled evaluation can lead to improvements in performance, particularly when the number of objectives is large.