  This paper studies an entropy-based multi-objective Bayesian optimization
(MBO). The entropy search is successful approach to Bayesian optimization.
However, for MBO, existing entropy-based methods ignore trade-off among
objectives or introduce unreliable approximations. We propose a novel
entropy-based MBO called Pareto-frontier entropy search (PFES) by considering
the entropy of Pareto-frontier, which is an essential notion of the optimality
of the multi-objective problem. Our entropy can incorporate the trade-off
relation of the optimal values, and further, we derive an analytical formula
without introducing additional approximations or simplifications to the
standard entropy search setting. We also show that our entropy computation is
practically feasible by using a recursive decomposition technique which has
been known in studies of the Pareto hyper-volume computation. 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 adapt to the decoupled setting by
considering the entropy of the marginal density for each output dimension. This
approach incorporates dependency among objectives conditioned on
Pareto-frontier, which is ignored by the existing method. Our numerical
experiments show effectiveness of PFES through several benchmark datasets.
