Bayesian optimization (BO) has emerged during the last few years as an effective approach to optimize black-box functions where direct queries of the objective are expensive. We consider the case where direct access to the function is not possible, but information about user preferences is. Such scenarios arise in problems where human preferences are modeled, such as A/B tests or recommender systems. We present a new framework for this scenario that we call Preferential Bayesian Optimization (PBO) and that allows to find the optimum of a latent function that can only be queried through pairwise comparisons, so-called duels. PBO extend the applicability of standard BO ideas and generalizes previous discrete dueling approaches by modeling the probability of the the winner of each duel by means of Gaussian process model with a Bernoulli likelihood. The latent preference function is used to define a family of acquisition functions that extend usual policies used in BO. We illustrate the benefits of PBO in a variety of experiments in which we show how the way correlations are modeled is the key ingredient to drastically reduce the number of comparisons to find the optimum of the latent function of interest.
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