Learning Modulo Theories for constructive preference elicitation

This paper introduces CLEO, a novel preference elicitation algorithm capable of recommending complex configurable objects characterized by both discrete and continuous attributes and constraints defined over them. While existing preference elicitation techniques focus on searching for the best instance in a database of candidates, CLEO takes a constructive approach to recommendation through interactive optimization in a space of feasible configurations. The algorithm assumes minimal initial information, i.e., a set of catalog attributes, and defines decisional features as logic formulae combining Boolean and algebraic constraints over the attributes. The (unknown) utility of the decision maker is modeled as a weighted combination of features. CLEO iteratively alternates a preference elicitation step, where pairs of candidate configurations are selected based on the current utility model, and a refinement step where the utility is refined by incorporating the feedback received. The elicitation step leverages a Max-SMT solver to return optimal configurations according to the current utility model. The refinement step is implemented as learning to rank, and a sparsifying norm is used to favor the selection of few informative features in the combinatorial space of candidate decisional features. A major feature of CLEO is that it can recommend optimal configurations in hybrid domains (i.e., including both Boolean and numeric attributes), thanks to the use of Max-SMT technology, while retaining uncertainty in the decision-maker's utility and noisy feedback. In so doing, it adapts the recently introduced learning modulo theory framework to the preference elicitation setting. The combinatorial formulation of the utility function coupled with the feature selection capabilities of 1-norm regularization allow to effectively deal with the uncertainty in the DM utility while retaining high expressiveness. Experimental results on complex recommendation tasks show the ability of CLEO to quickly identify optimal configurations, as well as its capacity to recover from suboptimal initial choices. Our empirical evaluation highlights how CLEO outperforms a state-of-the-art Bayesian preference elicitation algorithm when applied to a purely discrete task