Learning Modulo Theories for preference elicitation in hybrid domains

This paper introduces CLEO, a novel preference elicitation algorithm capable of recommending complex objects in hybrid domains, characterized by both discrete and continuous attributes and constraints defined over them. 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 (DM) is modelled as a weighted combination of features. CLEO iteratively alternates a preference elicitation step, where pairs of candidate solutions 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 hybrid solutions according to the current utility model. The refinement step is implemented as learning to rank, and a sparsifying norm is used to favour the selection of few informative features in the combinatorial space of candidate decisional features. CLEO is the first preference elicitation algorithm capable of dealing with hybrid domains, thanks to the use of Max-SMT technology, while retaining uncertainty in the DM utility and noisy feedback. Experimental results on complex recommendation tasks show the ability of CLEO to quickly focus towards optimal solutions, as well as its capacity to recover from suboptimal initial choices. While no competitors exist in the hybrid setting, CLEO outperforms a state-of-the-art Bayesian preference elicitation algorithm when applied to a purely discrete task.