Recommender systems with non-binary grades

We consider the interactive model of recommender systems, in which users are asked about just a few of their preferences, and in return the system outputs an approximation of all their preferences. The measure of performance is the probe complexity of the algorithm, defined to be the maximal number of answers any user should provide (probe complexity typically depends inversely on the number of users with similar preferences and on the quality of the desired approximation). Previous interactive recommendation algorithms assume that user preferences are binary, meaning that each object is either "liked" or "disliked" by each user. In this paper we consider the general case in which users may have a more refined scale of preference, namely more than two possible grades. We show how to reduce the non-binary case to the binary one, proving the following results. For discrete grades with s possible values, we give a simple deterministic reduction that preserves the approximation properties of the binary algorithm at the cost of increasing probe complexity by factor s. Our main result is for the general case, where we assume that user grades are arbitrary real numbers. For this case we present an algorithm that preserves the approximation properties of the binary algorithm while incurring only polylogarithmic overhead.