Adaptive collaborating filtering: The low noise regime

In this paper, we study collaborative filters that adapt future recommendations based on feedback from users. We consider discrete time and at each time a random user seeks a recommendation. The collaborative filter uses all past data available to make a recommendation, the user then provides binary feedback indicating whether he liked the item (rating 1) or not (rating 0), and this feedback is used by the collaborative filter for future decisions. In this setting, ideally the goal is to maximize the long run time average of the ratings, but practical considerations lead us to a moving horizon approximation. Our main result identifies a collaborative filter that optimizes a moving horizon cost in the limit as the noise in the ratings vanishes.