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We consider an online model for recommendation systems, with each user beingrecommended an item at each time-step and providing 'like' or 'dislike'feedback. Each user may be recommended a given item at most once. A latentvariable model specifies the user preferences: both users and items areclustered into types. All users of a given type have identical preferences forthe items, and similarly, items of a given type are either all liked or alldisliked by a given user. We assume that the matrix encoding the preferences ofeach user type for each item type is randomly generated; in this way, the modelcaptures structure in both the item and user spaces, the amount of structuredepending on the number of each of the types. The measure of performance of therecommendation system is the expected number of disliked recommendations peruser, defined as expected regret. We propose two algorithms inspired byuser-user and item-item collaborative filtering (CF), modified to explicitlymake exploratory recommendations, and prove performance guarantees in terms oftheir expected regret. For two regimes of model parameters, with structure onlyin item space or only in user space, we prove information-theoretic lowerbounds on regret that match our upper bounds up to logarithmic factors. Ouranalysis elucidates system operating regimes in which existing CF algorithmsare nearly optimal.

Regret Bounds and Regimes of Optimality for User-User and Item-Item Collaborative Filtering