The probabilities that a user likes each item are unknown.

Importantly, an item cannot be recommended twice to the same user. The probabilities that a user likes each item are unknown.

We investigate the problem of online collaborative filtering under no-repetition constraints, whereby users need to be served content in an online fashion and a given user cannot be recommended the same content item more than once. We start by designing and analyzing an algorithm that works under biclustering assumptions on the user-item preference matrix, and show that this algorithm exhibits an optimal regret guarantee, while being fully adaptive, in that it is oblivious to any prior knowledge about the sequence of users, the universe of items, as well as the biclustering parameters of the preference matrix. We then propose a more robust version of this algorithm which operates with general matrices. Also this algorithm is parameter free, and we prove regret guarantees that scale with the amount by which the preference matrix deviates from a biclustered structure. To our knowledge, these are the first results on online collaborative filtering that hold at this level of generality and adaptivity under no-repetition constraints. Finally, we complement our theoretical findings with simple experiments on real-world datasets aimed at both validating the theory and empirically comparing to standard baselines. This comparison shows the competitive advantage of our approach over these baselines.

This paper proposes a theoretical analysis of recommendation systems in an online setting, where items are sequentially recommended to users over time. In each round, a user, randomly picked from a population of $m$ users, requests a recommendation. The decision-maker observes the user and selects an item from a catalogue of $n$ items. Importantly, an item cannot be recommended twice to the same user. The probabilities that a user likes each item are unknown. The performance of the recommendation algorithm is captured through its regret, considering as a reference an Oracle algorithm aware of these probabilities. We investigate various structural assumptions on these probabilities: we derive for each structure regret lower bounds, and devise algorithms achieving these limits. Interestingly, our analysis reveals the relative weights of the different components of regret: the component due to the constraint of not presenting the same item twice to the same user, that due to learning the chances users like items, and finally that arising when learning the underlying structure.