Learning to Rank from Incomplete Rankings

In domains such as recommender systems and information retrieval, learning from human-generated feedback is especially challenging because the information provided is often sparse and incomplete. In this work, we address the problem of learning the top-$k$ items from incomplete rankings. Most existing models for incomplete rankings rely on rigid assumptions regarding both the ranking model that generates the latent ranking and the censoring mechanism that determines which comparisons remain unobserved. On the one hand, the ranking model is often assumed to follow a Plackett-Luce (PL) or Mallows distribution. On the other hand, the censoring mechanism is typically assumed to be Missing Completely At Random (MCAR) or to exhibit well-behaved dependencies on the latent ranking, such as winner feedback or top-$h$ feedback. We introduce a new, general framework for learning from incomplete rankings that unifies and strictly generalizes the established frameworks in the literature. We consider the broad class of ranking models that satisfy the complete consensus property, which comprehends all widely adopted models, including PL and Mallows. Furthermore, we present a new preference-based feedback model, named positional censoring, which generalizes winner and top-$h$ feedback. We show that it is possible to learn in this general setting by presenting the PIRATE algorithm and providing a near-optimal instance-dependent bound to the sample complexity. Finally, we show that, under the PL ranking, PIRATE matches the sample complexity of state-of-the-art algorithms in the relevant scenarios of winner and top-$h$ feedback.