Online Ranking with Top-1 Feedback

We consider a setting where a system learns to rank a fixed set of m items. The goal is produce a good ranking for users with diverse interests who interact with the system for T rounds in an online fashion. We consider a novel top-1 feedback model for this problem: at the end of each round, the relevance score for only the top ranked object is revealed to the system. However, the performance of the system is judged on the entire ranked list. We provide a comprehensive set of results regarding learnability under this challenging setting. For popular ranking measures such as PairwiseLoss and DCG, we prove that the minimax regret is of order T^{2/3}. Moreover, the minimax regret is achievable using an efficient algorithmic strategy that only spends O(m log m) time per round. The same algorithmic strategy achieves O(T^{2/3}) regret for Precision@k. Surprisingly, we show that for normalized versions of these ranking measures, namely AUC, NDCG and MAP, no online ranking algorithm can have sub-linear regret.