Online Ranking: Discrete Choice, Spearman Correlation and Other Feedback

Given a set V of n objects, an online ranking system outputs at each time step a full ranking of the set, observes a feedback of some form and suffers a loss. We study the setting in which the (adversarial) feedback is an element in V , and the loss is the position (0th, 1st, 2nd...) of the item in the outputted ranking. More generally, we study a setting in which the feedback is a subset U of at most k elements in V , and the loss is the sum of the positions of those elements. We present an algorithm of expected regret O(n √ Tk) over a time horizon of T steps with respect to the best single ranking in hindsight. This improves previous algorithms and analyses either by a factor of either Ω( √ k), a factor of Ω( √ log n) or by improving running time from quadratic to O(n log n) per round. We also prove a matching lower bound. Our techniques also imply an improved regret bound for online rank aggregation over the Spearman correlation measure, and to other more complex ranking loss functions.