Online Ranking Given Discrete Choice 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 a choice of an item from $V$, and the loss is the position (1st, 2nd, 3rd...) of the item in the outputted ranking. For this simple problem we present an algorithm of expected regret $O(n^{3/2}\sqrt{T})$ for a time horizon of $T$ steps, with respect to the best single ranking in hindsight. We also show this bound is tight. The algorithm improves on previous approaches in two ways: (i) it shaves off at least a $\Omega(\sqrt{\log n})$ factor in the expected regret bound, and (ii) it is extremely simple and efficient to implement (in fact, some previously known algorithms it is not even clear how to execute in sub-exponential time). Our algorithm works for a more general class of ranking problems in which the feedback is a vector of values of elements in $V$, and the loss is the sum of magnitudes of pairwise inversions (also known as AUC in the literature). For these cases we get an even stronger impovement in regret bounds. The main tool is the use of randomized sorting algorithms that, restricted to any fixed pair of items, gives rise to a multiplicative weights update scheme on a binary action set consisting of their two ordering possibilities.