  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 $O(n^{3/2}\sqrt{Tk})$ 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
$\Omega(\sqrt{k})$, a factor of $\Omega(\sqrt{\log n})$ 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.
