  We consider a setting where a system learns to rank a fixed set of $m$ items.
The goal is produce good item rankings for users with diverse interests who
interact online with the system for $T$ rounds. We consider a novel top-$1$
feedback model: at the end of each round, the relevance score for only the top
ranked object is revealed. 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 PairwiseLoss and DCG, two
popular ranking measures, we prove that the minimax regret is
$\Theta(T^{2/3})$. Moreover, the minimax regret is achievable using an
efficient strategy that only spends $O(m \log m)$ time per round. The same
efficient strategy achieves $O(T^{2/3})$ regret for Precision@$k$.
Surprisingly, we show that for normalized versions of these ranking measures,
i.e., AUC, NDCG \& MAP, no online ranking algorithm can have sublinear regret.
