  We give nearly-tight upper and lower bounds for the improving multi-armed
bandits problem. An instance of this problem has $k$ arms, each of whose reward
function is a concave and increasing function of the number of times that arm
has been pulled so far. We show that for any randomized online algorithm, there
exists an instance on which it must suffer at least an $\Omega(\sqrt{k})$
approximation factor relative to the optimal reward. We then provide a
randomized online algorithm that guarantees an $O(\sqrt{k})$ approximation
factor, if it is told the maximum reward achievable by the optimal arm in
advance. We then show how to remove this assumption at the cost of an extra
$O(\log k)$ approximation factor, achieving an overall $O(\sqrt{k} \log k)$
approximation relative to optimal.
