  The multi-armed bandit is a concise model for the problem of iterated
decision-making under uncertainty. In each round, a gambler must pull one of
$K$ arms of a slot machine, without any foreknowledge of their payouts, except
that they are uniformly bounded. A standard objective is to minimize the
gambler's regret, defined as the gambler's total payout minus the largest
payout which would have been achieved by any fixed arm, in hindsight. Note that
the gambler is only told the payout for the arm actually chosen, not for the
unchosen arms.
  Almost all previous work on this problem assumed the payouts to be
non-adaptive, in the sense that the distribution of the payout of arm $j$ in
round $i$ is completely independent of the choices made by the gambler on
rounds $1, \dots, i-1$. In the more general model of adaptive payouts, the
payouts in round $i$ may depend arbitrarily on the history of past choices made
by the algorithm.
  We present a new algorithm for this problem, and prove nearly optimal
guarantees for the regret against both non-adaptive and adaptive adversaries.
After $T$ rounds, our algorithm has regret $O(\sqrt{T})$ with high probability
(the tail probability decays exponentially). This dependence on $T$ is best
possible, and matches that of the full-information version of the problem, in
which the gambler is told the payouts for all $K$ arms after each round.
  Previously, even for non-adaptive payouts, the best high-probability bounds
known were $O(T^{2/3})$, due to Auer, Cesa-Bianchi, Freund and Schapire. The
expected regret of their algorithm is $O(T^{1/2}) for non-adaptive payouts, but
as we show, $\Omega(T^{2/3})$ for adaptive payouts.
