  We define a novel family of algorithms for the adversarial multi-armed bandit
problem, and provide a simple analysis technique based on convex smoothing. We
prove two main results. First, we show that regularization via the
\emph{Tsallis entropy}, which includes EXP3 as a special case, achieves the
$\Theta(\sqrt{TN})$ minimax regret. Second, we show that a wide class of
perturbation methods achieve a near-optimal regret as low as $O(\sqrt{TN \log
N})$ if the perturbation distribution has a bounded hazard rate. For example,
the Gumbel, Weibull, Frechet, Pareto, and Gamma distributions all satisfy this
key property.
