  In this paper we investigate the problem of stochastic multi-armed bandits
(MAB) in the (local) differential privacy (DP/LDP) model. Unlike previous
results that assume bounded/sub-Gaussian reward distributions, we focus on the
setting where each arm's reward distribution only has $(1+v)$-th moment with
some $v\in (0, 1]$. In the first part, we study the problem in the central
$\epsilon$-DP model. We first provide a near-optimal result by developing a
private and robust Upper Confidence Bound (UCB) algorithm. Then, we improve the
result via a private and robust version of the Successive Elimination (SE)
algorithm. Finally, we establish the lower bound to show that the
instance-dependent regret of our improved algorithm is optimal. In the second
part, we study the problem in the $\epsilon$-LDP model. We propose an algorithm
that can be seen as locally private and robust version of SE algorithm, which
provably achieves (near) optimal rates for both instance-dependent and
instance-independent regret. Our results reveal differences between the problem
of private MAB with bounded/sub-Gaussian rewards and heavy-tailed rewards. To
achieve these (near) optimal rates, we develop several new hard instances and
private robust estimators as byproducts, which might be used to other related
problems. Finally, experiments also support our theoretical findings and show
the effectiveness of our algorithms.
