Optimal Rates of (Locally) Differentially Private Heavy-tailed Multi-Armed Bandits

In this paper we study the problem of stochastic multi-armed bandits (MAB) in the (local) differential privacy (DP/LDP) model. Unlike the previous results which need to assume bounded reward distributions, here we mainly focus on the case the reward distribution of each arm 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 show that the instance-dependent regret bound of our improved algorithm is optimal by showing its lower bound. In the second part of the paper, we study the problem in the $\epsilon$-LDP model. We propose an algorithm which could be seen as locally private and robust version of the SE algorithm, and show it could achieve (near) optimal rates for both instance-dependent and instance-independent regrets. All of the above results can also reveal the differences between the problem of private MAB with bounded 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 could be used to other related problems. Finally, experimental results also support our theoretical analysis and show the effectiveness of our algorithms.