We study private and robust multi-armed bandits (MABs), where the agent receives Huber's contaminated heavy-tailed rewards and meanwhile needs to ensure differential privacy. We first present its minimax lower bound, characterizing the information-theoretic limit of regret with respect to privacy budget, contamination level and heavy-tailedness. Then, we propose a meta-algorithm that builds on a private and robust mean estimation sub-routine \texttt{PRM} that essentially relies on reward truncation and the Laplace mechanism only. For two different heavy-tailed settings, we give specific schemes of \texttt{PRM}, which enable us to achieve nearly-optimal regret. As by-products of our main results, we also give the first minimax lower bound for private heavy-tailed MABs (i.e., without contamination). Moreover, our two proposed truncation-based \texttt{PRM} achieve the optimal trade-off between estimation accuracy, privacy and robustness. Finally, we support our theoretical results with experimental studies.

翻译：我们研究隐私且鲁棒的多臂赌博机（MABs），其中智能体接收休伯（Huber）污染的厚尾奖励，同时需保障差分隐私。我们首先提出其极小化极大下界，刻画了隐私预算、污染程度和厚尾特性对遗憾的信息论极限。随后，我们提出一种元算法，该算法基于隐私鲁棒均值估计子例程\texttt{PRM}，其核心仅依赖于奖励截断和拉普拉斯机制。针对两种不同的厚尾设定，我们给出了\texttt{PRM}的具体方案，从而实现了近乎最优的遗憾。作为主要研究的副产品，我们还首次给出了隐私厚尾MABs（即无污染情形）的极小化极大下界。此外，我们提出的两种基于截断的\texttt{PRM}实现了估计精度、隐私性和鲁棒性之间的最优权衡。最后，我们通过实验研究验证了理论结果。