On Private and Robust Bandits

TL;DR

This paper introduces a new approach for private and robust multi-armed bandit algorithms that handle contaminated, heavy-tailed rewards while ensuring differential privacy, providing theoretical bounds and practical schemes.

Contribution

It presents the first minimax lower bound for private heavy-tailed MABs and proposes a meta-algorithm with nearly-optimal regret using reward truncation and Laplace mechanisms.

Findings

Achieves nearly-optimal regret bounds in heavy-tailed, contaminated, private settings.

Provides the first minimax lower bound for private heavy-tailed MABs.

Demonstrates the effectiveness of truncation-based PRM schemes through experiments.

Abstract

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.