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

TL;DR

This paper develops and analyzes differentially private algorithms for heavy-tailed multi-armed bandit problems, achieving near-optimal regret rates in both central and local privacy models, and introduces new estimators and hard instances.

Contribution

It introduces the first near-optimal private algorithms for heavy-tailed MABs under both central and local differential privacy models, with new estimators and lower bounds.

Findings

Algorithms achieve near-optimal regret rates.

Heavy-tailed rewards require different privacy techniques.

Experimental results support theoretical guarantees.

Abstract

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.