# Achieving Privacy in the Adversarial Multi-Armed Bandit

## TL;DR

This paper presents a novel approach combining differential privacy mechanisms with the EXP3 algorithm to achieve improved privacy guarantees and regret bounds in adversarial multi-armed bandit problems, supported by empirical validation.

## Contribution

It introduces a new method that enhances privacy in adversarial bandits by integrating Laplace and exponential mechanisms with EXP3, improving regret bounds and privacy levels.

## Key findings

- Achieves $ ilde{O}(rac{
oot{2} ext{T} ext{ln T}}{ ext{ε}})$ regret bound with differential privacy.
- Improves privacy from linear leakage to $ ilde{O}(
oot{ln T})$-DP in EXP3.
- Demonstrates empirical results aligning with theoretical improvements.

## Abstract

In this paper, we improve the previously best known regret bound to achieve $\epsilon$-differential privacy in oblivious adversarial bandits from $\mathcal{O}{(T^{2/3}/\epsilon)}$ to $\mathcal{O}{(\sqrt{T} \ln T /\epsilon)}$. This is achieved by combining a Laplace Mechanism with EXP3. We show that though EXP3 is already differentially private, it leaks a linear amount of information in $T$. However, we can improve this privacy by relying on its intrinsic exponential mechanism for selecting actions. This allows us to reach $\mathcal{O}{(\sqrt{\ln T})}$-DP, with a regret of $\mathcal{O}{(T^{2/3})}$ that holds against an adaptive adversary, an improvement from the best known of $\mathcal{O}{(T^{3/4})}$. This is done by using an algorithm that run EXP3 in a mini-batch loop. Finally, we run experiments that clearly demonstrate the validity of our theoretical analysis.