Faster Rates for Private Adversarial Bandits

TL;DR

This paper introduces new differentially private algorithms for adversarial bandits and bandits with expert advice, achieving improved regret bounds and the first private algorithms for the latter.

Contribution

It provides a simple conversion method for private adversarial bandits and introduces the first private algorithms for bandits with expert advice, with improved regret bounds.

Findings

Achieves regret bound of $O(rac{ oot{2} {KT}}{ oot{2} { ext{ extepsilon}}})$ for private adversarial bandits.

First private algorithms for bandits with expert advice with sublinear regret.

Establishes separation between central and local differential privacy in adversarial bandits.

Abstract

We design new differentially private algorithms for the problems of adversarial bandits and bandits with expert advice. For adversarial bandits, we give a simple and efficient conversion of any non-private bandit algorithm to a private bandit algorithm. Instantiating our conversion with existing non-private bandit algorithms gives a regret upper bound of $O\left(\frac{\sqrt{KT}}{\sqrt{\epsilon}}\right)$, improving upon the existing upper bound $O\left(\frac{\sqrt{KT \log(KT)}}{\epsilon}\right)$ for all $\epsilon \leq 1$. In particular, our algorithms allow for sublinear expected regret even when $\epsilon \leq \frac{1}{\sqrt{T}}$, establishing the first known separation between central and local differential privacy for this problem. For bandits with expert advice, we give the first differentially private algorithms, with expected regret $O\left(\frac{\sqrt{NT}}{\sqrt{\epsilon}}\right), O\left(\frac{\sqrt{KT\log(N)}\log(KT)}{\epsilon}\right)$, and $\tilde{O}\left(\frac{N^{1/6}K^{1/2}T^{2/3}\log(NT)}{\epsilon ^{1/3}} + \frac{N^{1/2}\log(NT)}{\epsilon}\right)$, where $K$ and $N$ are the number of actions and experts respectively. These rates allow us to get sublinear regret for different combinations of small and large $K, N$ and $\epsilon.$