Concentrated Differential Privacy for Bandits

TL;DR

This paper introduces new algorithms for differentially private bandit learning, demonstrating that privacy constraints have negligible impact on regret and providing both upper and lower bounds with experimental validation.

Contribution

It formalizes DP adaptations for bandits, proposes three private algorithms with a generic blueprint, and establishes regret bounds and minimax lower bounds under zCDP.

Findings

Privacy costs are asymptotically negligible compared to non-private regret.

Proposed algorithms achieve a good privacy-utility trade-off.

First minimax lower bounds for bandits with zCDP are established.

Abstract

Bandits serve as the theoretical foundation of sequential learning and an algorithmic foundation of modern recommender systems. However, recommender systems often rely on user-sensitive data, making privacy a critical concern. This paper contributes to the understanding of Differential Privacy (DP) in bandits with a trusted centralised decision-maker, and especially the implications of ensuring zero Concentrated Differential Privacy (zCDP). First, we formalise and compare different adaptations of DP to bandits, depending on the considered input and the interaction protocol. Then, we propose three private algorithms, namely AdaC-UCB, AdaC-GOPE and AdaC-OFUL, for three bandit settings, namely finite-armed bandits, linear bandits, and linear contextual bandits. The three algorithms share a generic algorithmic blueprint, i.e. the Gaussian mechanism and adaptive episodes, to ensure a good privacy-utility trade-off. We analyse and upper bound the regret of these three algorithms. Our analysis shows that in all of these settings, the prices of imposing zCDP are (asymptotically) negligible in comparison with the regrets incurred oblivious to privacy. Next, we complement our regret upper bounds with the first minimax lower bounds on the regret of bandits with zCDP. To prove the lower bounds, we elaborate a new proof technique based on couplings and optimal transport. We conclude by experimentally validating our theoretical results for the three different settings of bandits.