Interactive and Concentrated Differential Privacy for Bandits

Achraf Azize, Debabrota Basu

Keywords: Differential Privacy, Multi-armed Bandits, Lower bounds, Regret Analysis, Interactive Privacy

TL;DR: We derive regret lower bounds and design algorithms with matching upper bounds for finite-armed and linear bandits preserving Interactive Concentrated Differential Privacy.

Abstract: Bandits play a crucial role in interactive learning schemes and modern recommender systems. However, these systems often rely on sensitive user data, making privacy a critical concern. This paper investigates privacy in bandits with a trusted centralized decision-maker through the lens of interactive Differential Privacy (DP). While bandits under pure $\epsilon$-global DP have been well-studied, we contribute to the understanding of bandits under zero Concentrated DP (zCDP). We provide minimax and problem-dependent lower bounds on regret for finite-armed and linear bandits, which quantify the cost of $\rho$-global zCDP in these settings. These lower bounds reveal two hardness regimes based on the privacy budget $\rho$ and suggest that $\rho$-global zCDP incurs less regret than pure $\epsilon$-global DP. We propose two $\rho$-global zCDP bandit algorithms, AdaC-UCB and AdaC-GOPE, for finite-armed and linear bandits respectively. Both algorithms use a common recipe of Gaussian mechanism and adaptive episodes. We analyze the regret of these algorithms to show that AdaC-UCB achieves the problem-dependent regret lower bound up to multiplicative constants, while AdaC-GOPE achieves the minimax regret lower bound up to poly-logarithmic factors. Finally, we provide experimental validation of our theoretical results under different settings.