FLIPHAT: Joint Differential Privacy for High Dimensional Sparse Linear Bandits

TL;DR

This paper introduces FLIPHAT, a differentially private algorithm for high-dimensional sparse linear bandits, balancing privacy and regret with optimal theoretical guarantees.

Contribution

The paper proposes FLIPHAT, a novel efficient algorithm that achieves optimal regret bounds under joint differential privacy for high-dimensional sparse linear bandits.

Findings

FLIPHAT attains near-optimal regret bounds considering privacy constraints.

The analysis includes a new refined estimation error bound for N-IHT.

Theoretical lower bounds on regret under privacy constraints are established.

Abstract

High dimensional sparse linear bandits serve as an efficient model for sequential decision-making problems (e.g. personalized medicine), where high dimensional features (e.g. genomic data) on the users are available, but only a small subset of them are relevant. Motivated by data privacy concerns in these applications, we study the joint differentially private high dimensional sparse linear bandits, where both rewards and contexts are considered as private data. First, to quantify the cost of privacy, we derive a lower bound on the regret achievable in this setting. To further address the problem, we design a computationally efficient bandit algorithm, \textbf{F}orgetfu\textbf{L} \textbf{I}terative \textbf{P}rivate \textbf{HA}rd \textbf{T}hresholding (FLIPHAT). Along with doubling of episodes and episodic forgetting, FLIPHAT deploys a variant of Noisy Iterative Hard Thresholding (N-IHT) algorithm as a sparse linear regression oracle to ensure both privacy and regret-optimality. We show that FLIPHAT achieves optimal regret in terms of privacy parameters $\epsilon, \delta$, context dimension $d$, and time horizon $T$ up to a linear factor in model sparsity and logarithmic factor in $d$. We analyze the regret by providing a novel refined analysis of the estimation error of N-IHT, which is of parallel interest.