Differential Privacy in Kernelized Contextual Bandits via Random Projections

TL;DR

This paper introduces a differentially private algorithm for kernelized contextual bandits that leverages random projections to reduce sensitivity, achieving optimal regret bounds while preserving privacy.

Contribution

The paper proposes a novel private kernel-ridge regression estimator using private covariance estimation and random projections, enabling state-of-the-art privacy-preserving bandit performance.

Findings

Achieves regret bounds of (\u007f\u007f\u007f\u007f\u007f\u007f\u007f\u007f\u007f\u007f\u007f\u007f\u007f\u007f\u007f\u007f\u007f\u007f\u007f\u007f\u007f\u007f",

State-of-the-art performance guarantees in different privacy models.

Abstract

We consider the problem of contextual kernel bandits with stochastic contexts, where the underlying reward function belongs to a known Reproducing Kernel Hilbert Space. We study this problem under an additional constraint of Differential Privacy, where the agent needs to ensure that the sequence of query points is differentially private with respect to both the sequence of contexts and rewards. We propose a novel algorithm that achieves the state-of-the-art cumulative regret of $\widetilde{\mathcal{O}}(\sqrt{\gamma_TT}+\frac{\gamma_T}{\varepsilon_{\mathrm{DP}}})$ and $\widetilde{\mathcal{O}}(\sqrt{\gamma_TT}+\frac{\gamma_T\sqrt{T}}{\varepsilon_{\mathrm{DP}}})$ over a time horizon of $T$ in the joint and local models of differential privacy, respectively, where $\gamma_T$ is the effective dimension of the kernel and $\varepsilon_{\mathrm{DP}} > 0$ is the privacy parameter. The key ingredient of the proposed algorithm is a novel private kernel-ridge regression estimator which is based on a combination of private covariance estimation and private random projections. It offers a significantly reduced sensitivity compared to its classical counterpart while maintaining a high prediction accuracy, allowing our algorithm to achieve the state-of-the-art performance guarantees.