Cascading Bandit under Differential Privacy

Abstract: This paper studies \emph{differential privacy (DP)} and \emph{local differential privacy (LDP)} in cascading bandits. Under DP, we propose an algorithm which guarantees $\epsilon$-indistinguishability and a regret of $\mathcal{O}((\frac{\log T}{\epsilon})<sup>{1+\xi})$ for an arbitrarily small $\xi$. This is a significant improvement from the previous work of $\mathcal{O}(\frac{\log³ T}{\epsilon})$ regret. Under ($\epsilon$,$\delta$)-LDP, we relax the $K<sup>2$ dependence through the tradeoff between privacy budget $\epsilon$ and error probability $\delta$, and obtain a regret of $$\mathcal{O}(\frac{K\log (1/\delta) \log T}{\epsilon<sup>2})$$, where $K$ is the size of the arm subset. This result holds for both Gaussian mechanism and Laplace mechanism by analyses on the composition. Our results extend to combinatorial semi-bandit. We show respective lower bounds for DP and LDP cascading bandits. Extensive experiments corroborate our theoretic findings.