Open Problem: What is the Complexity of Joint Differential Privacy in Linear Contextual Bandits?

Contextual bandits serve as a theoretical framework to design recommender systems, which often rely on user-sensitive data, making privacy a critical concern. However, a significant gap remains between the known upper and lower bounds on the regret achievable in linear contextual bandits under Joint Differential Privacy (JDP), which is a popular privacy definition used in this setting. In particular, the best regret upper bound is known to be $O\left(d \sqrt{T} \log(T) + \textcolor{blue}{d^{3/4} \sqrt{T \log(1/\delta)} / \sqrt{\epsilon}} \right)$, while the lower bound is $\Omega \left(\sqrt{d T \log(K)} + \textcolor{blue}{d/(\epsilon + \delta)}\right)$. We discuss the recent progress on this problem, both from the algorithm design and lower bound techniques, and posit the open questions.