Open Problem: Optimal Rates for Stochastic Decision-Theoretic Online Learning Under Differentially Privacy

For the stochastic variant of decision-theoretic online learning with $K$ actions, $T$ rounds, and minimum gap $\Delta_{\min}$, the optimal, gap-dependent rate of the pseudo-regret is known to be $O \left( \frac{\log K}{\Delta_{\min}} \right)$. We ask to settle the optimal gap-dependent rate for the problem under $\varepsilon$-differential privacy.