  In this paper, we improve the previously best known regret bound to achieve
$\epsilon$-differential privacy in oblivious adversarial bandits from
$\mathcal{O}{(T^{2/3}/\epsilon)}$ to $\mathcal{O}{(\sqrt{T} \ln T /\epsilon)}$.
This is achieved by combining a Laplace Mechanism with EXP3. We show that
though EXP3 is already differentially private, it leaks a linear amount of
information in $T$. However, we can improve this privacy by relying on its
intrinsic exponential mechanism for selecting actions. This allows us to reach
$\mathcal{O}{(\sqrt{\ln T})}$-DP, with a regret of $\mathcal{O}{(T^{2/3})}$
that holds against an adaptive adversary, an improvement from the best known of
$\mathcal{O}{(T^{3/4})}$. This is done by using an algorithm that run EXP3 in a
mini-batch loop. Finally, we run experiments that clearly demonstrate the
validity of our theoretical analysis.
