We design new differentially private algorithms for the problems of adversarial bandits and bandits with expert advice. For adversarial bandits, we give a simple and efficient conversion of any non-private bandit algorithms to private bandit algorithms. Instantiating our conversion with existing non-private bandit algorithms gives a regret upper bound of O(KTε)Oleft(frac{sqrt{KT}}{sqrt{varepsilon}}right)O(εKT), improving upon the existing upper bound O(KTlog(KT)ε)Oleft(frac{sqrt{KT log(KT)}}{varepsilon}right)O(εKTlog(KT)) in all privacy regimes. In particular, our algorithms…