Abstract: We present a provably optimal differentially private algorithm for the stochastic multi-arm bandit problem, as opposed to the private analogue of the UCB-algorithm (Mishra and Thakurta, 2015; Tossou and Dimitrakakis, 2016) which doesn't meet the recently discovered lower-bound of $\Omega \left(\frac{K\log(T)}{\epsilon} \right)$ (Shariff and Sheffet, 2018). Our construction is based on a different algorithm, Successive Elimination (Even-Dar et al., 2002), that repeatedly pulls all remaining arms until an arm is found to be suboptimal and is then eliminated. In order to devise a private analogue of Successive Elimination we visit the problem of private \emph{stopping rule}, that takes as input a stream of i.i.d samples from an unknown distribution and returns a \emph{multiplicative} $(1 \pm \alpha)$-approximation of the distribution's mean, and prove the optimality of our private stopping rule. We then present the private Successive Elimination algorithm which meets both the non-private lower bound (Lai and Robbins, 1985) and the above-mentioned private lower bound. We also compare empirically the performance of our algorithm with the private UCB algorithm.