  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 stopping
rule, that takes as input a stream of i.i.d samples from an unknown
distribution and returns a 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.
