  We consider a non-stationary formulation of the stochastic multi-armed bandit
where the rewards are no longer assumed to be identically distributed. For the
best-arm identification task, we introduce a version of Successive Elimination
based on random shuffling of the $K$ arms. We prove that under a novel and mild
assumption on the mean gap $\Delta$, this simple but powerful modification
achieves the same guarantees in term of sample complexity and cumulative regret
than its original version, but in a much wider class of problems, as it is not
anymore constrained to stationary distributions. We also show that the original
{\sc Successive Elimination} fails to have controlled regret in this more
general scenario, thus showing the benefit of shuffling. We then remove our
mild assumption and adapt the algorithm to the best-arm identification task
with switching arms. We adapt the definition of the sample complexity for that
case and prove that, against an optimal policy with $N-1$ switches of the
optimal arm, this new algorithm achieves an expected sample complexity of
$O(\Delta^{-2}\sqrt{NK\delta^{-1} \log(K \delta^{-1})})$, where $\delta$ is the
probability of failure of the algorithm, and an expected cumulative regret of
$O(\Delta^{-1}{\sqrt{NTK \log (TK)}})$ after $T$ time steps.
