  We introduce GLR-klUCB, a novel algorithm for the piecewise iid
non-stationary bandit problem with bounded rewards. This algorithm combines an
efficient bandit algorithm, kl-UCB, with an efficient, parameter-free,
changepoint detector, the Bernoulli Generalized Likelihood Ratio Test, for
which we provide new theoretical guarantees of independent interest. Unlike
previous non-stationary bandit algorithms using a change-point detector,
GLR-klUCB does not need to be calibrated based on prior knowledge on the arms'
means. We prove that this algorithm can attain a $O(\sqrt{TA
\Upsilon_T\log(T)})$ regret in $T$ rounds on some "easy" instances, where A is
the number of arms and $\Upsilon_T$ the number of change-points, without prior
knowledge of $\Upsilon_T$. In contrast with recently proposed algorithms that
are agnostic to $\Upsilon_T$, we perform a numerical study showing that
GLR-klUCB is also very efficient in practice, beyond easy instances.
