  We consider minimisation of dynamic regret in non-stationary bandits with a
slowly varying property. Namely, we assume that arms' rewards are stochastic
and independent over time, but that the absolute difference between the
expected rewards of any arm at any two consecutive time-steps is at most a
drift limit $\delta > 0$. For this setting that has not received enough
attention in the past, we give a new algorithm which extends naturally the
well-known Successive Elimination algorithm to the non-stationary bandit
setting. We establish the first instance-dependent regret upper bound for
slowly varying non-stationary bandits. The analysis in turn relies on a novel
characterization of the instance as a detectable gap profile that depends on
the expected arm reward differences. We also provide the first minimax regret
lower bound for this problem, enabling us to show that our algorithm is
essentially minimax optimal. Also, this lower bound we obtain matches that of
the more general total variation-budgeted bandits problem, establishing that
the seemingly easier former problem is at least as hard as the more general
latter problem in the minimax sense. We complement our theoretical results with
experimental illustrations.
