  There has been a recent surge of interest in nonparametric bandit algorithms
based on subsampling. One drawback however of these approaches is the
additional complexity required by random subsampling and the storage of the
full history of rewards. Our first contribution is to show that a simple
deterministic subsampling rule, proposed in the recent work of Baudry et al.
(2020) under the name of ''last-block subsampling'', is asymptotically optimal
in one-parameter exponential families. In addition, we prove that these
guarantees also hold when limiting the algorithm memory to a polylogarithmic
function of the time horizon. These findings open up new perspectives, in
particular for non-stationary scenarios in which the arm distributions evolve
over time. We propose a variant of the algorithm in which only the most recent
observations are used for subsampling, achieving optimal regret guarantees
under the assumption of a known number of abrupt changes. Extensive numerical
simulations highlight the merits of this approach, particularly when the
changes are not only affecting the means of the rewards.
