  An online reinforcement learning algorithm is anytime if it does not need to
know in advance the horizon T of the experiment. A well-known technique to
obtain an anytime algorithm from any non-anytime algorithm is the "Doubling
Trick". In the context of adversarial or stochastic multi-armed bandits, the
performance of an algorithm is measured by its regret, and we study two
families of sequences of growing horizons (geometric and exponential) to
generalize previously known results that certain doubling tricks can be used to
conserve certain regret bounds. In a broad setting, we prove that a geometric
doubling trick can be used to conserve (minimax) bounds in $R\_T = O(\sqrt{T})$
but cannot conserve (distribution-dependent) bounds in $R\_T = O(\log T)$. We
give insights as to why exponential doubling tricks may be better, as they
conserve bounds in $R\_T = O(\log T)$, and are close to conserving bounds in
$R\_T = O(\sqrt{T})$.
