This paper studies the deviations of the regret in a stochastic multi-armed
bandit problem. When the total number of plays n is known beforehand by the
agent, Audibert et al. (2009) exhibit a policy such that with probability at
least 1-1/n, the regret of the policy is of order log(n). They have also shown
that such a property is not shared by the popular ucb1 policy of Auer et al.
(2002). This work first answers an open question: it extends this negative
result to any anytime policy. The second contribution of this paper is to
design anytime robust policies for specific multi-armed bandit problems in
which some restrictions are put on the set of possible distributions of the
different arms
