  We revisit lower bounds on the regret in the case of multi-armed bandit
problems. We obtain non-asymptotic, distribution-dependent bounds and provide
straightforward proofs based only on well-known properties of Kullback-Leibler
divergences. These bounds show in particular that in an initial phase the
regret grows almost linearly, and that the well-known logarithmic growth of the
regret only holds in a final phase. The proof techniques come to the essence of
the information-theoretic arguments used and they are deprived of all
unnecessary complications.
