KL-UCB-switch: optimal regret bounds for stochastic bandits from both a distribution-dependent and a distribution-free viewpoints

In the context of K-armed stochastic bandits with distribution only assumed to be supported by [0,1], we introduce the first algorithm, called KL-UCB-switch, that enjoys simultaneously a distribution-free regret bound of optimal order $\sqrt{KT}$ and a distribution-dependent regret bound of optimal order as well, that is, matching the $\kappa\ln T$ lower bound by Lai & Robbins (1985) and Burnetas & Katehakis (1996). This self-contained contribution simultaneously presents state-of-the-art techniques for regret minimization in bandit models, and an elementary construction of non-asymptotic confidence bounds based on the empirical likelihood method for bounded distributions.