Regret lower bounds and extended Upper Confidence Bounds policies in stochastic multi-armed bandit problem

This paper is devoted to regret lower bounds in the classical model of stochastic multi-armed bandit. A well-known result of Lai and Robbins, which has then been extended by Burnetas and Katehakis, has established the presence of a logarithmic bound for all consistent policies. We relax the notion of consistence, and exhibit a generalisation of the logarithmic bound. We also show the non existence of logarithmic bound in the general case of Hannan consistency. To get these results, we study variants of popular Upper Confidence Bounds (ucb) policies. As a by-product, we prove that it is impossible to design an adaptive policy that would select the best of two algorithms by taking advantage of the properties of the environment.

Robustness of Anytime Bandit Policies

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.