ABSTRACT. In the stochastic multi-armed bandit problem we consider a modification of the UCB algorithm of Auer et al. [4]. For this modified algorithm we give an improved bound on the regret with respect to the optimal reward. While for the original UCB algorithm the regret in Karmed bandits after T trials is bounded by const ·, where ∆ measures the distance between a suboptimal arm and the optimal arm, for the modified UCB algorithm we show an upper bound on the regret of const · K log(T ∆ 2 ) ∆ .

Abstract. Considering one-dimensional continuum-armed bandit problems, we propose an improvement of an algorithm of Kleinberg and a new set of conditions which give rise to improved rates. In particular, we introduce a novel assumption that is complementary to the previous smoothness conditions, while at the same time smoothness of the mean payoff function is required only at the maxima. Under these new assumptions new bounds on the expected regret are derived. In particular, we show that apart from logarithmic factors, the expected regret scales with the square-root of the number of trials, provided that the mean payoff function has finitely many maxima and its second derivatives are continuous and non-vanishing at the maxima. This improves a previous result of Cope by weakening the assumptions on the function. We also derive matching lower bounds. To complement the bounds on the expected regret, we provide high probability bounds which exhibit similar scaling.

We consider the restless Markov bandit problem, in which the state of each arm evolves according to a Markov process independently of the learner's actions. We suggest an algorithm, that first represents the setting as an MDP which exhibits some special structural properties. In order to grasp this information we introduce the notion of ε-structured MDPs, which are a generalization of concepts like (approximate) state aggregation and MDP homomorphisms. We propose a general algorithm for learning ε-structured MDPs and show regret bounds that demonstrate that additional structural information enhances learning. Applied to the restless bandit setting, this algorithm achieves after any T steps regret of order Õ ( √ T ) with respect to the best policy that knows the distributions of all arms. We make no assumptions on the Markov chains underlying each arm except that they are irreducible. In addition, we show that index-based policies are necessarily suboptimal for the considered problem.