Kullback-Leibler Upper Confidence Bounds for Optimal Sequential Allocation

Olivier Cappé (1) , Aurélien Garivier (2) , Odalric-Ambrym Maillard (3) , Rémi Munos (4) , Gilles Stoltz (5, 6)

1 LTCI - Laboratoire Traitement et Communication de l'Information
2 IMT - Institut de Mathématiques de Toulouse UMR5219
3 MUL - Montanuniversität Leoben
4 SEQUEL - Sequential Learning
5 CLASSIC - Computational Learning, Aggregation, Supervised Statistical, Inference, and Classification
6 GREGH - Groupement de Recherche et d'Etudes en Gestion à HEC

We consider optimal sequential allocation in the context of the so-called stochastic multi-armed bandit model. We describe a generic index policy, in the sense of Gittins (1979), based on upper confidence bounds of the arm payoffs computed using the Kullback-Leibler divergence. We consider two classes of distributions for which instances of this general idea are analyzed: The kl-UCB algorithm is designed for one-parameter exponential families and the empirical KL-UCB algorithm for bounded and finitely supported distributions. Our main contribution is a unified finite-time analysis of the regret of these algorithms that asymptotically matches the lower bounds of Lai and Robbins (1985) and Burnetas and Katehakis (1996), respectively. We also investigate the behavior of these algorithms when used with general bounded rewards, showing in particular that they provide significant improvements over the state-of-the-art.

Keywords: Multi-armed bandit problems; Upper confidence bound; Kullback-Leibler divergence; Sequential testing

Domains: Probability [math.PR]; Statistics [math.ST]; Statistics Theory [stat.TH]; Machine Learning [cs.LG]; Machine Learning [stat.ML]

Olivier Cappé, Aurélien Garivier, Odalric-Ambrym Maillard, Rémi Munos, Gilles Stoltz. Kullback-Leibler Upper Confidence Bounds for Optimal Sequential Allocation. Annals of Statistics, 2013, 41 (3), pp.1516-1541. ⟨10.1214/13-AOS1119⟩. ⟨hal-00738209v2⟩