Online Optimization in X-Armed Bandits

Sébastien Bubeck 1 Rémi Munos 1 Gilles Stoltz 2, 3 Csaba Szepesvari 4
1 SEQUEL - Sequential Learning
LIFL - Laboratoire d'Informatique Fondamentale de Lille, LAGIS - Laboratoire d'Automatique, Génie Informatique et Signal, Inria Lille - Nord Europe
Abstract : We consider a generalization of stochastic bandit problems where the set of arms, X, is allowed to be a generic topological space. We constraint the mean-payoff function with a dissimilarity function over X in a way that is more general than Lipschitz. We construct an arm selection policy whose regret improves upon previous result for a large class of problems. In particular, our results imply that if X is the unit hypercube in a Euclidean space and the mean-payoff function has a finite number of global maxima around which the behavior of the function is locally Holder with a known exponent, then the expected regret is bounded up to a logarithmic factor by $\sqrt{n}$, i.e., the rate of the growth of the regret is independent of the dimension of the space. Moreover, we prove the minimax optimality of our algorithm for the class of mean-payoff functions we consider.
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Sébastien Bubeck, Rémi Munos, Gilles Stoltz, Csaba Szepesvari. Online Optimization in X-Armed Bandits. Twenty-Second Annual Conference on Neural Information Processing Systems, Dec 2008, Vancouver, Canada. ⟨inria-00329797⟩

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