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We lay the foundations of a non-parametric theory of best-arm identificationin multi-armed bandits with a fixed budget T. We consider general, possiblynon-parametric, models D for distributions over the arms; an overarchingexample is the model D = P(0,1) of all probability distributions over [0,1]. Wepropose upper bounds on the average log-probability of misidentifying theoptimal arm based on information-theoretic quantities that correspond to infimaover Kullback-Leibler divergences between some distributions in D and a givendistribution. This is made possible by a refined analysis of thesuccessive-rejects strategy of Audibert, Bubeck, and Munos (2010). We finallyprovide lower bounds on the same average log-probability, also in terms of thesame new information-theoretic quantities; these lower bounds are larger whenthe (natural) assumptions on the considered strategies are stronger. All thesenew upper and lower bounds generalize existing bounds based, e.g., on gapsbetween distributions.

On Best-Arm Identification with a Fixed Budget in Non-Parametric Multi-Armed Bandits