Antoine Barrier

Antoine Barrier, Aurélien Garivier, Gilles Stoltz

We lay the foundations of a non-parametric theory of best-arm identification in multi-armed bandits with a fixed budget T. We consider general, possibly non-parametric, models D for distributions over the arms; an overarching example is the model D = P(0,1) of all probability distributions over [0,1]. We propose upper bounds on the average log-probability of misidentifying the optimal arm based on information-theoretic quantities that correspond to infima over Kullback-Leibler divergences between some distributions in D and a given distribution. This is made possible by a refined analysis of the successive-rejects strategy of Audibert, Bubeck, and Munos (2010). We finally provide lower bounds on the same average log-probability, also in terms of the same new information-theoretic quantities; these lower bounds are larger when the (natural) assumptions on the considered strategies are stronger. All these new upper and lower bounds generalize existing bounds based, e.g., on gaps between distributions.

Antoine Barrier, Aurélien Garivier, Tomáš Kocák

We propose a new strategy for best-arm identification with fixed confidence of Gaussian variables with bounded means and unit variance. This strategy, called Exploration-Biased Sampling, is not only asymptotically optimal: it is to the best of our knowledge the first strategy with non-asymptotic bounds that asymptotically matches the sample complexity.But the main advantage over other algorithms like Track-and-Stop is an improved behavior regarding exploration: Exploration-Biased Sampling is biased towards exploration in a subtle but natural way that makes it more stable and interpretable. These improvements are allowed by a new analysis of the sample complexity optimization problem, which yields a faster numerical resolution scheme and several quantitative regularity results that we believe of high independent interest.