We consider a generalization of stochastic bandits where the set of arms, $\cX$, is allowed to be a generic measurable space and the mean-payoff function is ``locally Lipschitz'' with respect to a dissimilarity function that is known to the decision maker. Under this condition we construct an arm selection policy, called HOO hierarchical optimistic optimization), with improved regret bounds compared to previous results for a large class of problems. In particular, our results imply that if $\cX$ 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 Hölder continuous with a known exponent, then the expected of HOO regret is bounded up to a logarithmic factor by $\sqrt{n}$, i.e., the rate of growth of the regret is independent of the dimension of the space. We also prove the minimax optimality of our algorithm when the dissimilarity is a metric.