  The paper considers the problem of global optimization in the setup of
stochastic process bandits. We introduce an UCB algorithm which builds a
cascade of discretization trees based on generic chaining in order to render
possible his operability over a continuous domain. The theoretical framework
applies to functions under weak probabilistic smoothness assumptions and also
extends significantly the spectrum of application of UCB strategies. Moreover
generic regret bounds are derived which are then specialized to Gaussian
processes indexed on infinite-dimensional spaces as well as to quadratic forms
of Gaussian processes. Lower bounds are also proved in the case of Gaussian
processes to assess the optimality of the proposed algorithm.
