  In this paper, we consider the problem of stochastic optimization under a
bandit feedback model. We generalize the GP-UCB algorithm [Srinivas and al.,
2012] to arbitrary kernels and search spaces. To do so, we use a notion of
localized chaining to control the supremum of a Gaussian process, and provide a
novel optimization scheme based on the computation of covering numbers. The
theoretical bounds we obtain on the cumulative regret are more generic and
present the same convergence rates as the GP-UCB algorithm. Finally, the
algorithm is shown to be empirically more efficient than its natural
competitors on simple and complex input spaces.
