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Consider the sequential optimization of a continuous, possibly non-convex,and expensive to evaluate objective function $f$. The problem can be cast as aGaussian Process (GP) bandit where $f$ lives in a reproducing kernel Hilbertspace (RKHS). The state of the art analysis of several learning algorithmsshows a significant gap between the lower and upper bounds on the simple regretperformance. When $N$ is the number of exploration trials and $\gamma_N$ is themaximal information gain, we prove an $\tilde{\mathcal{O}}(\sqrt{\gamma_N/N})$bound on the simple regret performance of a pure exploration algorithm that issignificantly tighter than the existing bounds. We show that this bound isorder optimal up to logarithmic factors for the cases where a lower bound onregret is known. To establish these results, we prove novel and sharpconfidence intervals for GP models applicable to RKHS elements which may be ofbroader interest.

Optimal Order Simple Regret for Gaussian Process Bandits