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We consider Bayesian optimization using Gaussian Process models, alsoreferred to as kernel-based bandit optimization. We study the methodology ofexploring the domain using random samples drawn from a distribution. We showthat this random exploration approach achieves the optimal error rates. Ouranalysis is based on novel concentration bounds in an infinite dimensionalHilbert space established in this work, which may be of independent interest.We further develop an algorithm based on random exploration with domainshrinking and establish its order-optimal regret guarantees under bothnoise-free and noisy settings. In the noise-free setting, our analysis closesthe existing gap in regret performance and thereby resolves a COLT openproblem. The proposed algorithm also enjoys a computational advantage overprevailing methods due to the random exploration that obviates the expensiveoptimization of a non-convex acquisition function for choosing the query pointsat each iteration.

Random Exploration in Bayesian Optimization: Order-Optimal Regret and Computational Efficiency