- Abstract: Many applications require a learner to make sequential decisions given uncertainty regarding both the system’s payoff function and safety constraints. When learning algorithms are used in safety-critical systems, it is paramount that the learner’s actions do not violate the safety constraints at any stage of the learning process. In this paper, we study a stochastic bandit optimization problem where the system’s unknown payoff and constraint functions are sampled from Gaussian Processes (GPs). We develop a safe variant of the proposed algorithm by Srinivas et al. (2010), GP-UCB, called SGP-UCB, with necessary modifications to respect safety constraints at every round. Our most important contribution is to derive the first sub-linear regret bounds for this problem.