  Enforcing safety is a key aspect of many problems pertaining to sequential
decision making under uncertainty, which require the decisions made at every
step to be both informative of the optimal decision and also safe. For example,
we value both efficacy and comfort in medical therapy, and efficiency and
safety in robotic control. We consider this problem of optimizing an unknown
utility function with absolute feedback or preference feedback subject to
unknown safety constraints. We develop an efficient safe Bayesian optimization
algorithm, StageOpt, that separates safe region expansion and utility function
maximization into two distinct stages. Compared to existing approaches which
interleave between expansion and optimization, we show that StageOpt is more
efficient and naturally applicable to a broader class of problems. We provide
theoretical guarantees for both the satisfaction of safety constraints as well
as convergence to the optimal utility value. We evaluate StageOpt on both a
variety of synthetic experiments, as well as in clinical practice. We
demonstrate that StageOpt is more effective than existing safe optimization
approaches, and is able to safely and effectively optimize spinal cord
stimulation therapy in our clinical experiments.
