  Black-box zero-th order optimization is a central primitive for applications
in fields as diverse as finance, physics, and engineering. In a common
formulation of this problem, a designer sequentially attempts candidate
solutions, receiving noisy feedback on the value of each attempt from the
system. In this paper, we study scenarios in which feedback is also provided on
the safety of the attempted solution, and the optimizer is constrained to limit
the number of unsafe solutions that are tried throughout the optimization
process. Focusing on methods based on Bayesian optimization (BO), prior art has
introduced an optimization scheme -- referred to as SAFEOPT -- that is
guaranteed not to select any unsafe solution with a controllable probability
over feedback noise as long as strict assumptions on the safety constraint
function are met. In this paper, a novel BO-based approach is introduced that
satisfies safety requirements irrespective of properties of the constraint
function. This strong theoretical guarantee is obtained at the cost of allowing
for an arbitrary, controllable but non-zero, rate of violation of the safety
constraint. The proposed method, referred to as SAFE-BOCP, builds on online
conformal prediction (CP) and is specialized to the cases in which feedback on
the safety constraint is either noiseless or noisy. Experimental results on
synthetic and real-world data validate the advantages and flexibility of the
proposed SAFE-BOCP.
