Bayesian Optimization With Formal Safety Guarantees via Online Conformal Prediction
Yunchuan Zhang, Sangwoo Park, Osvaldo Simeone
Abstract
Black-box zero-th order optimizationis 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 <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">safety</i> 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 <sc xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">SafeOpt</small> – that is guaranteed not to select <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">any</i> 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 <sc xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Safe-Bocp</small>, 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 <sc xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Safe-Bocp</small>.