Benefits of Monotonicity in Safe Exploration with Gaussian Processes

Arpan Losalka, Jonathan Scarlett

Keywords: bayesian optimization, safe bayesian optimization, safety, clinical trial, safe exploration, gaussian process, safe optimization, sublinear, regret, level set

Abstract: We consider the problem of sequentially maximising an unknown function over a set of actions while ensuring that every sampled point has a function value below a given safety threshold. We model the function using kernel-based and Gaussian process methods, while differing from previous works in our assumption that the function is monotonically increasing with respect to a safety variable. This assumption is motivated by various practical applications such as adaptive clinical trial design and robotics. Taking inspiration from the GP-UCB and SAFEOPT algorithms, we propose an algorithm, monotone safe UCB (M-SafeUCB) for this task. We show that M-SafeUCB enjoys theoretical guarantees in terms of safety, a suitably-defined regret notion, and approximately finding the entire safe boundary. In addition, we illustrate that the monotonicity assumption yields significant benefits in terms of the guarantees obtained, as well as algorithmic simplicity and efficiency. We support our theoretical findings by performing empirical evaluations on a variety of functions, including a simulated clinical trial experiment.