PACSBO: Probably approximately correct safe Bayesian optimization

TL;DR

PACSBO introduces a safe Bayesian optimization algorithm that estimates the RKHS norm from data, reducing conservatism and providing safety guarantees without requiring prior smoothness assumptions, demonstrated through experiments.

Contribution

The paper proposes a novel safe BO algorithm that estimates the RKHS norm locally, improving safety guarantees and applicability over existing methods.

Findings

PACSBO effectively estimates the RKHS norm from data.

The algorithm reduces conservatism compared to global approaches.

Numerical and hardware experiments show improved safety and performance.

Abstract

Safe Bayesian optimization (BO) algorithms promise to find optimal control policies without knowing the system dynamics while at the same time guaranteeing safety with high probability. In exchange for those guarantees, popular algorithms require a smoothness assumption: a known upper bound on a norm in a reproducing kernel Hilbert space (RKHS). The RKHS is a potentially infinite-dimensional space, and it is unclear how to, in practice, obtain an upper bound of an unknown function in its corresponding RKHS. In response, we propose an algorithm that estimates an upper bound on the RKHS norm of an unknown function from data and investigate its theoretical properties. Moreover, akin to Lipschitz-based methods, we treat the RKHS norm as a local rather than a global object, and thus reduce conservatism. Integrating the RKHS norm estimation and the local interpretation of the RKHS norm into a safe BO algorithm yields PACSBO, an algorithm for probably approximately correct safe Bayesian optimization, for which we provide numerical and hardware experiments that demonstrate its applicability and benefits over popular safe BO algorithms.