Regret bounds for meta Bayesian optimization with an unknown Gaussian process prior

TL;DR

This paper develops regret bounds for meta Bayesian optimization when the Gaussian process prior is unknown, using empirical Bayes to estimate the prior from offline data, leading to near-zero regret guarantees.

Contribution

It introduces a method to estimate the Gaussian process prior from offline data and proves near-zero regret bounds for Bayesian optimization algorithms under this setting.

Findings

Achieves near-zero regret bounds that decrease with more data.

Empirically validated on robotic task and motion planning problems.

Abstract

Bayesian optimization usually assumes that a Bayesian prior is given. However, the strong theoretical guarantees in Bayesian optimization are often regrettably compromised in practice because of unknown parameters in the prior. In this paper, we adopt a variant of empirical Bayes and show that, by estimating the Gaussian process prior from offline data sampled from the same prior and constructing unbiased estimators of the posterior, variants of both GP-UCB and probability of improvement achieve a near-zero regret bound, which decreases to a constant proportional to the observational noise as the number of offline data and the number of online evaluations increase. Empirically, we have verified our approach on challenging simulated robotic problems featuring task and motion planning.