Regret bounds for meta Bayesian optimization with an unknown Gaussian process prior
This work addresses the gap between theoretical guarantees and practical performance in Bayesian optimization for applications like robotics, though it is incremental as it builds on existing methods like GP-UCB.
The paper tackles the problem of unknown parameters in Bayesian optimization priors by using empirical Bayes to estimate a Gaussian process prior from offline data, achieving a near-zero regret bound that decreases with more data and evaluations.
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.