Tight Regret Bounds for Bayesian Optimization in One Dimension

Abstract

We consider the problem of Bayesian optimization (BO) in one dimension, under a Gaussian process prior and Gaussian sampling noise. We provide a theoretical analysis showing that, under fairly mild technical assumptions on the kernel, the best possible cumulative regret up to time $T$ behaves as $\hat{I}\text{c◯}(\sqrt{T})$ and $O(\sqrt{T\log T})$. This gives a tight characterization up to a $\sqrt{\log T}$ factor, and includes the first non-trivial lower bound for noisy BO. Our assumptions are satisfied, for example, by the squared exponential and Matérn-$\hat{I}½$ kernels, with the latter requiring $\hat{I}½>2$. Our results certify the near-optimality of existing bounds (Srinivas {\em et al.}, 2009) for the SE kernel, while proving them to be strictly suboptimal for the Matérn kernel with $\hat{I}½>2$.

ICML 2018 + supplementary material. This version also includes an 'Errata' section correcting two minor mistakes