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 Ω(√T) and O(√Tlog T). This gives a tight characterization up to a √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 Mat\'ern-ν kernels, with the latter requiring ν > 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 Mat\'ern kernel with ν > 2.