Abstract

We consider black box optimization of an unknown function in the nonparametric Gaussian process setting when the noise in the observed function values can be heavy tailed. This is in contrast to existing literature that typically assumes subGaussian noise distributions for queries. Under the assumption that the unknown function belongs to the Reproducing Kernel Hilbert Space (RKHS) induced by a kernel, we first show that an adaptation ofthe well-known  GP-UCB algorithm  with reward truncation enjoys sublinear regret even with only the (1 + )-th moments, (0, 1], of the reward distribution being bounded ( hides logarithmic factors). However, for the common squared exponential (SE) and Matérn kernels, this is seen to be significantly larger than a fundamental lower bound on regret. We resolve this gap by developing novel Bayesian optimization algorithms, based on kernel approximation techniques, with regret bounds matching the lower bound in order for the SE kernel. We numerically benchmark the algorithms on environments based on both synthetic models and real-world data sets.