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 sub-Gaussian 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 of the
well-known GP-UCB algorithm with reward truncation enjoys sublinear
$\tilde{O}(T^{\frac{2 + \alpha}{2(1+\alpha)}})$ regret even with only the
$(1+\alpha)$-th moments, $\alpha \in (0,1]$, of the reward distribution being
bounded ($\tilde{O}$ hides logarithmic factors). However, for the common
squared exponential (SE) and Mat\'{e}rn kernels, this is seen to be
significantly larger than a fundamental $\Omega(T^{\frac{1}{1+\alpha}})$ 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.

本文通过对高斯过程黑盒优化中噪音模型的研究，提出了一种可用于重尾噪音的、基于核函数的 UCB 算法以及基于核函数近似的贝叶斯优化算法，解决了现有算法在重尾噪音下表现较差的问题，并在人造和真实数据集上验证了算法性能。