In this paper, we consider the problem of black-box optimization using Gaussian Process (GP) bandit optimization with a small number of batches. Assuming the unknown function has a low norm in the Reproducing Kernel Hilbert Space (RKHS), we introduce a batch algorithm inspired by batched finite-arm bandit algorithms, and show that it achieves the cumulative regret upper bound $O^\ast(\sqrt{T\gamma_T})$ using $O(\log\log T)$ batches within time horizon $T$, where the $O^\ast(\cdot)$ notation hides dimension-independent logarithmic factors and $\gamma_T$ is the maximum information gain associated with the kernel. This bound is near-optimal for several kernels of interest and improves on the typical $O^\ast(\sqrt{T}\gamma_T)$ bound, and our approach is arguably the simplest among algorithms attaining this improvement. In addition, in the case of a constant number of batches (not depending on $T$), we propose a modified version of our algorithm, and characterize how the regret is impacted by the number of batches, focusing on the squared exponential and Mat\'ern kernels. The algorithmic upper bounds are shown to be nearly minimax optimal via analogous algorithm-independent lower bounds.

翻译：在本文中, 我们考虑使用 Gaussian 进程( GP) 土匪优化来优化黑盒问题。 如果在复制 Kernel Hilbert 空间( RKHS) 中, 未知函数的常态值较低, 我们引入了由批量有限武器土匪算法启发的批量算法, 并展示它利用美元( log\log\logT) 在时间范围内直系交易的批次在一定范围内用美元T$( log\log T) 来实现累积的遗憾 $O ast( log\log\log T) 优化。 美元( codt) $( count (\ codt) $( $\ gamma_ T) 在时间范围内, 隐藏维度独立的对调调调调调调调调调调调调调调调调调时, $\ gammamama_ T$( legle) legal comendal rubes) 的算算出一个不变的版本。