  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.
