Title: Lower Bounds on Regret for Noisy Gaussian Process Bandit Optimization,
Abstract: In this paper, we consider the problem of sequentially optimizing a black-box
function $f$ based on noisy samples and bandit feedback. We assume that $f$ is
smooth in the sense of having a bounded norm in some reproducing kernel Hilbert
space (RKHS), yielding a commonly-considered non-Bayesian form of Gaussian
process bandit optimization. We provide algorithm-independent lower bounds on
the simple regret, measuring the suboptimality of a single point reported after
$T$ rounds, and on the cumulative regret, measuring the sum of regrets over the
$T$ chosen points. For the isotropic squared-exponential kernel in $d$
dimensions, we find that an average simple regret of $\epsilon$ requires $T =
\Omega\big(\frac{1}{\epsilon^2} (\log\frac{1}{\epsilon})^{d/2}\big)$, and the
average cumulative regret is at least $\Omega\big( \sqrt{T(\log T)^{d/2}}
\big)$, thus matching existing upper bounds up to the replacement of $d/2$ by
$2d+O(1)$ in both cases. For the Matérn-$\nu$ kernel, we give analogous
bounds of the form $\Omega\big( (\frac{1}{\epsilon})^{2+d/\nu}\big)$ and
$\Omega\big( T^{\frac{\nu + d}{2\nu + d}} \big)$, and discuss the resulting
gaps to the existing upper bounds.