Lower Bounds on Regret for Noisy Gaussian Process Bandit Optimization

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 $ϵ$ requires $T = \Omega\big(\frac{1}{\epsilon^2} (\log\frac{1}{\epsilon})^{d/2}\big)$, and the average cumulative regret is at least $\Omega\big( \sqrt{T(\log T)^{d/2}} \big)$,thusmatchingexistingupperboundsuptothereplacementof$d/2$by$2d+O(1)$inbothcases\mathrm{.}FortheMat\acute{e}rn-$\nu$ kernel, we give analogous bounds of the form $\mathrm{\Omega }((\frac{1}{ϵ}{)}^{2+d\mathrm{/}\nu })$ and $\mathrm{\Omega }\left(T^{\frac{\nu +d}{2\nu +d}}\right)$, and discuss the resulting gaps to the existing upper bounds.