Jonathan Scarlett, Ilijia Bogunovic, Volkan Cevher
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 $T = Ω\big(\frac{1}{ε^2} (\log\frac{1}ε)^{d/2}\big)$, and the average cumulative regret is at least $Ω\big( \sqrt{T(\log T)^{d/2}} \big)$, thus matching existing upper bounds up to the replacement of $d/2$ by $2d+O(1)$ in both cases. For the Matérn-$ν$ kernel, we give analogous bounds of the form $Ω\big( (\frac{1}ε)^{2+d/ν}\big)$ and $Ω\big( T^{\frac{ν+ d}{2ν+ d}} \big)$, and discuss the resulting gaps to the existing upper bounds.