Lower Bounds on Regret for Noisy Gaussian Process Bandit Optimization

摘要

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 $T = Ωig(rac1ε^2 (lograc1ε)^d/2ig)$, and the average cumulative regret is at least $Ωig( sqrt{T}(log T)^d ig)$, thus matching existing upper bounds up to the replacement of $d/2$ by $d+O(1)$ in both cases. For the Matérn-$ν$ kernel, we give analogous bounds of the form $Ωig( (rac1ε)^2+d/νig)$ and $Ωig( T^racν+ d2ν+ d ig)$, and discuss the resulting gaps to the existing upper bounds.