Lower Bounds on Regret for Noisy Gaussian Process Bandit Optimization · arXivDesk
1706.00090May 31, 2017Appearing in COLT 2017. This version corrects a few minor mistakes in Table I, which summarizes the new and existing regret bounds
Lower Bounds on Regret for Noisy Gaussian Process Bandit Optimization
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
Nearby in the stack
dimensions, we find that an average simple regret of
ε
requires
T=Ω(ε21(logε1)d/2)
, and the average cumulative regret is at least
Ω(T(logT)d/2)
, 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((ε1)2+d/ν)
and
Ω(T2ν+dν+d)
, and discuss the resulting gaps to the existing upper bounds.