779 research outputs found
Lower Bounds on Regret for Noisy Gaussian Process Bandit Optimization
In this paper, we consider the problem of sequentially optimizing a black-box
function based on noisy samples and bandit feedback. We assume that 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
rounds, and on the cumulative regret, measuring the sum of regrets over the
chosen points. For the isotropic squared-exponential kernel in
dimensions, we find that an average simple regret of requires , and the
average cumulative regret is at least , thus matching existing upper bounds up to the replacement of by
in both cases. For the Mat\'ern- kernel, we give analogous
bounds of the form and
, and discuss the resulting
gaps to the existing upper bounds.Comment: Appearing in COLT 2017. This version corrects a few minor mistakes in
Table I, which summarizes the new and existing regret bound
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