Lower Bounds on Regret for Noisy Gaussian Process Bandit Optimization

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Jonathan Scarlett, Ilija Bogunovic, Volkan Cevher ;
Proceedings of the 2017 Conference on Learning Theory, PMLR 65:1723-1742, 2017.

Abstract

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 $T = Ω\big(\frac1ε^2 (\log\frac1ε)^d/2\big)$, and the average cumulative regret is at least $Ω\big( \sqrt{T}(\log T)^d \big)$, 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 $Ω\big( (\frac1ε)^2+d/ν\big)$ and $Ω\big( T^\fracν+ d2ν+ d \big)$, and discuss the resulting gaps to the existing upper bounds.

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