Lower Bounds on Regret for Noisy Gaussian Process Bandit Optimization
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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(1/ϵ^2 (1/ϵ)^d/2), and the average cumulative regret is at least Ω( √(T( T)^d)), 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( (1/ϵ)^2+d/ν) and Ω( T^ν + d/2ν + d), and discuss the resulting gaps to the existing upper bounds.
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