Lower Bounds on Regret for Noisy Gaussian Process Bandit Optimization
This work addresses theoretical gaps in optimization algorithms for researchers in machine learning and statistics, though it is incremental as it refines known bounds.
The paper tackles the problem of noisy Gaussian process bandit optimization by providing algorithm-independent lower bounds on simple and cumulative regret for isotropic squared-exponential and Matérn kernels, matching existing upper bounds up to logarithmic factors in dimensions.
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 $T = Ω\big(\frac{1}{ε^2} (\log\frac{1}ε)^{d/2}\big)$, and the average cumulative regret is at least $Ω\big( \sqrt{T(\log T)^{d/2}} \big)$, 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 $Ω\big( (\frac{1}ε)^{2+d/ν}\big)$ and $Ω\big( T^{\frac{ν+ d}{2ν+ d}} \big)$, and discuss the resulting gaps to the existing upper bounds.