We aim to optimize a black-box function $f:\mathcal{X} \mapsto \mathbb{R}$
under the assumption that $f$ is H\"older smooth and has bounded norm in the
RKHS associated with a given kernel $K$. This problem is known to have an
agnostic Gaussian Process (GP) bandit interpretation in which an appropriately
constructed GP surrogate model with kernel $K$ is used to obtain an upper
confidence bound (UCB) algorithm. In this paper, we propose a new algorithm
(\texttt{LP-GP-UCB}) where the usual GP surrogate model is augmented with Local
Polynomial (LP) estimators of the H\"older smooth function $f$ to construct a
multi-scale UCB guiding the search for the optimizer. We analyze this algorithm
and derive high probability bounds on its simple and cumulative regret. We then
prove that the elements of many common RKHS are H\"older smooth and obtain the
corresponding H\"older smoothness parameters, and hence, specialize our regret
bounds for several commonly used kernels. When specialized to the Squared
Exponential (SE) kernel, \texttt{LP-GP-UCB} matches the optimal performance,
while for the case of Mat\'ern kernels $(K_{\nu})_{\nu>0}$, it results in
uniformly tighter regret bounds for all values of the smoothness parameter
$\nu>0$. Most notably, for certain ranges of $\nu$, the algorithm achieves
near-optimal bounds on simple and cumulative regrets, matching the
algorithm-independent lower bounds up to polylog factors, and thus closing the
large gap between the existing upper and lower bounds for these values of
$\nu$. Additionally, our analysis provides the first explicit regret bounds, in
terms of the budget $n$, for the Rational-Quadratic (RQ) and Gamma-Exponential
(GE). Finally, experiments with synthetic functions as well as a CNN
hyperparameter tuning task demonstrate the practical benefits of our
multi-scale partitioning approach over some existing algorithms numerically.

提出一种新算法，通过在通用高斯过程代理模型中增加本地多项式估计函数 $f$ 来构建多尺度 UCB 以指导优化器的搜索，从而在预算 $n$ 的情况下获得理论上的累积失望次数界限。