# Multi-Scale Zero-Order Optimization of Smooth Functions in an RKHS

### Abstract

We aim to optimize a black-box function f : X→ R under the assumption that f is H¨older smooth and has bounded norm in the Reproducing Kernel Hilbert Space (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 (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 upper confidence bound 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 reproducing kernel Hilbert spaces are H¨older smooth and obtain the corresponding H¨older smoothness parameters, and hence, specialize our regret bounds for several commonly used and practically relevant kernels. When specialized to the Squared Exponential (SE) kernel, LP-GP-UCB matches the optimal performance, while for the case of Mat´ern kernels (Kν)ν>0, it results in uniformly tighter regret bounds for all values of the smoothness parameter ν > 0. Most notably, for certain ranges of ν, the algorithm achieves near-optimal bounds on simple and cumulative regrets, matching the algorithm-independent lower bounds up to poly-logarithmic factors, and thus closing the large gap between the existing upper and lower bounds for these values of ν. 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 Convolutional Neural Network hyperparameter tuning task demonstrate the practical benefits of our multi-scale partitioning approach over some existing algorithms numerically.

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