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

arXiv: https://arxiv.org/abs/2005.04832 · HTML: https://mlsanity.com/paper/2005.04832/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 May 2020.

## Built on by (most cited first, 16 of 16)

- [Gaussian Process Bandit Optimization with Few Batches](https://mlsanity.com/paper/2110.07788/) (2021)
- [Optimal Order Simple Regret for Gaussian Process Bandits](https://mlsanity.com/paper/2108.09262/) (2021)
- [On the Sublinear Regret of GP-UCB](https://mlsanity.com/paper/2307.07539/) (2023)
- [Improved Regret Analysis in Gaussian Process Bandits: Optimality for Noiseless Reward, RKHS norm, and Non-Stationary Variance](https://mlsanity.com/paper/2502.06363/) (2025)
- [Open Problem: Tight Online Confidence Intervals for RKHS Elements](https://mlsanity.com/paper/2110.15458/) (2021)
- [Lenient Regret and Good-Action Identification in Gaussian Process Bandits](https://mlsanity.com/paper/2102.05793/) (2021)
- [Ada-BKB: Scalable Gaussian Process Optimization on Continuous Domains by Adaptive Discretization](https://mlsanity.com/paper/2106.08598/) (2021)
- [Instance-Dependent Regret Analysis of Kernelized Bandits](https://mlsanity.com/paper/2203.06297/) (2022)
- [Adaptation to Misspecified Kernel Regularity in Kernelised Bandits](https://mlsanity.com/paper/2304.13830/) (2023)
- [On Average-Case Error Bounds for Kernel-Based Bayesian Quadrature](https://mlsanity.com/paper/2202.10615/) (2022)
- [Regret Bounds for Noise-Free Cascaded Kernelized Bandits](https://mlsanity.com/paper/2211.05430/) (2022)
- [Differentially Private Kernelized Contextual Bandits](https://mlsanity.com/paper/2501.07046/) (2025)
- [Batched Kernelized Bandits: Refinements and Extensions](https://mlsanity.com/paper/2603.12627/) (2026)
- [Consequences of Kernel Regularity for Bandit Optimization](https://mlsanity.com/paper/2512.05957/) (2025)
- [No-Regret Algorithms for Safe Bayesian Optimization with Monotonicity Constraints](https://mlsanity.com/paper/2406.03264/) (2024)
- [Nearly-Optimal Algorithm for Adversarial Kernelized Bandits](https://mlsanity.com/paper/2605.10299/) (2026)

## Abstract (arXiv)

We aim to optimize a black-box function f:X mapsto 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 (textttLP-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, textttLP-GP-UCB matches the optimal performance, while for the case of Mat'ern kernels (Knu)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.