Consequences of Kernel Regularity for Bandit Optimization

TL;DR

This paper explores how the spectral decay of various kernels influences the performance of bandit optimization algorithms, unifying global and local approaches through spectral analysis.

Contribution

It characterizes the spectral properties of multiple kernels and links these to regret bounds, providing a unified framework for analyzing kernel and local methods in bandit optimization.

Findings

Spectral decay determines asymptotic regret bounds.

Unified analysis framework for kernel and local methods.

Order-optimal hybrid algorithm combining global and local approaches.

Abstract

In this work we investigate the relationship between kernel regularity and algorithmic performance in the bandit optimization of RKHS functions. While reproducing kernel Hilbert space (RKHS) methods traditionally rely on global kernel regressors, it is also common to use a smoothness-based approach that exploits local approximations. We show that these perspectives are deeply connected through the spectral properties of isotropic kernels. In particular, we characterize the Fourier spectra of the Mat\'ern, square-exponential, rational-quadratic, $\gamma$-exponential, piecewise-polynomial, and Dirichlet kernels, and show that the decay rate determines asymptotic regret from both viewpoints. For kernelized bandit algorithms, spectral decay yields upper bounds on the maximum information gain, governing worst-case regret, while for smoothness-based methods, the same decay rates establish H\"older space embeddings and Besov space norm-equivalences, enabling local continuity analysis. These connections show that kernel-based and locally adaptive algorithms can be analyzed within a unified framework. This allows us to derive explicit regret bounds for each kernel family, obtaining novel results in several cases and providing improved analysis for others. Furthermore, we analyze LP-GP-UCB, an algorithm that combines both approaches, augmenting global Gaussian process surrogates with local polynomial estimators. While the hybrid approach does not uniformly dominate specialized methods, it achieves order-optimality across multiple kernel families.