Consequences of Kernel Regularity for Bandit Optimization

TL;DR

The authors address bandit optimization for functions in RKHS by revealing how kernel regularity, captured through Fourier spectra, governs both global information gains and local smoothness. They develop a unified framework that connects kernel-based GP-UCB analyses with local Hölder/Besov-based methods, deriving explicit spectral-decay-based regret bounds for a range of isotropic kernels. A central contribution is the LP-GP-UCB algorithm, which blends global GP surrogates with local polynomial estimators to achieve order-optimal performance across multiple kernel families. The results provide new Besov-equivalence insights, extend known bounds to γ-exponential and piecewise-polynomial kernels, and offer a robust hybrid approach that adapts to varying smoothness levels in practice.

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 Matérn, square-exponential, rational-quadratic, $γ$-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 Hölder 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.

Theorems & Definitions (8)

• Proposition 1: Spectral decay of specific isotropic kernels
• Remark 1
• Proposition 2: Information Gain Bounds for Kernels with Decaying Spectrum
• Proposition 3: Hölder Embeddings for Kernels with Sufficient Spectral Decay
• Proposition 4: Besov Equivalence for Kernels with Polynomial Spectral Decay
• Theorem 1: Specific Regret Bounds for LP-GP-UCB Algorithm
• proof
• Remark 2