Consequences of Kernel Regularity for Bandit Optimization

2025-12-05

AI Summary

This paper investigates how the regularity of kernel functions affects the performance of RKHS-based functions in Bayesian optimization (bandit setting). We propose a unified analytical framework grounded in the Fourier spectral decay of isotropic kernels, establishing—for the first time—a systematic connection between global kernel regression and local smoothness methods (e.g., local polynomial estimation) in regret analysis. Leveraging tools including Fourier analysis, maximal information gain characterization, and embeddings into Hölder/Besov spaces, we derive tight asymptotic regret upper bounds for Matérn and squared-exponential kernels—some of which are novel. Furthermore, we design LP-GP-UCB, an algorithm that integrates Gaussian process priors with local polynomial estimation, achieving order-optimal regret bounds across multiple kernel classes. This approach significantly enhances optimization efficiency and theoretical interpretability for non-stationary objective functions.

Technology Category

• Machine Learning: Kernel Methods
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Learning to Search

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

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.

Problem

• Investigates kernel regularity's impact on bandit optimization performance.
• Connects kernel spectral decay to regret bounds and smoothness analysis.
• Derives explicit regret bounds for various kernel families and hybrid algorithms.

Innovation

• Unified framework connects kernel spectral decay with regret bounds
• Hybrid algorithm LP-GP-UCB combines global Gaussian processes with local polynomials
• Spectral analysis of kernels determines asymptotic regret for both global and local methods