Frequentist Regret Analysis of Gaussian Process Thompson Sampling via Fractional Posteriors

2026-02-16

AI Summary

This work addresses the lack of a discretization-free frequentist regret analysis for Gaussian process Thompson sampling (GP-TS) in continuous compact action spaces. The authors propose a unified analytical framework based on fractional posteriors, interpreting variance inflation as sampling from a temperature-parameterized fractional posterior, thereby circumventing the need for action space discretization. By introducing the fractional Gaussian process posterior, information gain γₜ, and posterior contraction rate εₜ, and leveraging kernel-agnostic techniques, they derive discretization-free regret bounds applicable to the squared exponential, Matérn-ν, and rational quadratic kernels. These bounds achieve Õ(T^{1/2}) and Õ(T^{(2ν+3d)/(2(2ν+d))}) rates, respectively, significantly broadening the theoretical applicability of GP-TS.

Technology Category

• Reasoning under Uncertainty: Sequential Decision Making
• Machine Learning: Calibration & Uncertainty Quantification
• Search and Optimization: Mixed Discrete/Continuous 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
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We study Gaussian Process Thompson Sampling (GP-TS) for sequential decision-making over compact, continuous action spaces and provide a frequentist regret analysis based on fractional Gaussian process posteriors, without relying on domain discretization as in prior work. We show that the variance inflation commonly assumed in existing analyses of GP-TS can be interpreted as Thompson Sampling with respect to a fractional posterior with tempering parameter $\alpha \in (0,1)$. We derive a kernel-agnostic regret bound expressed in terms of the information gain parameter $\gamma_t$ and the posterior contraction rate $\epsilon_t$, and identify conditions on the Gaussian process prior under which $\epsilon_t$ can be controlled. As special cases of our general bound, we recover regret of order $\tilde{\mathcal{O}}(T^{\frac{1}{2}})$ for the squared exponential kernel, $\tilde{\mathcal{O}}(T^{\frac{2\nu+3d}{2(2\nu+d)}} )$ for the Mat\'ern-$\nu$ kernel, and a bound of order $\tilde{\mathcal{O}}(T^{\frac{2\nu+3d}{2(2\nu+d)}})$ for the rational quadratic kernel. Overall, our analysis provides a unified and discretization-free regret framework for GP-TS that applies broadly across kernel classes.

Problem

• Gaussian Process Thompson Sampling
• frequentist regret analysis
• continuous action spaces
• fractional posteriors
• kernel-agnostic regret

Innovation

• Gaussian Process Thompson Sampling
• fractional posterior
• frequentist regret analysis
• kernel-agnostic bound
• posterior contraction rate