Sharper Guarantees for Misspecified Kernelized Bandit Optimization

TL;DR

This paper improves theoretical guarantees for misspecified kernelized bandit optimization by reducing the impact of misspecification to logarithmic or polylogarithmic growth through spectral localization techniques.

Contribution

It introduces new offline and online bounds that significantly lower the misspecification penalty for a broad class of kernels, using spectral localization and domain-splitting methods.

Findings

Offline simple-regret bounds with spectral Lebesgue constant control.

Logarithmic amplification for 1D monotone spectra.

Polylogarithmic amplification for multivariate Fourier-diagonal kernels.

Abstract

Existing guarantees for misspecified kernelized bandit optimization pay for misspecification through kernel complexity: in generic offline bounds, the misspecification level $\varepsilon$ is multiplied by $\sqrt{d_\mathrm{eff}}$, where $d_\mathrm{eff}$ is the kernel effective dimension, while in online regret bounds, the corresponding penalty is $\sqrt{\gamma_n}\,n\varepsilon$, where $\gamma_n$ is the maximum information gain after $n$ rounds of interaction. In this work, we show that, for a large class of kernels, the misspecification amplification can be reduced to logarithmic or polylogarithmic growth. In the offline setting, we first prove high-probability simple-regret bounds whose misspecification term is governed by a spectral Lebesgue constant. This yields logarithmic amplification for one-dimensional monotone spectra and polylogarithmic amplification for multivariate Fourier-diagonal product kernels. In the online setting, we modify a domain-splitting algorithm and prove a cumulative regret bound of $\widetilde{\mathcal O}(\sqrt{\gamma_n n}+n\varepsilon)$ under mild localized eigendecay assumptions, removing the extra $\sqrt{\gamma_n}$ factor from the misspecification term. The common principle is localization: spectral localization controls the Lebesgue constant of the offline approximation operator, while domain splitting implements the spatial analogue of this mechanism in the online setting, preventing local misspecification errors from being amplified globally.