A Novel Self-Normalized Bernstein-Like Dimension-Free Inequality and Regret Bounds for Generalized Kernelized Bandits

Alberto Maria Metelli, Simone Drago, Marco Mussi

Keywords: Self-normalized inequality, concentration inequality, regret bounds, generalized linear bandits, kernelized bandits

Abstract: We study the regret minimization problem in the novel setting of *generalized kernelized bandits* (GKBs), where we optimize an unknown function $f\^{\*}$ belonging to a *reproducing kernel Hilbert space* (RKHS) having access to samples generated by an *exponential family* (EF) noise model whose mean is a non-linear function $\\mu(f^{\*})$. This model extends both *kernelized bandits* (KBs) and *generalized linear bandits* (GLBs). We propose an optimistic algorithm, GKB-UCB, and we explain why existing self-normalized concentration inequalities do not allow to provide tight regret guarantees. For this reason, we devise a novel self-normalized Bernstein-like dimension-free inequality resorting to Freedman's inequality and a stitching argument, which represents a contribution of independent interest. Based on it, we conduct a regret analysis of GKB-UCB, deriving a regret bound of order $\widetilde{O}( \gamma\_T \sqrt{T/\kappa\_{\*}})$, being $T$ the learning horizon, ${\gamma}\_T$ the maximal information gain, and $\kappa\_{\*}$ a term characterizing the magnitude the reward nonlinearity. Our result matches, up to multiplicative constants and logarithmic terms, the state-of-the-art bounds for both KBs and GLBs and provides a *unified view* of both settings.

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Track: Regular Track: unpublished work

Submission Number: 102