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

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) reward model whose mean is a non-linear function $\mu(f^*)$. This setting extends both kernelized bandits (KBs) and generalized linear bandits (GLBs), providing a unified view of both settings. We propose an optimistic regret minimization algorithm, GKB-UCB, and we explain why existing self-normalized concentration inequalities used for KBs and GLBs do not allow to provide tight regret guarantees. For this reason, we devise a novel self-normalized Bernstein-like dimension-free inequality that applies to a Hilbert space of functions with bounded norm, representing a contribution of independent interest. Based on it, we analyze 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 of the expected reward non-linearity. Our result is tight in its dependence on $T$, $\gamma_T$, and $\kappa_*$ for both KBs and GLBs. Finally, we present a tractable version GKB-UCB, Trac-GKB-UCB, which attains similar regret guarantees, and we discuss its time and space complexity.