Improved Self-Normalized Concentration in Hilbert Spaces: Sublinear Regret for GP-UCB

In the kernelized bandit problem, a learner aims to sequentially compute the optimum of a function lying in a reproducing kernel Hilbert space given only noisy evaluations at sequentially chosen points. In particular, the learner aims to minimize regret, which is a measure of the suboptimality of the choices made. Arguably the most popular algorithm is the Gaussian Process Upper Confidence Bound (GP-UCB) algorithm, which involves acting based on a simple linear estimator of the unknown function. Despite its popularity, existing analyses of GP-UCB give a suboptimal regret rate, which fails to be sublinear for many commonly used kernels such as the Mat\'ern kernel. This has led to a longstanding open question: are existing regret analyses for GP-UCB tight, or can bounds be improved by using more sophisticated analytical techniques? In this work, we resolve this open question and show that GP-UCB enjoys nearly optimal regret. In particular, our results directly imply sublinear regret rates for the Mat\'ern kernel, improving over the state-of-the-art analyses and partially resolving a COLT open problem posed by Vakili et al. Our improvements rely on two key technical results. First, we use modern supermartingale techniques to construct a novel, self-normalized concentration inequality that greatly simplifies existing approaches. Second, we address the importance of regularizing in proportion to the smoothness of the underlying kernel $k$. Together, these new technical tools enable a simplified, tighter analysis of the GP-UCB algorithm.