  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 yield 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 a
key technical contribution -- regularizing kernel ridge estimators in
proportion to the smoothness of the underlying kernel $k$. Applying this key
idea together with a largely overlooked concentration result in separable
Hilbert spaces (for which we provide an independent, simplified derivation), we
are able to provide a tighter analysis of the GP-UCB algorithm.
