Summary: Metric Entropy and Gaussian Bandits
S. Gr¨unew¨alder
UCL
J.Y. Audibert
Universit´e ParisEst
M. Opper
TUBerlin
J. ShaweTaylor
UCL
Abstract
Metric entropy and generic chaining methods are powerful tools from probabil
ity theory that can be used to study pathwise properties of stochastic processes.
Despite this fact they have largely been ignored in machine learning. We demon
strate their power in this work in applying them to a bandit problem with a
Gaussian process prior. The difficulty of the setting lies in the fact that we are
dealing with a continuous space of arms and we need to control the supremum of
a reward process on the arms. We apply the so called Dudley integral to reduce
the problem of controlling the supremum of a "difficult" stochastic process to the
problem of bounding a canonical metric that is based solely on the covariance
function (which is an analytical and thus "simple" object). We consider the sce
