  Boosting is a widely used machine learning approach based on the idea of
aggregating weak learning rules. While in statistical learning numerous
boosting methods exist both in the realizable and agnostic settings, in online
learning they exist only in the realizable case. In this work we provide the
first agnostic online boosting algorithm; that is, given a weak learner with
only marginally-better-than-trivial regret guarantees, our algorithm boosts it
to a strong learner with sublinear regret.
  Our algorithm is based on an abstract (and simple) reduction to online convex
optimization, which efficiently converts an arbitrary online convex optimizer
to an online booster.
  Moreover, this reduction extends to the statistical as well as the online
realizable settings, thus unifying the 4 cases of statistical/online and
agnostic/realizable boosting.
