  In the recent years, a number of parameter-free algorithms have been
developed for online linear optimization over Hilbert spaces and for learning
with expert advice. These algorithms achieve optimal regret bounds that depend
on the unknown competitors, without having to tune the learning rates with
oracle choices.
  We present a new intuitive framework to design parameter-free algorithms for
\emph{both} online linear optimization over Hilbert spaces and for learning
with expert advice, based on reductions to betting on outcomes of adversarial
coins. We instantiate it using a betting algorithm based on the
Krichevsky-Trofimov estimator. The resulting algorithms are simple, with no
parameters to be tuned, and they improve or match previous results in terms of
regret guarantee and per-round complexity.
