  Numerous machine learning problems require an exploration basis - a mechanism
to explore the action space. We define a novel geometric notion of exploration
basis with low variance, called volumetric spanners, and give efficient
algorithms to construct such a basis.
  We show how efficient volumetric spanners give rise to the first efficient
and optimal regret algorithm for bandit linear optimization over general convex
sets. Previously such results were known only for specific convex sets, or
under special conditions such as the existence of an efficient self-concordant
barrier for the underlying set.
