  Stochastic optimization is one of the central problems in Machine Learning
and Theoretical Computer Science. In the standard model, the algorithm is given
a fixed distribution known in advance. In practice though, one may acquire at a
cost extra information to make better decisions. In this paper, we study how to
buy information for stochastic optimization and formulate this question as an
online learning problem. Assuming the learner has an oracle for the original
optimization problem, we design a $2$-competitive deterministic algorithm and a
$e/(e-1)$-competitive randomized algorithm for buying information. We show that
this ratio is tight as the problem is equivalent to a robust generalization of
the ski-rental problem, which we call super-martingale stopping.
  We also consider an adaptive setting where the learner can choose to buy
information after taking some actions for the underlying optimization problem.
We focus on the classic optimization problem, Min-Sum Set Cover, where the goal
is to quickly find an action that covers a given request drawn from a known
distribution. We provide an $8$-competitive algorithm running in polynomial
time that chooses actions and decides when to buy information about the
underlying request.
