  We examine a stochastic formulation for data-driven optimization wherein the
decision-maker is not privy to the true distribution, but has knowledge that it
lies in some hypothesis set and possesses a historical data set, from which
information about it can be gleaned. We define a prescriptive solution as a
decision rule mapping such a data set to decisions. As there does not exist
prescriptive solutions that are generalizable over the entire hypothesis set,
we define out-of-sample optimality as a local average over a neighbourhood of
hypotheses, and averaged over the sampling distribution. We prove sufficient
conditions for local out-of-sample optimality, which reduces to functions of
the sufficient statistic of the hypothesis family. We present an optimization
problem that would solve for such an out-of-sample optimal solution, and does
so efficiently by a combination of sampling and bisection search algorithms.
Finally, we illustrate our model on the newsvendor model, and find strong
performance when compared against alternatives in the literature. There are
potential implications of our research on end-to-end learning and Bayesian
optimization.
