  We consider generic stochastic optimization problems in the presence of side
information which enables a more insightful decision. The side information
constitutes observable exogenous covariates that alter the conditional
probability distribution of the random problem parameters. A decision maker who
adapts her decisions according to the observed side information solves an
optimization problem where the objective function is specified by the
conditional expectation of the random cost. If the joint probability
distribution is unknown, then the conditional expectation can be approximated
in a data-driven manner using the Nadaraya-Watson (NW) kernel regression. While
the emerging approximation scheme has found successful applications in diverse
decision problems under uncertainty, it is largely unknown whether the scheme
can provide any reasonable out-of-sample performance guarantees. In this paper,
we establish guarantees for the generic problems by leveraging techniques from
moderate deviations theory. Our analysis motivates the use of a variance-based
regularization scheme which, in general, leads to a non-convex optimization
problem. We adopt ideas from distributionally robust optimization to obtain
tractable formulations. We present numerical experiments for newsvendor and
wind energy commitment problems to highlight the effectiveness of our
regularization scheme.
