  We investigate a simple approximation scheme, based on overlapping linear
decision rules, for solving data-driven two-stage distributionally robust
optimization problems with the type-$\infty$ Wasserstein ambiguity set. Our
main result establishes that this approximation scheme is asymptotically
optimal for two-stage stochastic linear optimization problems; that is, under
mild assumptions, the optimal cost and optimal first-stage decisions obtained
by approximating the robust optimization problem converge to those of the
underlying stochastic problem as the number of data points grows to infinity.
These guarantees notably apply to two-stage stochastic problems that do not
have relatively complete recourse, which arise frequently in applications. In
this context, we show through numerical experiments that the approximation
scheme is practically tractable and produces decisions which significantly
outperform those obtained from state-of-the-art data-driven alternatives.
