  This paper studies a class of multiagent stochastic optimization problems
where the objective is to minimize the expected value of a function which
depends on a random variable. The probability distribution of the random
variable is unknown to the agents, so each one gathers samples of it. The
agents aim to cooperatively find, using their data, a solution to the
optimization problem with guaranteed out-of-sample performance. The approach is
to formulate a data-driven distributionally robust optimization problem using
Wasserstein ambiguity sets, which turns out to be equivalent to a convex
program. We reformulate the latter as a distributed optimization problem and
identify a convex-concave augmented Lagrangian function whose saddle points are
in correspondence with the optimizers provided a min-max interchangeability
criteria is met. Our distributed algorithm design then consists of the
saddle-point dynamics associated to the augmented Lagrangian. We formally
establish that the trajectories of the dynamics converge asymptotically to a
saddle point and hence an optimizer of the problem. Finally, we provide a class
of functions that meet the min-max interchangeability criteria. Simulations
illustrate our results.
