  We present a computationally efficient framework, called $\texttt{FlowDRO}$,
for solving flow-based distributionally robust optimization (DRO) problems with
Wasserstein uncertainty sets while aiming to find continuous worst-case
distribution (also called the Least Favorable Distribution, LFD) and sample
from it. The requirement for LFD to be continuous is so that the algorithm can
be scalable to problems with larger sample sizes and achieve better
generalization capability for the induced robust algorithms. To tackle the
computationally challenging infinitely dimensional optimization problem, we
leverage flow-based models and continuous-time invertible transport maps
between the data distribution and the target distribution and develop a
Wasserstein proximal gradient flow type algorithm. In theory, we establish the
equivalence of the solution by optimal transport map to the original
formulation, as well as the dual form of the problem through Wasserstein
calculus and Brenier theorem. In practice, we parameterize the transport maps
by a sequence of neural networks progressively trained in blocks by gradient
descent. We demonstrate its usage in adversarial learning, distributionally
robust hypothesis testing, and a new mechanism for data-driven distribution
perturbation differential privacy, where the proposed method gives strong
empirical performance on high-dimensional real data.
