On the Interplay of Optimal Transport and Distributionally Robust Optimization

Optimal Transport (OT) seeks the most efficient way to morph one probability distribution into another
one, and Distributionally Robust Optimization (DRO) studies worst-case risk minimization problems under
distributional ambiguity. It is well known that OT gives rise to a rich class of data-driven DRO models,
where the decision-maker plays a zero-sum game against nature who can adversely reshape the empirical
distribution of the uncertain problem parameters within a prescribed transportation budget. Even though
generic OT problems are computationally hard, the Nash strategies of the decision-maker and nature in
OT-based DRO problems can often be computed efficiently. In this talk we will uncover deep connections
between robustification and regularization, and we will disclose striking properties of nature’s Nash
strategy, which implicitly constructs an adversarial training dataset. Finally, we will argue that, while OT
is useful for DRO, ideas from DRO can also help us to solve challenging OT problems.

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