The mathematics of distributionally robust optimization: a PDE and gradient flow perspective

Distributionally robust optimization (DRO), and its applications to control and machine learning, have become impactful research topics. This talk presents two recent works in the use of PDEs and gradient flows for sampling and inference, which contribute to a new perspective in DRO. I will introduce the concept of gradient flows in the space of probability measures, highlighting their role as powerful tools in modern analysis and statistical inference, especially tailored for DRO. Then, I will present two works that 1) use PDE gradient flows as a mathematically principled framework for solving general DRO problems, and 2) study the Gaussian gradient flows in the Wasserstein-Fisher-Rao space, which is the foundation for recent distributionally robust linear quadratic control methods.