Several recent developments in stochastic control and learning problems lead to solving measure-valued PDEs. Solutions for such PDEs can in turn be realized as gradient descent of suitable measure-valued functionals w.r.t. the Wasserstein metric related to the optimal mass transport. This opens up the possibility to numerically solve such PDEs via variational algorithms, and developing such algorithms has been a topic of burgeoning interest across many research communities.
Time-stepping via these algorithms realizes the Wasserstein proximal updates, which generalize the concept of gradient steps to the manifold of probability measures. In this talk, we will clarify this circle of ideas. We will then propose a new distributed algorithm performing the Wasserstein proximal updates. The proposed algorithm generalizes the finite dimensional Euclidean consensus ADMM to the measure-valued Wasserstein ADMM, and then to its entropy-regularized Sinkhorn variant. We will explain how the proposed algorithm differs compared to the Euclidean ADMM, and will provide numerical case-studies.
Abhishek Halder is an Associate Professor in the Department of Aerospace Engineering, and in the Translational AI Center at Iowa State University. He served as an Assistant Professor in the Department of Applied Mathematics, and an affiliated faculty in the Department of Electrical and Computer Engineering at University of California, Santa Cruz. Before that he held postdoctoral positions in the Department of Mechanical and Aerospace Engineering at University of California, Irvine, and in the Department of Electrical and Computer Engineering at Texas A&M University. He obtained his Bachelors and Masters from Indian Institute of Technology Kharagpur in 2008, and Ph.D. from Texas A&M University in 2014, all in Aerospace Engineering. His research interests are in stochastic systems, control and optimization with application focus on large scale cyber-physical systems. He is a co-founder of the annual NorCal Control Workshop that brings together systems-control researchers from academia and industry in the Northern California region fostering collaboration and professional networking. He is the creator and instructor for the course "Feedback Control" in the California State Summer School for Mathematics & Science (COSMOS) which teaches feedback control theory to 8-11 graders without using calculus or linear algebra. Abhishek is a Senior Member of IEEE.