A Generalized Sinkhorn Algorithm for Mean-Field Schrödinger Bridge
This work addresses the problem of designing controllers for large-scale multi-agent systems, representing an incremental advancement in solving nonconvex mean-field interactions.
The paper tackles the computationally challenging mean-field Schrödinger bridge problem, which involves controlling a diffusion process with nonlocal interactions, by proposing a generalized Hopf-Cole transform and a Sinkhorn-type algorithm, showing convergence guarantees under mild assumptions and illustrating results with numerical examples.
The mean-field Schrödinger bridge (MFSB) problem concerns designing a minimum-effort controller that guides a diffusion process with nonlocal interaction to reach a given distribution from another by a fixed deadline. Unlike the standard Schrödinger bridge, the dynamical constraint for MFSB is the mean-field limit of a population of interacting agents with controls. It serves as a natural model for large-scale multi-agent systems. The MFSB is computationally challenging because the nonlocal interaction makes the problem nonconvex. We propose a generalization of the Hopf-Cole transform for MFSB and, building on it, design a Sinkhorn-type recursive algorithm to solve the associated system of integro-PDEs. Under mild assumptions on the interaction potential, we discuss convergence guarantees for the proposed algorithm. We present numerical examples with repulsive and attractive interactions to illustrate the theoretical contributions.