A Generalized Sinkhorn Algorithm for Mean-Field Schr\"odinger Bridge

TL;DR

This paper introduces a generalized Sinkhorn algorithm for the mean-field Schrödinger bridge, enabling efficient computation for large-scale multi-agent systems with nonlocal interactions.

Contribution

It extends the Hopf-Cole transform and develops a Sinkhorn-type recursive algorithm with convergence guarantees for solving MFSB problems.

Findings

Algorithm successfully handles repulsive and attractive interactions.

Numerical examples demonstrate the effectiveness of the proposed method.

Convergence guarantees are established under mild assumptions.

Abstract

The mean-field Schr\"odinger 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 Schr\"odinger 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.