A Generalized Sinkhorn Algorithm for Mean-Field Schrödinger Bridge

2026-04-07

AI Summary

This work addresses the minimum-control problem of steering a diffusion process with nonlocal interactions from an initial distribution to a target distribution within a fixed time horizon—a task rendered challenging by inherent nonconvexity. The authors propose, for the first time, an extension of the Sinkhorn algorithmic framework to the mean-field Schrödinger bridge problem incorporating nonlocal interactions. By generalizing the Hopf–Cole transformation, they construct a Sinkhorn-type iterative scheme to solve the associated system of integro-partial differential equations. Under mild conditions, the method enjoys provable convergence guarantees and demonstrates strong empirical performance in numerical experiments involving both repulsive and attractive interaction kernels, effectively achieving precise terminal distribution control in large-scale multi-agent systems.

Abstract

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.

Problem

• Mean-Field Schrödinger Bridge
• nonlocal interaction
• diffusion process
• nonconvex optimization
• multi-agent systems

Innovation

• Mean-Field Schrödinger Bridge
• Generalized Hopf-Cole Transform
• Sinkhorn Algorithm
• Integro-PDEs
• Nonlocal Interaction