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

2026-04-09 · Source: stat.ML updates on arXiv.org · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Robotics & Autonomous Systems · Depth: Expert, extended

Summary

Asmaa Eldesoukey, Yongxin Chen, and Abhishek Halder introduce a generalized Sinkhorn algorithm to solve the mean-field Schrödinger bridge (MFSB) problem, which models minimum-effort control for large-scale multi-agent systems with nonlocal interactions. The MFSB problem is computationally challenging due to its nonconvex nature. The authors propose a generalization of the Hopf-Cole transform to derive a new Schrödinger system, upon which their Sinkhorn-type recursive algorithm is built. This algorithm addresses the associated system of integro-PDEs and is proven to have local convergence guarantees under mild assumptions on the interaction potential. Numerical examples demonstrate the algorithm's effectiveness for both repulsive and attractive interaction potentials, showing rapid convergence and high accuracy in steering initial probability distributions to target distributions.

Key takeaway

Research scientists working on optimal control for large-scale multi-agent systems should consider this generalized Sinkhorn algorithm. It provides a robust, numerically stable approach to solve the nonconvex mean-field Schrödinger bridge problem, offering local convergence guarantees. You can apply this method to design minimum-effort feedback control strategies for systems with complex interactions, such as those found in crowd dynamics or swarm robotics, by carefully modeling the interaction potential.

Key insights

A generalized Sinkhorn algorithm solves the nonconvex mean-field Schrödinger bridge problem for multi-agent system control.

Principles

• Nonlocal interactions introduce nonconvexity in mean-field control.
• Hopf-Cole transform can simplify complex PDE systems.
• Sinkhorn iterations offer robust numerical solutions for optimal transport problems.

Method

The method involves transforming the MFSB problem into a Schrödinger system using a generalized Hopf-Cole transform, then applying a nested fixed-point Sinkhorn-type algorithm to solve the resulting integro-PDEs, with damping for stability.

In practice

• Model crowd dynamics using MFSB with interaction potentials.
• Apply to population biology and swarm robotics control.
• Initialize with non-interacting SB solutions for better convergence.

Topics

• Mean-Field Schrödinger Bridge
• Generalized Sinkhorn Algorithm
• Hopf-Cole Transform
• Multi-Agent Systems
• Stochastic Optimal Control

Best for: Research Scientist, AI Scientist

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Editorial summary, takeaway, and curation by AIssential. Original article published by stat.ML updates on arXiv.org.