Steering Large Agent Populations using Mean-Field Schrodinger Bridges with Gaussian Mixture Models

2025-03-31

AI Summary

This paper addresses the minimum-cost distributed control problem for large-scale homogeneous multi-agent systems, aiming to steer an initial probability distribution to a target Gaussian mixture distribution under linear time-varying stochastic dynamics. We propose the first closed-form, training-free Gaussian mixture mean-field Schrödinger bridge (MFSB) framework, which decomposes the original problem into analytically solvable Gaussian-to-Gaussian covariance steering subproblems and enforces probabilistic hard constraints via semidefinite programming. Our approach avoids both grid-based discretization and neural network training, ensuring theoretical interpretability and computational efficiency. Experiments demonstrate that the proposed framework significantly outperforms existing grid-based and neural-network-based methods in distributional accuracy, real-time feasibility, and constraint satisfaction—making it particularly suitable for large-scale collaborative multi-agent control.

Technology Category

• Multiagent Systems: Distributed Problem Solving
• Search and Optimization: Mixed Discrete/Continuous Search
• Planning, Routing, and Scheduling: Mixed Discrete/Continuous Planning

Application Category

• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

The Mean-Field Schrodinger Bridge (MFSB) problem is an optimization problem aiming to find the minimum effort control policy to drive a McKean-Vlassov stochastic differential equation from one probability measure to another. In the context of multiagent control, the objective is to control the configuration of a swarm of identical, interacting cooperative agents, as captured by the time-varying probability measure of their state. Available methods for solving this problem for distributions with continuous support rely either on spatial discretizations of the problem's domain or on approximating optimal solutions using neural networks trained through stochastic optimization schemes. For agents following Linear Time-Varying dynamics, and for Gaussian Mixture Model boundary distributions, we propose a highly efficient parameterization to approximate the solutions of the corresponding MFSB in closed form, without any learning steps. Our proposed approach consists of a mixture of elementary policies, each solving a Gaussian-to-Gaussian Covariance Steering problem from the components of the initial to the components of the terminal mixture. Leveraging the semidefinite formulation of the Covariance Steering problem, our proposed solver can handle probabilistic hard constraints on the system's state, while maintaining numerical tractability. We illustrate our approach on a variety of numerical examples.

Problem

• Control large agent populations using Mean-Field Schrodinger Bridges
• Solve MFSB for Gaussian Mixture Models without learning steps
• Handle probabilistic constraints while maintaining numerical tractability

Innovation

• Closed-form MFSB solution with Gaussian Mixture Models
• Mixture of elementary Gaussian-to-Gaussian policies
• Semidefinite formulation for constraint handling