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Optimal Transport-Based Decentralized Multi-Agent Distribution Matching

Kooktae Lee

TL;DR

This work tackles terminal distribution matching for a finite multi-agent system by formulating the objective with the squared 2-Wasserstein distance $W_2^2(\mu,\nu)$ and developing a decentralized control framework built on optimal transport. To overcome the intractability of global OT, it introduces a two-phase cycle with sequential local target assignment and parallel trajectory updates, supplemented by memory-based corrections to cope with intermittent communication. The authors provide cycle-level convergence guarantees via a surrogate transport cost and show that the true Wasserstein distance remains controlled under centralized or memory-enabled decentralized implementations. Simulation results for both linear and nonlinear dynamics demonstrate scalable, robust distribution matching under realistic communication constraints. This work tightly couples OT-based distribution modeling with decentralized feedback control for finite-agent systems and offers a practical path to precise terminal configurations.

Abstract

This paper presents a decentralized control framework for distribution matching in multi-agent systems (MAS), where agents collectively achieve a prescribed terminal spatial distribution. The problem is formulated using optimal transport (Wasserstein distance), which provides a principled measure of distributional discrepancy and serves as the basis for the control design. To avoid solving the global optimal transport problem directly, the distribution-matching objective is reformulated into a tractable per-agent decision process, enabling each agent to identify its desired terminal locations using only locally available information. A sequential weight-update rule is introduced to construct feasible local transport plans, and a memory-based correction mechanism is incorporated to maintain reliable operation under intermittent and range-limited communication. Convergence guarantees are established, showing cycle-wise improvement of a surrogate transport cost under both linear and nonlinear agent dynamics. Simulation results demonstrate that the proposed framework achieves effective and scalable distribution matching while operating fully in a decentralized manner.

Optimal Transport-Based Decentralized Multi-Agent Distribution Matching

TL;DR

This work tackles terminal distribution matching for a finite multi-agent system by formulating the objective with the squared 2-Wasserstein distance and developing a decentralized control framework built on optimal transport. To overcome the intractability of global OT, it introduces a two-phase cycle with sequential local target assignment and parallel trajectory updates, supplemented by memory-based corrections to cope with intermittent communication. The authors provide cycle-level convergence guarantees via a surrogate transport cost and show that the true Wasserstein distance remains controlled under centralized or memory-enabled decentralized implementations. Simulation results for both linear and nonlinear dynamics demonstrate scalable, robust distribution matching under realistic communication constraints. This work tightly couples OT-based distribution modeling with decentralized feedback control for finite-agent systems and offers a practical path to precise terminal configurations.

Abstract

This paper presents a decentralized control framework for distribution matching in multi-agent systems (MAS), where agents collectively achieve a prescribed terminal spatial distribution. The problem is formulated using optimal transport (Wasserstein distance), which provides a principled measure of distributional discrepancy and serves as the basis for the control design. To avoid solving the global optimal transport problem directly, the distribution-matching objective is reformulated into a tractable per-agent decision process, enabling each agent to identify its desired terminal locations using only locally available information. A sequential weight-update rule is introduced to construct feasible local transport plans, and a memory-based correction mechanism is incorporated to maintain reliable operation under intermittent and range-limited communication. Convergence guarantees are established, showing cycle-wise improvement of a surrogate transport cost under both linear and nonlinear agent dynamics. Simulation results demonstrate that the proposed framework achieves effective and scalable distribution matching while operating fully in a decentralized manner.
Paper Structure (18 sections, 5 theorems, 29 equations, 3 figures)

This paper contains 18 sections, 5 theorems, 29 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Approach: Reducing Wasserstein Distance via Sequential Local Updates
  4. Sample Selection Phase: Sequential Target Assignment
  5. Control Execution Phase: Parallel Trajectory Update
  6. Wasserstein Distance Convergence Mapping via Optimal Control
  7. Linear Time-Invariant System Case
  8. Control-Affine Nonlinear System Case
  9. Fully Decentralized Implementation via Local Memory
  10. Sequential Update Scheme in a Decentralized Setup
  11. Local Memory Construction
  12. Weight Update with Memory Decay
  13. Cycle-Level Convergence Guarantees
  14. Simulations
  15. LTI System Case
  16. ...and 3 more sections

Key Result

Lemma 1

Consider the weighted least-squares problem where $\pi_j \ge 0$ and $y^* := (\sum_j \pi_j y_j)/(\sum_j \pi_j)$ is the weighted average target. If the equation $Ax + Bu = y^*$ is feasible, then every solution of this equation achieves the minimum value of the weighted least-squares cost. Among these minimizers, the minimum-norm one is obtained This is a standard result for quadratic objectives wit

Figures (3)

  • Figure 1: Distribution matching for the LTI system
  • Figure 2: Distribution matching for the unicycle model.
  • Figure 3: Cycle-wise evolution of $\sqrt{\Psi^{(\ell)}(k_{\ell})}$, $\sqrt{\Psi^{(\ell)}(k_{\ell+1})}$, and $\mathcal{W}_2(\mu_{k_{\ell+1}},\nu)$.

Theorems & Definitions (13)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4: Centralized Communication vs. Distributed Control
  • Lemma 1: Conversion to a Constrained Minimum-Norm Form
  • Lemma 2: Minimum-norm solution under affine constraint
  • Theorem 1: Optimal Control for Linear Time-Invariant Systems
  • proof
  • Proposition 1: Optimality Conditions for the Nonlinear Case
  • Remark 5
  • ...and 3 more