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Distributed Online Optimization for Multi-Agent Optimal Transport

Vishaal Krishnan, Sonia Martínez

Abstract

We propose a scalable, distributed algorithm for the optimal transport of large-scale multi-agent systems. We formulate the problem as one of steering the collective towards a target probability measure while minimizing the total cost of transport, with the additional constraint of distributed implementation. Using optimal transport theory, we realize the solution as an iterative transport based on a proximal descent scheme. At each stage of the transport, the agents implement an online, distributed primal-dual algorithm to obtain local estimates of the Kantorovich potential for optimal transport from the current distribution of the collective to the target distribution. Using these estimates as their local objective functions, the agents then implement the transport by proximal descent. This two-step process is carried out recursively by the agents to converge asymptotically to the target distribution. We rigorously establish the underlying theoretical framework for the algorithm and test its behavior via numerical experiments.

Distributed Online Optimization for Multi-Agent Optimal Transport

Abstract

We propose a scalable, distributed algorithm for the optimal transport of large-scale multi-agent systems. We formulate the problem as one of steering the collective towards a target probability measure while minimizing the total cost of transport, with the additional constraint of distributed implementation. Using optimal transport theory, we realize the solution as an iterative transport based on a proximal descent scheme. At each stage of the transport, the agents implement an online, distributed primal-dual algorithm to obtain local estimates of the Kantorovich potential for optimal transport from the current distribution of the collective to the target distribution. Using these estimates as their local objective functions, the agents then implement the transport by proximal descent. This two-step process is carried out recursively by the agents to converge asymptotically to the target distribution. We rigorously establish the underlying theoretical framework for the algorithm and test its behavior via numerical experiments.

Paper Structure

This paper contains 15 sections, 4 theorems, 43 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Mathematical preliminaries
  3. Notation
  4. Monge and Kantorovich formulations of optimal transport
  5. Optimal iterative transport of measures
  6. Multi-agent implementation
  7. Multi-agent optimal transport algorithm
  8. Numerical experiments
  9. Conclusion
  10. Proofs of Lemmas
  11. Proof of Lemma \ref{['lemma:cconvex_phic']}
  12. Proof of Lemma \ref{['lemma:OT_stage_sequence']}
  13. Proofs of Theorems
  14. Proof of Theorem \ref{['thm:KKT_Kantorovich_duality']}
  15. Proof of Theorem \ref{['prop:law_Kant_duality_update_RV']}

Key Result

Lemma 1

Let $c: \Omega \times \Omega \rightarrow {\mathbb{R}}_{\ge 0}$ be a metric on $\Omega$ conformal to the Euclidean distance, with a strictly positive conformal factor $\xi \in C^1(\Omega)$. (a) The conjugate of the Kantorovich potential in eq:conjugate_pair satisfies $\phi^c = - \phi$ and $|\phi(x) -

Figures (2)

  • Figure 1: Positions of agents at three different stages (time instants $k=0, 50, 100, 300$) of transport by Algorithm \ref{['alg:dist_opt_transport']}, i.e., the iterative process \ref{['eq:Kantorovich_duality_update_RV']} with local estimates of Kantorovich potential supplied by \ref{['eq:p-d_Kantorovich_discrete']} (with $n=1$ iterations of the primal-dual algorithm \ref{['eq:p-d_Kantorovich_discrete']} for every transport step \ref{['eq:p-d_Kantorovich_discrete']}; Target probability measure shown in grayscale with a darker shade indicating a region of higher target density; The plots show convergence in time of the agents to full coverage of the target coverage profile (represented by the target probability distribution).
  • Figure 2: Variance in target mass $\mu^*(\mathcal{V}_i)$ across the partition vs time for iteration steps $n=1,10$ of the primal-dual algorithm \ref{['eq:p-d_Kantorovich_discrete']} for every transport step \ref{['eq:Kantorovich_duality_update_RV']}. The plot shows the (empirical) mean along with $95\%$ confidence bounds of the variance of $\mu^*(\mathcal{V}_i)$ from $10$ trials for each $n$ from the same initial condition.

Theorems & Definitions (5)

  • Lemma 1: Local Lipschitz bound
  • Theorem 1: First-order optimality conditions
  • Lemma 2: Decomposition of optimal transport cost
  • Theorem 2: Stochastic process for optimal iterative transport
  • Remark 1: Optimize-then-discretize vs discretize-then-optimize