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Dynamic control of stochastic matching systems in heavy traffic: An effective computational method for high-dimensional problems

Baris Ata, Yaosheng Xu

TL;DR

This paper addresses dynamic bipartite matching under uncertainty in high dimensions by deriving a heavy-traffic Brownian control problem (drift-rate control of a reflected Brownian motion) and solving it with a deep BSDE-based neural network framework. The authors produce a tractable, data-driven policy that is applicable to pre-limit queueing systems and demonstrate its superiority over standard benchmark policies across nine test problems, including up to 120 dimensions. Key contributions include the explicit HJB formulation, the drift-rate control characterization, and a computational pipeline that leverages neural networks to approximate the value function and its gradient for high-dimensional settings. The results indicate significant performance gains and practical potential for dynamic dispatch and matching platforms, with theoretical grounding in diffusion approximations and stochastic processing networks.

Abstract

Bipartite matching systems arise in many settings where agents or tasks from two distinct sets must be paired dynamically under compatibility constraints. We consider a high-dimensional bipartite matching system under uncertainty and seek an effective dynamic control policy that maximizes the expected discounted total value generated by the matches minus the congestion-related costs. To derive a tractable approximation, we focus attention on balanced, high-volume systems, i.e., the heavy-traffic regime, and derive an approximating Brownian control problem. We then develop a computational method that relies on deep neural network technology for solving this problem. To show the effectiveness of the policy derived from our computational method, we compare it to the benchmark policies available in the extant literature in the context of the original matching problem. In the test problems attempted thus far, our proposed policy outperforms the benchmarks, and its derivation is computationally feasible for dimensions up to 100 or more.

Dynamic control of stochastic matching systems in heavy traffic: An effective computational method for high-dimensional problems

TL;DR

This paper addresses dynamic bipartite matching under uncertainty in high dimensions by deriving a heavy-traffic Brownian control problem (drift-rate control of a reflected Brownian motion) and solving it with a deep BSDE-based neural network framework. The authors produce a tractable, data-driven policy that is applicable to pre-limit queueing systems and demonstrate its superiority over standard benchmark policies across nine test problems, including up to 120 dimensions. Key contributions include the explicit HJB formulation, the drift-rate control characterization, and a computational pipeline that leverages neural networks to approximate the value function and its gradient for high-dimensional settings. The results indicate significant performance gains and practical potential for dynamic dispatch and matching platforms, with theoretical grounding in diffusion approximations and stochastic processing networks.

Abstract

Bipartite matching systems arise in many settings where agents or tasks from two distinct sets must be paired dynamically under compatibility constraints. We consider a high-dimensional bipartite matching system under uncertainty and seek an effective dynamic control policy that maximizes the expected discounted total value generated by the matches minus the congestion-related costs. To derive a tractable approximation, we focus attention on balanced, high-volume systems, i.e., the heavy-traffic regime, and derive an approximating Brownian control problem. We then develop a computational method that relies on deep neural network technology for solving this problem. To show the effectiveness of the policy derived from our computational method, we compare it to the benchmark policies available in the extant literature in the context of the original matching problem. In the test problems attempted thus far, our proposed policy outperforms the benchmarks, and its derivation is computationally feasible for dimensions up to 100 or more.

Paper Structure

This paper contains 26 sections, 1 theorem, 65 equations, 5 figures, 10 tables, 6 algorithms.

Table of Contents

  1. Introduction
  2. Literature review
  3. Model
  4. Approximating Brownian Control Problem
  5. HJB equation and the optimal policy for the Brownian control problem
  6. Optimal drift-rate control policy for the Brownian control problem.
  7. Computational Method
  8. Proposed Matching Policy for the Pre-limit System
  9. Benchmark Policies
  10. Test Problems
  11. X Models
  12. Zigzag Models
  13. High Dimensional Models
  14. Computational Results
  15. X model.
  16. ...and 11 more sections

Key Result

Proposition 1

If $V: \mathbb{R}_{+}^I \rightarrow \mathbb{R}$ is a $C^2$ function with polynomial growth that satisfies the $HJB$ equation (eq:hjb) and (eq:hjbb), then it also satisfies the following identity almost surely for any $T>0$:

Figures (5)

  • Figure 1: A bipartite matching model.
  • Figure 2: The X Model.
  • Figure 3: Zigzag models under different configurations.
  • Figure 4: Distribution of Average Gradients Across Abandonment Levels
  • Figure 5: 24-dimensional models.

Theorems & Definitions (2)

  • Proposition 1
  • proof