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An Efficient Distributed Nash Equilibrium Seeking with Compressed and Event-triggered Communication

Xiaomeng Chen, Wei Huo, Yuchi Wu, Subhrakanti Dey, Ling Shi

TL;DR

This work addresses the challenge of high communication costs in distributed Nash equilibrium seeking over directed graphs by introducing compressed and event-triggered strategies. The ETC-DNES algorithm combines difference compression with deterministic event triggers, requiring only row-stochastic adjacency and achieving convergence under restricted strong monotonicity and Lipschitz-gradients, with a linear rate under exponential or diminishing trigger thresholds. The SETC-DNES extension adds stochastic triggering, delivering further communication savings while preserving linear convergence through a contraction matrix $oldsymbol{C}$. Empirical results on a 50-agent network show substantial reductions in transmitted bits (up to orders of magnitude) without sacrificing accuracy, and SETC-DNES often outperforms deterministic counterparts in communication efficiency. The framework broadens applicability to directed graphs and various compressors and offers practical guidance for scalable, communication-efficient NE seeking.

Abstract

Distributed Nash equilibrium (NE) seeking problems for networked games have been widely investigated in recent years. Despite the increasing attention, communication expenditure is becoming a major bottleneck for scaling up distributed approaches within limited communication bandwidth between agents. To reduce communication cost, an efficient distributed NE seeking (ETC-DNES) algorithm is proposed to obtain an NE for games over directed graphs, where the communication efficiency is improved by event-triggered exchanges of compressed information among neighbors. ETC-DNES saves communication costs in both transmitted bits and rounds of communication. Furthermore, our method only requires the row-stochastic property of the adjacency matrix, unlike previous approaches that hinged on doubly-stochastic communication matrices. We provide convergence guarantees for ETC-DNES on games with restricted strongly monotone mappings and testify its efficiency with no sacrifice on the accuracy. The algorithm and analysis are extended to a compressed algorithm with stochastic event-triggered mechanism (SETC-DNES). In SETC-DNES, we introduce a random variable in the triggering condition to further enhance algorithm efficiency. We demonstrate that SETC-DNES guarantees linear convergence to the NE while achieving even greater reductions in communication costs compared to ETC-DNES. Finally, numerical simulations illustrate the effectiveness of the proposed algorithms.

An Efficient Distributed Nash Equilibrium Seeking with Compressed and Event-triggered Communication

TL;DR

This work addresses the challenge of high communication costs in distributed Nash equilibrium seeking over directed graphs by introducing compressed and event-triggered strategies. The ETC-DNES algorithm combines difference compression with deterministic event triggers, requiring only row-stochastic adjacency and achieving convergence under restricted strong monotonicity and Lipschitz-gradients, with a linear rate under exponential or diminishing trigger thresholds. The SETC-DNES extension adds stochastic triggering, delivering further communication savings while preserving linear convergence through a contraction matrix . Empirical results on a 50-agent network show substantial reductions in transmitted bits (up to orders of magnitude) without sacrificing accuracy, and SETC-DNES often outperforms deterministic counterparts in communication efficiency. The framework broadens applicability to directed graphs and various compressors and offers practical guidance for scalable, communication-efficient NE seeking.

Abstract

Distributed Nash equilibrium (NE) seeking problems for networked games have been widely investigated in recent years. Despite the increasing attention, communication expenditure is becoming a major bottleneck for scaling up distributed approaches within limited communication bandwidth between agents. To reduce communication cost, an efficient distributed NE seeking (ETC-DNES) algorithm is proposed to obtain an NE for games over directed graphs, where the communication efficiency is improved by event-triggered exchanges of compressed information among neighbors. ETC-DNES saves communication costs in both transmitted bits and rounds of communication. Furthermore, our method only requires the row-stochastic property of the adjacency matrix, unlike previous approaches that hinged on doubly-stochastic communication matrices. We provide convergence guarantees for ETC-DNES on games with restricted strongly monotone mappings and testify its efficiency with no sacrifice on the accuracy. The algorithm and analysis are extended to a compressed algorithm with stochastic event-triggered mechanism (SETC-DNES). In SETC-DNES, we introduce a random variable in the triggering condition to further enhance algorithm efficiency. We demonstrate that SETC-DNES guarantees linear convergence to the NE while achieving even greater reductions in communication costs compared to ETC-DNES. Finally, numerical simulations illustrate the effectiveness of the proposed algorithms.
Paper Structure (14 sections, 9 theorems, 53 equations, 3 figures, 2 tables, 3 algorithms)

This paper contains 14 sections, 9 theorems, 53 equations, 3 figures, 2 tables, 3 algorithms.

Table of Contents

  1. INTRODUCTION
  2. PRELIMINARIES AND PROBLEM STATEMENT
  3. Network Model
  4. Compressors
  5. Problem Formulation
  6. Event-triggered and Compressed Distributed NE Seeking Algorithm (ETC-DNES)
  7. Algorithm Development
  8. Convergence Analysis
  9. Stochastic Event-triggered and Compressed Distributed NE Seeking Algorithm (SETC-DNES)
  10. Algorithm Development
  11. Convergence Analysis
  12. SIMULATIONS
  13. CONCLUSION AND FUTURE WORK
  14. APPENDIX

Key Result

Lemma 1

(Lemma 1 in tatarenko2020geometric1) Given Assumption asp2, the augmented mapping $\mathbf{F}_a$ of game $\Gamma$ is Lipschitz continuous with $L_F=\eta L_m+||I-W||_{\text{F}}$, where $L_m=\max_iL_i$.

Figures (3)

  • Figure 1: Evolution of Residual with respect to (a) the number of iterations and (b) the number of transmitted bits.
  • Figure 2: (a) Triggering instants for ET-NE xu2021event; (b) Triggering instants for SETC-NE huo2023distributed; (c) Triggering instants for ETC-DNES-1; (d) Triggering instants for ETC-DNES-2; (e) Triggering instants for SETC-DNES; (f) A comparison on the residuals and communication rate (averaged trigger iterations/total iterations) by stopping the algorithms at $k = 5000$.
  • Figure 3: Total transmitted bits of different algorithms to obtain $\mathcal{R}\leq 0.01$.

Theorems & Definitions (22)

  • Definition 1
  • Definition 2
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • ...and 12 more