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Communication-efficient and Differentially-private Distributed Nash Equilibrium Seeking with Linear Convergence

Xiaomeng Chen, Wei Huo, Kemi Ding, Subhrakanti Dey, Ling Shi

TL;DR

This work addresses distributed Nash equilibrium seeking under simultaneous privacy and communication constraints on directed graphs. It proposes CDP-NES, a unified framework that fuses difference compression with Laplacian noise and constant-step updates to achieve linear convergence to a neighborhood of the NE while guaranteeing $\epsilon$-differential privacy. Theoretical results show a linear convergence rate $\mathcal{O}(\rho(\mathbf{A})^k)$ with $\rho(\mathbf{A})<1$ and a mean-square NE error scaling as $\bar{\theta}^2$, with $\bar{\theta}$ tied to privacy budgets; privacy is ensured by a Laplace mechanism with $\theta_i > 2\gamma\eta K M/\epsilon_i$ under a bounded-gradient assumption. Simulations on a connectivity-control game demonstrate substantial communication savings from compression and competitive convergence under privacy constraints, highlighting practical impact for large-scale distributed decision-making with privacy guarantees.

Abstract

The distributed computation of a Nash equilibrium (NE) for non-cooperative games is gaining increased attention recently. Due to the nature of distributed systems, privacy and communication efficiency are two critical concerns. Traditional approaches often address these critical concerns in isolation. This work introduces a unified framework, named CDP-NES, designed to improve communication efficiency in the privacy-preserving NE seeking algorithm for distributed non-cooperative games over directed graphs. Leveraging both general compression operators and the noise adding mechanism, CDP-NES perturbs local states with Laplacian noise and applies difference compression prior to their exchange among neighbors. We prove that CDP-NES not only achieves linear convergence to a neighborhood of the NE in games with restricted monotone mappings but also guarantees $ε$-differential privacy, addressing privacy and communication efficiency simultaneously. Finally, simulations are provided to illustrate the effectiveness of the proposed method.

Communication-efficient and Differentially-private Distributed Nash Equilibrium Seeking with Linear Convergence

TL;DR

This work addresses distributed Nash equilibrium seeking under simultaneous privacy and communication constraints on directed graphs. It proposes CDP-NES, a unified framework that fuses difference compression with Laplacian noise and constant-step updates to achieve linear convergence to a neighborhood of the NE while guaranteeing -differential privacy. Theoretical results show a linear convergence rate with and a mean-square NE error scaling as , with tied to privacy budgets; privacy is ensured by a Laplace mechanism with under a bounded-gradient assumption. Simulations on a connectivity-control game demonstrate substantial communication savings from compression and competitive convergence under privacy constraints, highlighting practical impact for large-scale distributed decision-making with privacy guarantees.

Abstract

The distributed computation of a Nash equilibrium (NE) for non-cooperative games is gaining increased attention recently. Due to the nature of distributed systems, privacy and communication efficiency are two critical concerns. Traditional approaches often address these critical concerns in isolation. This work introduces a unified framework, named CDP-NES, designed to improve communication efficiency in the privacy-preserving NE seeking algorithm for distributed non-cooperative games over directed graphs. Leveraging both general compression operators and the noise adding mechanism, CDP-NES perturbs local states with Laplacian noise and applies difference compression prior to their exchange among neighbors. We prove that CDP-NES not only achieves linear convergence to a neighborhood of the NE in games with restricted monotone mappings but also guarantees -differential privacy, addressing privacy and communication efficiency simultaneously. Finally, simulations are provided to illustrate the effectiveness of the proposed method.
Paper Structure (16 sections, 6 theorems, 38 equations, 2 figures, 1 algorithm)

This paper contains 16 sections, 6 theorems, 38 equations, 2 figures, 1 algorithm.

Table of Contents

  1. INTRODUCTION
  2. PRELIMINARIES AND PROBLEM STATEMENT
  3. Network Model
  4. Compressors
  5. Differential Privacy
  6. Problem Definition
  7. CDP-NES Algorithm
  8. Convergence Analysis
  9. Supporting Lemmas
  10. Main results
  11. Privacy analysis
  12. SIMULATIONS
  13. CONCLUSION AND FUTURE WORK
  14. Appendix
  15. Proof of Lemma \ref{['mainl']}
  16. ...and 1 more sections

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 (2)

  • Figure 1: Residual ($\mathcal{R}=||\mathbf{X}^k-\mathbf{X}^\star||_{\text{F}}$) v.s. number of iterations for the differential-private NE seeking in Wang and Başar wang2022ensuring, the compressed NE seeking (C-DNES) chen2022linear and the proposed CDP-NES algorithm with different privacy budgets.
  • Figure 2: Total numbers of transmitted bits of different algorithms to obtain $\mathcal{R}\leq 0.02$.

Theorems & Definitions (14)

  • Remark 1
  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Theorem 1
  • ...and 4 more