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Ensure Differential Privacy and Convergence Accuracy in Consensus Tracking and Aggregative Games with Coupling Constraints

Yongqiang Wang

TL;DR

The paper tackles fully distributed generalized Nash equilibrium (GNE) seeking with shared coupling constraints under differential privacy (DP). It co-designs the GNE mechanism with a DP-noise injection scheme to guarantee both provable convergence to the GNE and $\epsilon$-DP, even as iterations grow unbounded, and introduces a robust DP consensus-tracking algorithm to support accurate tracking under persistent DP-noise. The core theoretical contribution is a convergence result for stochastically-perturbed nonstationary fixed-point iterations, enabling analysis of the proposed algorithms under diminishing stepsizes while preserving privacy budgets. The methodology is validated through simulations on a Nash-Cournot game, showing that the proposed DP-GNE seeking algorithm outperforms existing DP approaches in accuracy while maintaining rigorous privacy guarantees. The work advances privacy-preserving distributed optimization by accommodating coupling constraints and providing finite cumulative privacy budgets on infinite horizons, with implications for privacy-aware resource-sharing and networked decision-making.

Abstract

We address differential privacy for fully distributed aggregative games with shared coupling constraints. By co-designing the generalized Nash equilibrium (GNE) seeking mechanism and the differential-privacy noise injection mechanism, we propose the first GNE seeking algorithm that can ensure both provable convergence to the GNE and rigorous epsilon-differential privacy, even with the number of iterations tending to infinity. As a basis of the co-design, we also propose a new consensus-tracking algorithm that can achieve rigorous epsilon-differential privacy while maintaining accurate tracking performance, which, to our knowledge, has not been achieved before. To facilitate the convergence analysis, we also establish a general convergence result for stochastically-perturbed nonstationary fixed-point iteration processes, which lie at the core of numerous optimization and variational problems. Numerical simulation results confirm the effectiveness of the proposed approach.

Ensure Differential Privacy and Convergence Accuracy in Consensus Tracking and Aggregative Games with Coupling Constraints

TL;DR

The paper tackles fully distributed generalized Nash equilibrium (GNE) seeking with shared coupling constraints under differential privacy (DP). It co-designs the GNE mechanism with a DP-noise injection scheme to guarantee both provable convergence to the GNE and -DP, even as iterations grow unbounded, and introduces a robust DP consensus-tracking algorithm to support accurate tracking under persistent DP-noise. The core theoretical contribution is a convergence result for stochastically-perturbed nonstationary fixed-point iterations, enabling analysis of the proposed algorithms under diminishing stepsizes while preserving privacy budgets. The methodology is validated through simulations on a Nash-Cournot game, showing that the proposed DP-GNE seeking algorithm outperforms existing DP approaches in accuracy while maintaining rigorous privacy guarantees. The work advances privacy-preserving distributed optimization by accommodating coupling constraints and providing finite cumulative privacy budgets on infinite horizons, with implications for privacy-aware resource-sharing and networked decision-making.

Abstract

We address differential privacy for fully distributed aggregative games with shared coupling constraints. By co-designing the generalized Nash equilibrium (GNE) seeking mechanism and the differential-privacy noise injection mechanism, we propose the first GNE seeking algorithm that can ensure both provable convergence to the GNE and rigorous epsilon-differential privacy, even with the number of iterations tending to infinity. As a basis of the co-design, we also propose a new consensus-tracking algorithm that can achieve rigorous epsilon-differential privacy while maintaining accurate tracking performance, which, to our knowledge, has not been achieved before. To facilitate the convergence analysis, we also establish a general convergence result for stochastically-perturbed nonstationary fixed-point iteration processes, which lie at the core of numerous optimization and variational problems. Numerical simulation results confirm the effectiveness of the proposed approach.
Paper Structure (16 sections, 15 theorems, 68 equations, 3 figures)

This paper contains 16 sections, 15 theorems, 68 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Problem Formulation and Preliminaries
  3. Monotone Operators
  4. On GNE Seeking
  5. On Differential Privacy
  6. Differentially-private consensus tracking
  7. Convergence Analysis
  8. Differential-privacy Analysis
  9. Differentially-private GNE seeking
  10. Convergence Analysis of Algorithm 2
  11. Finding a non-perturbed fixed-point iteration process
  12. Algorithm 2 is a stochastically-perturbed version of (\ref{['eq:iterate_algo3']})
  13. Combine preceding two subsections to prove the final result
  14. Privacy analysis of Algorithm 2
  15. Numerical Simulations
  16. ...and 1 more sections

Key Result

Lemma 1

belgioioso2020distributed Under Assumption ass:Kset, 1) if ${\rm zer}(T)\neq \emptyset$ and ${ \rm col}(x^{\ast},\lambda^{\ast})\in{\rm zer}(T)$, then $x^{\ast}$ is a variational GNE and $\lambda_1^{\ast}=\lambda_2^{\ast}=\cdots=\lambda_m^{\ast}\in\mathbb{R}^{n}_{+}$; 2) if a variational GNE exists,

Figures (3)

  • Figure 1: Nash-Cournot game of 20 players (firms) competing over 7 locations (markets). Each firm is represented by a circular and each market is represented by a square. An edge between firm $i$$(1\leq i\leq 20)$ and market $j$ ($1\leq j\leq 7$) means that firm $i$ participates in market $j$.
  • Figure 2: The randomly generated interaction patten of the 20 firms.
  • Figure 3: Comparison of Algorithm 2 with the existing GNE seeking algorithm by Belgioioso et al. in belgioioso2020distributed (under the same noise) and the differential-privacy approach from Ye et al. in ye2021differentially after adaptation to GNE seeking (under the same privacy budget $\epsilon$).

Theorems & Definitions (42)

  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Definition 1
  • Remark 1
  • Definition 2
  • Remark 2
  • Lemma 4
  • proof
  • Remark 3
  • ...and 32 more