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A Privacy Preserving Randomized Gossip Algorithm via Controlled Noise Insertion

Filip Hanzely, Jakub Konečný, Nicolas Loizou, Peter Richtárik, Dmitry Grishchenko

TL;DR

The paper addresses privacy concerns in distributed average consensus by introducing an asynchronous, randomized gossip algorithm with controlled noise insertion to protect initial node values $c_i$. It adopts a primal–dual framework with the incidence matrix ${\bf A}$ to model AC and leverages dual updates that translate into primal progress toward the consensus. The core contribution is a novel noise-insertion scheme where each node adds correlated, decaying noise $w_i^{t_i}=\phi_i^{t_i}v_i^{t_i}-\phi_i^{t_i-1}v_i^{t_i-1}$, ensuring convergence to the true average $\bar{c}$ while obscuring private data; the authors provide convergence analysis and design guidelines for $\phi_i$. Empirical results demonstrate the method's behavior under different network topologies and noise decay rates, highlighting a trade-off between privacy and convergence speed and offering practical thresholds for maintaining performance. These insights advance privacy-preserving distributed optimization by enabling asynchronous operation with formal convergence guarantees.

Abstract

In this work we present a randomized gossip algorithm for solving the average consensus problem while at the same time protecting the information about the initial private values stored at the nodes. We give iteration complexity bounds for the method and perform extensive numerical experiments.

A Privacy Preserving Randomized Gossip Algorithm via Controlled Noise Insertion

TL;DR

The paper addresses privacy concerns in distributed average consensus by introducing an asynchronous, randomized gossip algorithm with controlled noise insertion to protect initial node values . It adopts a primal–dual framework with the incidence matrix to model AC and leverages dual updates that translate into primal progress toward the consensus. The core contribution is a novel noise-insertion scheme where each node adds correlated, decaying noise , ensuring convergence to the true average while obscuring private data; the authors provide convergence analysis and design guidelines for . Empirical results demonstrate the method's behavior under different network topologies and noise decay rates, highlighting a trade-off between privacy and convergence speed and offering practical thresholds for maintaining performance. These insights advance privacy-preserving distributed optimization by enabling asynchronous operation with formal convergence guarantees.

Abstract

In this work we present a randomized gossip algorithm for solving the average consensus problem while at the same time protecting the information about the initial private values stored at the nodes. We give iteration complexity bounds for the method and perform extensive numerical experiments.

Paper Structure

This paper contains 8 sections, 2 theorems, 6 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work on Privacy Preserving Average Consensus
  3. Main Contributions
  4. Technical Preliminaries
  5. Primal and Dual Problems
  6. Randomized Gossip Setup: Choosing ${\bf A}$.
  7. Private Gossip via Controlled Noise Insertion
  8. Experiments

Key Result

Theorem 2

Let us define $\rho \overset{\text{def}}{=} 1-\frac{\alpha({\cal G})}{2m}$ and $\psi^t \overset{\text{def}}{=} \frac{1}{\sum_{i=1}^n\left(d_i\sigma_i^2\right)}\sum_{i=1}^n d_i \sigma_i^2\left(1-\frac{d_i}{m}\left(1-\phi_i^2\right) \right)^{t},$ where $\alpha({\cal G})$ stands for algebraic connectiv

Figures (1)

  • Figure 1: Convergence of Algorithm \ref{['PrivacyGossip']}, on the cycle graph (left) and random geometric graph (right) for different values of $\phi$. The "Relative Error " on the vertical axis represents the $\frac{\|x^t-x^*\|^2}{\|x^0 - x^*\|^2}$

Theorems & Definitions (3)

  • Definition 1
  • Theorem 2
  • Corollary 3