Differentially Private Average Consensus: Obstructions, Trade-Offs, and Optimal Algorithm Design

TL;DR

This paper investigates the limitations and design of differentially private algorithms for multi-agent average consensus, proposing a novel Laplacian consensus method that balances privacy and convergence accuracy.

Contribution

It introduces a new differentially private Laplacian consensus algorithm with proven convergence and privacy guarantees, and characterizes its optimal parameters for minimal estimation error.

Findings

The impossibility of exact average convergence under differential privacy.

A new Laplacian consensus algorithm with almost sure convergence to an unbiased estimate.

Optimal parameters correspond to a one-shot initial state perturbation.

Abstract

This paper studies the multi-agent average consensus problem under the requirement of differential privacy of the agents' initial states against an adversary that has access to all the messages. We first establish that a differentially private consensus algorithm cannot guarantee convergence of the agents' states to the exact average in distribution, which in turn implies the same impossibility for other stronger notions of convergence. This result motivates our design of a novel differentially private Laplacian consensus algorithm in which agents linearly perturb their state-transition and message-generating functions with exponentially decaying Laplace noise. We prove that our algorithm converges almost surely to an unbiased estimate of the average of agents' initial states, compute the exponential mean-square rate of convergence, and formally characterize its differential privacy properties. We show that the optimal choice of our design parameters (with respect to the variance of the convergence point around the exact average) corresponds to a one-shot perturbation of initial states and compare our design with various counterparts from the literature. Simulations illustrate our results.