Source-linked AI summary
Differentially Private Iterative Synchronous Consensus
Zhenqi Huang, Sayan Mitra, Geir Dullerud
TL;DR
The paper studies how agents can reach average consensus while protecting their private initial values from adversaries observing messages and internal states. It adapts differential privacy, proposes client-server and fully distributed noise-based mechanisms, and establishes a privacy–accuracy tradeoff. The analysis identifies conditions for infinite-horizon privacy and convergence, while leaving broader dynamics and verification extensions for future work.
Problem
The paper addresses iterative consensus with privacy against adversaries that can observe exchanged messages and internal states, motivated by distributed coordination and control applications.
Method
The paper adapts differential privacy and proposes client-server and distributed mechanisms that add geometrically attenuating Laplace noise to messages before consensus updates.
Results
Privacy over an infinite horizon requires q > 1 −σ, while larger noise improves privacy but worsens accuracy, yielding an explicit privacy–accuracy tradeoff.
Takeaways & Limitations
The mechanisms provide probabilistic convergence toward the average while protecting initial values in client-server and compromised-client distributed settings.
Takeaways & Limitations
The analysis is limited to discrete linear system dynamics, and the authors identify continuous or nonlinear systems, lower bounds, and controller synthesis as future work.
Abstract
from arXiv · showhide
The iterative consensus problem requires a set of processes or agents with different initial values, to interact and update their states to eventually converge to a common value. Protocols solving iterative consensus serve as building blocks in a variety of systems where distributed coordination is required for load balancing, data aggregation, sensor fusion, filtering, clock synchronization and platooning of autonomous vehicles. In this paper, we introduce the private iterative consensus problem where agents are required to converge while protecting the privacy of their initial values from honest but curious adversaries. Protecting the initial states, in many applications, suffice to protect all subsequent states of the individual participants. First, we adapt the notion of differential privacy in this setting of iterative computation. Next, we present a server-based and a completely distributed randomized mechanism for solving private iterative consensus with adversaries who can observe the messages as well as the internal states of the server and a subset of the clients. Finally, we establish the tradeoff between privacy and the accuracy of the proposed randomized mechanism.
1 Introduction
The paper frames iterative consensus as repeated state updates toward a common value while protecting agents’ initial values from honest-but-curious adversaries. It proposes randomized client-server and distributed mechanisms and analyzes their privacy–accuracy tradeoff.
- Motivation: Iterative consensus repeatedly updates agents’ states until they converge to a common decision or value.Such mechanisms support distributed coordination applications including load balancing, filtering, sensor fusion, clock synchronization, and flocking.
- Problem: The private consensus problem protects initial values from adversaries observing exchanged messages and internal states while agents converge to their initial-value average.The paper derives its privacy notion from differential privacy for iterative computation.
- Mechanism: The client-server mechanism adds geometrically decaying Laplace noise to client messages, averages the noisy messages, and uses weighted feedback to update client states.The mechanism uses a weighted average of the server feedback and each client’s earlier state.
- Mechanism: The distributed mechanism extends the approach to adversaries observing messages and states of compromised clients, while providing privacy for good clients and a communication-based convergence condition.The paper presents the distributed setting as a generalization of the client-server mechanism.
- Privacy and accuracy: With ϵ-differential privacy, the randomized mechanism converges to the average probabilistically, with an accuracy radius determined by the privacy and participation parameters.The paper defines (b, r)-accuracy as convergence within radius r of the average with probability 1−b.
2 Preliminaries
The preliminaries define the client-server setting, its privacy threat model, and formal criteria for differential privacy, convergence, and accuracy. The framework requires convergence while limiting what an adversary can infer from messages and internal states.
- Problem setting: The system contains N clients with private initial values and a server that exchanges messages with clients in four phases each round.The phases are client messages, server computation, server responses, and client state updates.
- Threat model: The adversary may inspect all communication channels and the server’s internal state, and may also observe some clients’ states.The privacy objective concerns executions associated with adjacent initial global states.
- Privacy: The paper adapts differential privacy for streaming-style sequences of messages and internal states over adjacent initial global states.Adjacency is defined over vectors whose individual components differ according to the paper’s specified metric condition.
- Convergence: Convergence is defined by the expected squared difference between every pair of agents’ local states tending to zero over time.The expectation is taken over the mechanism’s random coin flips.
- Accuracy: Accuracy requires executions to converge within radius r of the average of the initial state, with failure probability bounded by b.The paper introduces accuracy alongside convergence and privacy as a formal objective.
3 A Client-Server Mechanism and its Analysis
The client-server mechanism adds geometrically decaying Laplace noise to client messages, then uses server averaging and client interpolation to achieve privacy, convergence, and accuracy.
- Mechanism: Each round, clients send their current states plus Laplace noise whose scale decays geometrically, the server averages messages, and clients interpolate toward the feedback.The mechanism uses parameters σ, c, and q, with noise drawn from Lap(cq^t).
- Privacy: The privacy proof constructs a bijection between executions from δ-adjacent initial states that produce the same observable message and server-state sequence.The corresponding executions have identical client reports and server feedback, making them indistinguishable to the adversary.
- Privacy: For q ∈(1−σ, 1), the mechanism guarantees ϵ-differential privacy with ϵ = q c(q+σ−1).The finite-prefix probability ratio converges to e^ϵδ as the horizon tends to infinity.
- Convergence: The mechanism converges exponentially because the disagreement potential satisfies P(t + 1) = aP(t), with a = (1 −σ)^2 ∈(0, 1).The shared random increment changes the average but not the relative distances between client states, so convergence is deterministic.
- Guarantees: The section formally establishes privacy, convergence, and accuracy for the centralized synchronous consensus mechanism.Accuracy is stated as Lemma 4, while the section concludes by noting a subsequent discussion of the privacy–accuracy trade-off.
4 A Distributed Mechanism
The distributed mechanism replaces the server with neighbor-to-neighbor communication and auxiliary states, while preserving privacy for uncompromised clients under graph and parameter conditions.
- Mechanism: Each client sends a noisy state to its neighbors, averages received messages into an auxiliary state, and interpolates its state toward that auxiliary value.Clients use independent interpolation factors σ_i and Laplace noise with scale c q^t.
- Privacy: Privacy is protected only for uncompromised clients when q ∈(1−σ_m, 1), with ϵ = q c(q+σ_m−1).Here σ_m is the minimum element of the clients’ interpolation-factor vector.
- Convergence: The distributed mechanism achieves convergence when the communication graph is connected, equivalently when its second Laplacian eigenvalue satisfies λ2 > 0.The convergence analysis uses the graph Laplacian and shows expected disagreement tends to zero as t →∞.
- Accuracy: The expected consensus value generally does not equal the initial average because node degree and evolution speed determine unequal weights.The paper characterizes the consensus using weights γ_i = 1/d_i, where d_i = |N(i)| + 1.
- Accuracy: The distributed mechanism achieves the stated accuracy guarantee, and its privacy–accuracy trade-off is similar to that of the client-server mechanism.The accuracy result is given as Lemma 8, followed by a joint discussion of the two mechanisms’ trade-offs.
5 Discussion on Results
The mechanisms achieve private iterative consensus over an infinite horizon by using geometrically decaying Laplace noise, with privacy increasing as noise becomes larger or decays more slowly while accuracy worsens. The resulting privacy–accuracy relationship is illustrated for the centralized mechanism.
- Mechanism parameters: The two mechanisms add noise streams whose standard deviation decreases toward the Dirac distribution at zero over an infinite horizon.The mechanisms use linear combination factor σ, initial noise c, and noise convergence rate q.
- Mechanism parameters: Privacy over an infinite horizon requires q > 1 −σ, so noise converges more slowly than the system’s inertia.This condition lets the noise cover the trail of system dynamics.
- Privacy–accuracy tradeoff: Larger c or q decreases ϵ, indicating higher privacy when the Laplace noise has larger parameters and standard deviation.The relationship is reported from Lemmas 1 and 5.
- Privacy–accuracy tradeoff: More dispersive noise produces worse accuracy, establishing a tradeoff between privacy and accuracy.Figure 1 illustrates this tradeoff as a function of the noise convergence rate q in the centralized mechanism.
- Privacy–accuracy tradeoff: For fixed q, ϵ-differential privacy with N agents and accuracy level b yields accuracy radius r = O(1/(ϵ√βN)).Figure 1 shows the dependence between ϵ and r for specific parameter values.
6 Related Work
The paper’s mechanism shares noise-based communication with prior aggregate-computation protocols but uses geometrically decaying Laplace noise and feedback-driven convergence. Related distributed protocols use different privacy architectures and computation targets.
- Noise-based protocols: The consensus mechanism resembles prior sum and inner-product protocols because all add noise to communicated participant states.Its distinctive noise is geometrically decaying Laplace noise.
- Noise-based protocols: Unlike those protocols, this mechanism uses computed outputs as feedback to update participant states and achieve convergence.The comparison concerns the role of communicated noisy states in an iterative control loop.
- Server assumptions: A secret-sharing framework preserves privacy when at least one of multiple servers is uncompromised, whereas this mechanism guarantees privacy even if its only server is compromised.The two approaches therefore differ in their server-compromise assumptions.
- Distributed protocols: A distributed ring protocol computes k maximum values by communicating a global vector and randomly replacing values under an exponentially decaying probability.Its computation target and update procedure differ from iterative consensus.
7 Conclusions and Future direction
The paper formalizes synchronized private consensus and proves server-based and distributed mechanisms using convergent Laplace noise. It identifies differential privacy–accuracy tradeoffs and points to broader control settings and automated controller verification as future directions.
- Contributions: The paper proposes client-server and purely distributed mechanisms that add convergent Laplace noise to client messages while proving privacy and convergence.The privacy proof constructs a bijective map between executions from adjacent initial states.
- Contributions: The work extends differential privacy to a control setting whose ultimate goal is convergence.The authors describe this as the first investigation of differential privacy in such control systems, to the best of their knowledge.
- Future directions: The analysis currently concerns discrete linear dynamics, with extensions to continuous or nonlinear systems identified as future work.The authors also identify establishing a lower bound as significant.
- Future directions: Automated verification and synthesis algorithms for privacy-preserving controllers remain an open direction for feedback control systems.The paper highlights extending verification ideas toward synthesis and feedback control.