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Resilient consensus for multi-agent systems subject to differential privacy requirements

Davide Fiore, Giovanni Russo

arXiv:1805.08458v2eess.SYcs.SY

TL;DR

The paper addresses consensus among non-faulty agents despite faulty participants, while protecting the agents’ initial conditions. It proposes DP-MSR and establishes consensus under a robustness condition, characterizes accuracy, and analyzes privacy guarantees in special cases.

  • Problem

    The paper asks how non-faulty agents can reach consensus despite faulty agents while satisfying differential privacy for their initial conditions.

    Source: §1 · p. 1, §3.3 · p. 5

  • Method

    DP-MSR modifies MSR by adding noise to transmitted messages and filtering neighbor messages before state updates.

    Source: §4 · p. 5

  • Results

    The paper establishes mean-square consensus under (2f +1)-robustness, characterizes accuracy, and proves differential privacy guarantees in two special cases.

    Source: §5.1 · p. 6, §5.2 · p. 7, §5.3 · p. 8

  • Takeaways & Limitations

    DP-MSR is presented as a distributed approach addressing differential privacy and fault tolerance together over directed graphs.

    Source: §1.1 · p. 2, §7 · p. 11

    Ideal (0, 0)-accuracy cannot be obtained; the radius r is a decreasing function of p.

    Source: §5.2 · p. 8

Abstract

from arXiv · show

We consider multi-agent systems interacting over directed network topologies where a subset of agents is adversary/faulty and where the non-faulty agents have the goal of reaching consensus, while fulfilling a differential privacy requirement on their initial conditions. To address this problem, we develop an update law for the non-faulty agents. Specifically, we propose a modification of the so-called Mean-Subsequence-Reduced (MSR) algorithm, the Differentially Private MSR (DP-MSR) algorithm, and characterize three important properties of the algorithm: correctness, accuracy and differential privacy. We show that if the network topology is $(2f +1)$-robust, then the algorithm allows the non-faulty agents to reach consensus despite the presence of up to $f$ faulty agents and we characterize the accuracy of the algorithm. Furthermore, we also show in two important cases that our distributed algorithm can be tuned to guarantees differential privacy of the initial conditions and the differential privacy requirement is related to the maximum network degree. The results are illustrated via simulations.

1 Introduction

The paper frames consensus in distributed systems as requiring both resilience to faulty agents and privacy for non-faulty agents’ initial conditions. It introduces DP-MSR to address these requirements together on directed graphs.

1 Introduction

  • Power grids and smart transportation motivate systems that must address both fault tolerance and privacy preservation.The introduction identifies failures or attacks and individual-information privacy as challenges in distributed engineering systems.Source: §1 · p. 1

1.1 Contributions of the paper

  • DP-MSR combines differential privacy and resilient consensus over directed graphs, with correctness, accuracy, and privacy properties characterized.The authors describe it as, to their knowledge, the first algorithm to handle both requirements over directed graphs.Source: §1.1 · p. 2
  • Prior differentially private consensus assumed undirected graphs without malicious agents, while resilient-consensus work considered directed graphs without differential-privacy mechanisms.The paper positions its contribution at the intersection of these two strands.Source: §1.1 · p. 2

2 Mathematical Preliminaries

The preliminaries establish notation and probability concepts, define directed-graph robustness, and state a stochastic-consensus result used in the paper.

2 Mathematical Preliminaries

  • The paper sets notation for real-valued vectors and matrices, norms, sets, and vector-valued sequences.It also introduces Borel sets, closed convex hulls, and identity and all-ones vectors.Source: §2 · p. 2

2.1 Probability theory

  • The probability preliminaries define random variables and expectation, and give the Laplace distribution’s density and variance.For Lap(b), the stated variance is 2b^2.Source: §2.1 · pp. 2–3

2.2 Directed graphs and r-robustness

  • In a directed graph, edges represent information flow; r-robustness characterizes local redundancy through r-reachable sets.A graph is r-robust when, for every pair of nonempty disjoint node subsets, at least one subset is r-reachable.Source: §2.2 · p. 3

2.3 Stochastic consensus

  • For the cited stochastic-consensus setup, any initial state with bounded variance yields mean-square consensus.The preliminaries also define consensus and the backward product used in the analysis.Source: §2.3 · p. 3

3 Problem Formulation

The formulation specifies the network, faulty-agent model, consensus and privacy requirements, and the accuracy criterion used to assess the algorithm.

3 Problem Formulation

  • Agents have internal states and exchange messages over a directed graph, with state updates based on their own states and received messages.The update rule may vary by agent, while all agents use the same communication policy.Source: §3 · pp. 3–4

3.1 Adversary model and resilient consensus

  • The faulty set contains at most f agents, who may be Byzantine or malicious and may update their states arbitrarily or collude.The paper uses “faulty agent” for both categories.Source: §3.1 · p. 4
  • Faulty agents can send arbitrary attack signals intended to steer the system toward a malicious consensus state.The signals may depend on that target and the present and past internal states of non-faulty agents.Source: §3.1 · p. 4
  • Resilient asymptotic consensus requires validity and convergence, with the consensus value between the smallest and largest non-faulty initial states.The paper calls an algorithm correct when it guarantees both conditions.Source: §3.1 · p. 4

3.2 Differential privacy

  • Differential privacy is defined by comparing observation distributions for neighboring initial conditions of non-faulty agents.The setup treats the message sequence as the observation generated by the initial state and noise.Source: §3.2 · p. 4
  • The network is (p, r)-accurate when each non-faulty state converges to a random limit within distance r of its expectation with probability at least 1 − p.The definition requires the limit’s expectation to be finite.Source: §3.2 · p. 4

3.3 Problem statement

  • The objective is resilient asymptotic consensus with (p, r)-accuracy and ε-differential privacy of non-faulty agents’ initial conditions.The privacy guarantee is required when faulty agents behave arbitrarily within the adopted communication policy.Source: §3.3 · p. 5

4 The DP-MSR algorithm

DP-MSR modifies MSR by adding noise to transmitted messages while retaining neighbor-message filtering and state updating.

4 The DP-MSR algorithm

  • The notation defines θi(k) as internal state and ηi(k) as independent zero-mean Laplace noise, with ηi(k) ∼ Lap(b(k)) and b(k)=c q^k, where c>0 and 1/2<q<1.Source: §4 · p. 5
  • Each agent receives messages from its in-neighbors and discards the smallest f and largest f values before updating.The retained messages are those from neighbors not removed by this filtering step.Source: §4 · p. 5
  • The update averages an agent’s internal state with the remaining neighbor messages, using weights determined by f and the graph topology.This preserves the MSR-style filtering while applying it to noisy messages.Source: §4 · p. 5

5 Properties of the DP-MSR algorithm

The section analyzes DP-MSR’s correctness, consensus accuracy, and differential-privacy guarantees, including a one-step privacy bound and the role of faulty agents in privacy. It also notes that the proof extends to deterministic algorithms in the stated setting.

5.1 Correctness

  • Under (2f + 1)-robustness, DP-MSR satisfies validity and brings all non-faulty agents to mean-square consensus on a common random variable.Validity means each non-faulty agent’s expected next state is a convex combination of the current expected states of non-faulty agents.Source: §5.1 · pp. 5–6

5.2 Accuracy

  • The consensus value has expectation w^T θ0 for a nonnegative weight vector summing to one; (3f + 1)-robustness ensures a nonzero variance lower bound.The variance upper bound requires only (2f + 1)-robustness, while (3f + 1)-robustness is needed for a nonzero lower bound independently of topology; ideal (0, 0)-accuracy is unattainable.Source: §5.2 · pp. 7–8

5.3 Differential privacy

  • With geometrically decaying Laplace noise and 1/2 < q < 1, the algorithm guarantees ε-differential privacy with ε = ¯ε + ¯δ f dout.The analysis considers both no faulty agents and faulty agents following the communication policy; those agents may adapt attack signals to non-faulty initial conditions.Source: §5.3 · p. 8
  • The proof establishes the differential-privacy result by relating probabilities of corresponding executions, integrating over executions, and applying the definition of differential privacy.Source: §5.3 · p. 10
  • When λ=0, ε = barε + bardelta f d_out max c; if bardelta=0, faulty agents do not perturb the system’s differential privacy.The one-step condition is identified as λ=0.Source: §5.3 · p. 10
  • For φ=0, the proof uses only deterministic operations on input data and therefore applies to any deterministic algorithm, despite Algorithm 1’s changing directed topology.Source: §5.3 · p. 10

6 Simulation example

The simulation illustrates DP-MSR on a 25-agent directed network with one faulty agent. Non-faulty agents reach mean-square consensus, and the observed variance of the consensus value falls within the stated theoretical bounds.

6 Simulation example

  • The example uses 25 agents, one faulty agent, and a 4-robust circulant digraph in which each agent communicates with eight agents ahead.The non-faulty agents’ initial conditions are drawn from a standard normal distribution and held constant across simulations.Source: §6 · p. 10
  • 0.05 was the numerically observed var(θ∞), within the theoretical bounds 0.0037 ≤ var(θ∞) ≤ 27.4286.The simulation ran 10^4 trials.Source: §6 · p. 10
  • Figure 1 tracks non-faulty agents’ internal states over time and marks the faulty agent’s attack signal with crosses.The expected consensus value is reported to lie between the non-faulty agents’ smallest and largest initial values.Source: Figure 1 · p. 10, §6 · p. 10
  • Figure 2 shows the statistical distribution of θ∞, with dashed lines marking the smallest and largest initial conditions of non-faulty agents.The distribution was obtained from 10^4 simulations.Source: §6 · p. 10, Figure 2 · p. 11

7 Conclusions

The paper introduces DP-MSR for consensus under directed communication, faulty agents, and differential-privacy requirements on non-faulty agents’ initial conditions. It characterizes the algorithm’s correctness, accuracy, and privacy properties, and identifies further settings for study.

7 Conclusions

  • DP-MSR is introduced for non-faulty agents seeking consensus under directed communication, faulty agents, and differential-privacy requirements on initial conditions.Source: §7 · p. 11
  • The paper characterizes DP-MSR’s correctness, accuracy, and differential-privacy properties and uses simulations to illustrate its results.Source: §7 · p. 11
  • Future work includes privacy analysis when faulty agents violate communication policy, asynchronous systems, and algorithms for detecting and isolating faulty agents.Source: §7 · p. 11
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