Differential privacy in consensus of multi-agent systems

Abstract

With the rise in reports on privacy disclosure of multi-agent systems (MASs), there is a pressing need for designing a privacy-preserving consensus algorithm. This thesis is concerned with the consensus problem of multi-agent systems subject to differential privacy. Three different classes of MASs are considered, that is, generic linear multivariable MASs on undirected graphs, first-order MASs on directed but not necessarily weight-balanced graphs, and MASs with positive agents on balanced directed graphs. Additive random noises have been widely utilized in randomized mechanisms to preserve differential privacy. By using this idea, two types of differentially private consensus problem are investigated. First, for generic linear multivariable MASs on undirected graphs, the differential private consensus problem is addressed. To guarantee differential privacy of agents, general independent multivariate random noises with zero mean are added to the shared information. Based on it, a novel differentially private consensus algorithm is introduced. It is shown that the growth rate of the additive noises plays an essential role in the achievement of mean-square consensus. By applying probability theory, a necessary and sufficient condition is established for ensuring ε-differential privacy of the states at any time instant over an infinite time horizon, which provides a guideline for designing multivariate additive noises that preserve ε-differential privacy. Second, for first-order MASs on directed but not necessarily weight-balanced graphs, the differentially private average consensus problem is considered. It is worth mentioning that the asymmetry of weight matrices may lead to the deviation of the consensus value from the averaged initial states. To solve this problem, a novel average consensus algorithm is designed, in which two variables are computed in parallel at each update and additive random noises with zero mean are imposed on the shared information. For the proposed algorithm, it is proved that mean-square average consensus is achieved if and only if the additive random noises exponentially decay with time. Furthermore, a few necessary and sufficient conditions are established to characterize the statistics of additive noises that preserve ε-differential privacy of the agents at any time instant over an infinite time horizon. For MASs with positive agents, the problem of achieving positive average consensus while preserving ε-differential privacy of agents is considered. It is observed that adding time-decaying random noises with zero mean to the exchanged information may impair the positivity and randomness of the states, although it has been extensively adopted in the existing approaches to solve the differentially private consensus problem. Motivated by this, multiplicative random noises are considered in a novel randomized mechanism to blur the shared state information while maintaining the positivity and randomness of the states. Then, time-decaying controller gains are designed to realize mean-square average consensus with the convergence error computed by the martingale convergence theory. By using probability theory and measure theory, it is shown that the proposed randomized mechanism preserves (ε,δ)-differential privacy of the agents.