Differentially private resilient distributed cooperative online estimation over digraphs

This article investigates resilient distributed online estimation (DOE) in unreliable directed networks with differential privacy requirements. In the network considered, some agents are subject to Byzantine attacks and thus could send arbitrary incorrect messages to their neighbors. The remaining agents aim to collaboratively estimate the value of an unknown vector parameter while protecting their private data. In this article, by adding private noises to mask the estimate, a stochastic approximation-type resilient differentially private DOE algorithm is proposed to protect the privacy of sensitive information. A time-varying step size is introduced to attenuate the divergence caused by the private noise, and furthermore, guarantees the convergence of the algorithm. When the directed graph is (2F$$ \Big(2F $$+1)-robust, the algorithm is shown to be both mean square and almost sure convergence in the sense of epsilon$$ \epsilon $$-differential privacy. A simulation example is given to verify the effectiveness and superiority of the algorithm.