Securely Solving Linear Algebraic Equations in a Distributed Framework Enhanced With Communication-Efficient Algorithms

Abstract

Solving linear algebraic equations (a.k.a., an LAE problem) distributedly in a network with multiple agents has wide applications in distributed control, estimation and signal processing. A consensus-based distributed computing framework is studied in this paper. Specifically, each agent knows only a subproblem of the LAE, i.e., a subset of all equations, and then all agents apply a consensus-based algorithm to update their estimates of the correct solution of the LAE problem iteratively. Under certain conditions, it has been shown that all the estimates converge to the exact solution exponentially fast. However, such a distributed paradigm is vulnerable to malicious behaviors in an adversarial environment. In this paper, we indicate a number of security threats in this process, and thus develop robust computing solutions against those attacks. With particular attention to low storage overhead, we develop a new alternating projection method to enhance the consensus algorithm. Furthermore, we design an innovative misbehavior detection mechanism by exploiting the homomorphic signature technique. Our method can detect misbehaving agents without the common, yet sometimes infeasible, assumption of local good majority. Another significant contribution presented in this paper is an original component dropping approach for mitigating the communication overhead during the consensus process. From both theoretical and engineering perspectives, we study how the consensus can still be reached when a big portion of elements in exchanged message vectors are dropped. In our preliminary numerical results, roughly 80% data transmission can be reduced per epoch at the expense of 35% extra epochs to reach the consensus, implying 73% reduction in terms of communication overhead.