Localization of Data Injection Attacks on Distributed M-Estimation

This paper examines data injection attacks on distributed statistical estimation. We consider a dynamically changing distributed network consisting of N agents exchanging information over time. The N agents share the common goal of minimizing a joint objective function, which is the average of the private objective functions in a distributed manner. The private objective function is a realization of an objective function known to all the agents, but uses private data known to the agent alone. The agents' data are independent and identically distributed. We have previously proposed a novel data injection attack on the Distributed Projected Gradient (DPG) algorithm which is performed locally by malicious nodes in the network that steer the network's final state to a state of their choice. The proposed attack cannot be detected using previously described techniques. We propose a new detection and localization scheme, performed in a single instance unlike other methods that require the algorithm to run for many instances to acquire statistics over time. This detection and localization scheme is performed by each agent and is purely local, and does not involve decisions made by other agents. Whenever an agent suspects another agent to be an attacker, it will block its data, and maintain convergence to the true optimal state. We provide exponential bounds for the probability of false alarm and probability of attacker detection and localization. Simulations show that when all the attackers are detected and isolated by each agent, the network will recover and converge to the true optimal state.