Convergence of the Distributed SG Algorithm Under Cooperative Excitation Condition

In this article, a distributed stochastic gradient (SG) algorithm is proposed where the estimators are aimed to collectively estimate an unknown time-invariant parameter from a set of noisy measurements obtained by distributed sensors. The proposed distributed SG algorithm combines the consensus strategy of the estimation of neighbors with the diffusion of regression vectors. For the theoretical investigation of the proposed algorithm, the main challenge lies in analyzing the influence of the Laplacian matrix on the state transition matrix and the properties of the product of nonindependent and nonstationary random matrices. Some analysis techniques such as graph theory and martingale theory are used to deal with the above issues. A cooperative excitation condition is introduced, under which the convergence of the distributed SG algorithm can be obtained without relying on the independency or stationarity assumptions of regression vectors which are commonly used in the existing literature. Furthermore, the convergence rate of the algorithm can be established. Finally, we show that all the sensors can cooperate to fulfill the estimation task even though any individual sensor cannot by a simulation example.