Decentralized Online Linear Regression With the Regularization Parameter and Noises

We analyze the convergence of decentralized regularized linear regression algorithm. At each time step, every node over the random time-varying graphs runs an online estimation algorithm consisting of an innovation term processing its own new measurement, a consensus term taking a weighted sum of estimations of its own and its neighbors with additive and multiplicative communication noises and a regularization term preventing over-fitting. The sample path spatio-temporal persistence of excitation condition is established for the almost sure convergence. Especially, it is shown that this condition holds if the graphs are uniformly conditionally jointly connected and conditionally balanced, and the regression models of all nodes are uniformly conditionally spatio-temporally jointly observable, under which the algorithm converges in mean square and almost surely.