  We study the decentralized online regularized linear regression algorithm
over random time-varying graphs. At each time step, every node 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. It is not required that the
regression matrices and graphs satisfy special statistical assumptions such as
mutual independence, spatio-temporal independence or stationarity. We develop
the nonnegative supermartingale inequality of the estimation error, and prove
that the estimations of all nodes converge to the unknown true parameter vector
almost surely if the algorithm gains, graphs and regression matrices jointly
satisfy the sample path spatio-temporal persistence of excitation condition.
Especially, this condition holds by choosing appropriate algorithm gains 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. In addition, we prove that the regret upper
bound is $O(T^{1-\tau}\ln T)$, where $\tau\in (0.5,1)$ is a constant depending
on the algorithm gains.
