🤖 AI Summary
This paper studies decentralized online regularized linear regression over random time-varying graphs. Each node performs parallel online updates incorporating a data-driven innovation term, a noisy distributed consensus term, and an ℓ₂-regularization term—without assuming independence, stationarity, or spatiotemporal decoupling of regression features, graph topology, or noise. We introduce the novel “sample-path spatiotemporal persistent excitation” condition and, for the first time, establish both global almost-sure convergence and mean-square convergence under jointly nonstationary, non-independent, and communication-noisy settings. Theoretically, we derive an upper bound of 𝒪(T^{1−τ} ln T) (with τ ∈ (0.5, 1)) on the estimation error regret, significantly broadening the applicability of decentralized online learning to dynamic, heterogeneous networks.
📝 Abstract
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 $O(T^{1- au}ln T)$, where $ auin (0.5,1)$ is a constant depending on the algorithm gains.