Decentralized Online Regularized Learning Over Random Time-Varying Graphs

TL;DR

This paper introduces a decentralized online regularized linear regression algorithm for time-varying graphs, proving convergence under general conditions and establishing a sublinear regret bound, applicable without strict statistical assumptions.

Contribution

It develops a convergence analysis for decentralized online regression over random graphs with minimal assumptions, and derives a sublinear regret bound for the algorithm.

Findings

All node estimations converge to the true parameter almost surely.

The algorithm achieves mean square convergence under certain graph conditions.

Regret grows sublinearly as $O(T^{1- au} ext{ln} T)$, indicating learning efficiency.

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 $O(T^{1-\tau}\ln T)$, where $\tau\in (0.5,1)$ is a constant depending on the algorithm gains.