Stability of adaptive network algorithms in multitask environments

Abstract

Multitask diffusion and consensus algorithms have been proposed to solve distributed optimization problems in real time from streaming data, where the nodes cooperate to estimate node-dependent parameters. A fundamental question motivating their development is: how do standard diffusion LMS and real time consensus (RTC) algorithms perform in such environments? Previously, only moment stability has been studied via energy conservation arguments which require unrealistic white regressor assumptions. We present realization-wise stability results for the first time, under both correlated regressors and noise. This allows us to give an explicit expression for the limit point of these algorithms in the slow adaptation regime. We see that the Adapt-Then-Combine (ATC), Combine-Then-Adapt (CTA) and RTC algorithms all converge to a common Pareto optimal solution of the distributed MMSE problem.