Learning over Multitask Graphs -- Part I: Stability Analysis

TL;DR

This paper introduces a multitask optimization framework over graphs with a diffusion strategy that adapts to streaming data, ensuring stability and small estimation errors under certain conditions.

Contribution

It formulates a new multitask inference approach incorporating graph smoothness and provides stability analysis for the proposed diffusion strategy.

Findings

The diffusion strategy converges under specific step-size conditions.

Estimation errors are small and proportional to the step-size.

The approach effectively incorporates graph structure into distributed inference.

Abstract

This paper formulates a multitask optimization problem where agents in the network have individual objectives to meet, or individual parameter vectors to estimate, subject to a smoothness condition over the graph. The smoothness condition softens the transition in the tasks among adjacent nodes and allows incorporating information about the graph structure into the solution of the inference problem. A diffusion strategy is devised that responds to streaming data and employs stochastic approximations in place of actual gradient vectors, which are generally unavailable. The approach relies on minimizing a global cost consisting of the aggregate sum of individual costs regularized by a term that promotes smoothness. We show in this Part I of the work, under conditions on the step-size parameter, that the adaptive strategy induces a contraction mapping and leads to small estimation errors on the order of the small step-size. The results in the accompanying Part II will reveal explicitly the influence of the network topology and the regularization strength on the network performance and will provide insights into the design of effective multitask strategies for distributed inference over networks.