# Decentralized Sparse Multitask RLS over Networks

## TL;DR

This paper introduces decentralized algorithms for sparse multitask recursive least squares over networks, enabling efficient, real-time collaborative estimation of multiple unknown vectors with theoretical convergence analysis.

## Contribution

It proposes novel decentralized online ADMM and subgradient algorithms for sparse multitask RLS, with simplified implementation and convergence guarantees.

## Key findings

- Algorithms effectively estimate multiple unknown vectors in networked systems.
- The subgradient method has proven convergence with an explicit error bound.
- Numerical simulations demonstrate the algorithms' accuracy and low complexity.

## Abstract

Distributed adaptive signal processing has attracted much attention in the recent decade owing to its effectiveness in many decentralized real-time applications in networked systems. Because many natural signals are highly sparse with most entries equal to zero, several decentralized sparse adaptive algorithms have been proposed recently. Most of them is focused on the single task estimation problems, in which all nodes receive data associated with the same unknown vector and collaborate to estimate it. However, many applications are inherently multitask oriented and each node has its own unknown vector different from others'. The related multitask estimation problem benefits from collaborations among the nodes as neighbor nodes usually share analogous properties and thus similar unknown vectors. In this work, we study the distributed sparse multitask recursive least squares (RLS) problem over networks. We first propose a decentralized online alternating direction method of multipliers (ADMM) algorithm for the formulated RLS problem. The algorithm is simplified for easy implementation with closed-form computations in each iteration and low storage requirements. Moreover, to further reduce the complexity, we present a decentralized online subgradient method with low computational overhead. We theoretically analyze the convergence behavior of the proposed subgradient method and derive an error bound related to the network topology and algorithm parameters. The effectiveness of the proposed algorithms is corroborated by numerical simulations and an accuracy-complexity tradeoff between the proposed two algorithms is highlighted.