# A decentralized proximal-gradient method with network independent step-sizes and separated convergence rates

## TL;DR

This paper introduces a decentralized proximal-gradient algorithm that allows agents to use independent step-sizes, achieves linear convergence under strong convexity, and separates convergence rate dependencies on objectives and network topology.

## Contribution

The proposed algorithm enables network-independent step-sizes and separates convergence rate dependencies, improving upon previous methods like PG-EXTRA.

## Key findings

- Agents can use uncoordinated step-sizes independent of network topology.
- Linear convergence is achievable without non-smooth terms under strong convexity.
- Convergence rate depends separately on objective functions and network structure.

## Abstract

This paper proposes a novel proximal-gradient algorithm for a decentralized optimization problem with a composite objective containing smooth and non-smooth terms. Specifically, the smooth and nonsmooth terms are dealt with by gradient and proximal updates, respectively. The proposed algorithm is closely related to a previous algorithm, PG-EXTRA \cite{shi2015proximal}, but has a few advantages. First of all, agents use uncoordinated step-sizes, and the stable upper bounds on step-sizes are independent of network topologies. The step-sizes depend on local objective functions, and they can be as large as those of the gradient descent. Secondly, for the special case without non-smooth terms, linear convergence can be achieved under the strong convexity assumption. The dependence of the convergence rate on the objective functions and the network are separated, and the convergence rate of the new algorithm is as good as one of the two convergence rates that match the typical rates for the general gradient descent and the consensus averaging. We provide numerical experiments to demonstrate the efficacy of the introduced algorithm and validate our theoretical discoveries.