# Exact Spectral-Like Gradient Method for Distributed Optimization

## TL;DR

This paper introduces a novel distributed spectral gradient method (DSG) for unconstrained optimization over networks, achieving R-linear convergence and demonstrating superior performance through simulations.

## Contribution

It develops the first exact distributed spectral gradient method with spectral-like step-sizes, enhancing convergence in distributed optimization.

## Key findings

- Achieves R-linear convergence under standard assumptions.
- Demonstrates improved performance in simulations.
- Introduces spectral-like step-size rules for distributed settings.

## Abstract

Since the initial proposal in the late 80s, spectral gradient methods continue to receive significant attention, especially due to their excellent numerical performance on various large scale applications. However, to date, they have not been sufficiently explored in the context of distributed optimization. In this paper, we consider unconstrained distributed optimization problems where $n$ nodes constitute an arbitrary connected network and collaboratively minimize the sum of their local convex cost functions. In this setting, building from existing exact distributed gradient methods, we propose a novel exact distributed gradient method wherein nodes' step-sizes are designed according to the novel rules akin to those in spectral gradient methods. We refer to the proposed method as Distributed Spectral Gradient method (DSG). The method exhibits R-linear convergence under standard assumptions for the nodes' local costs and safeguarding on the algorithm step-sizes. We illustrate the method's performance through simulation examples.