A Fast Distributed Asynchronous Newton-Based Optimization Algorithm

TL;DR

This paper introduces a novel asynchronous Newton-based algorithm for distributed convex optimization, achieving faster convergence rates and superior performance compared to existing asynchronous methods.

Contribution

The paper develops and analyzes the first asynchronous Newton-based method with proven linear and quadratic convergence rates for distributed optimization.

Findings

Algorithm guarantees almost sure convergence.

Achieves global linear and local quadratic convergence rates.

Numerical results show superior performance over existing methods.

Abstract

One of the most important problems in the field of distributed optimization is the problem of minimizing a sum of local convex objective functions over a networked system. Most of the existing work in this area focus on developing distributed algorithms in a synchronous setting under the presence of a central clock, where the agents need to wait for the slowest one to finish the update, before proceeding to the next iterate. Asynchronous distributed algorithms remove the need for a central coordinator, reduce the synchronization wait, and allow some agents to compute faster and execute more iterations. In the asynchronous setting, the only known algorithms for solving this problem could achieve either linear or sublinear rate of convergence. In this work, we built upon the existing literature to develop and analyze an asynchronous Newton-based method to solve a penalized version of the problem. We show that this algorithm guarantees almost sure convergence with global linear and local quadratic rate in expectation. Numerical studies confirm superior performance of our algorithm against other asynchronous methods.