A Newton Tracking Algorithm with Exact Linear Convergence Rate for Decentralized Consensus Optimization

TL;DR

This paper introduces a Newton tracking algorithm for decentralized consensus optimization that achieves exact solutions with linear convergence, leveraging local second-order information and neighbor communication.

Contribution

It proposes a novel Newton tracking method with proven linear convergence to the exact solution in decentralized settings, connecting it to existing gradient and second-order algorithms.

Findings

Converges linearly to the exact optimal solution.

Outperforms existing methods in numerical experiments.

Validated theoretical convergence rate through simulations.

Abstract

This paper considers the decentralized consensus optimization problem defined over a network where each node holds a second-order differentiable local objective function. Our goal is to minimize the summation of local objective functions and find the exact optimal solution using only local computation and neighboring communication. We propose a novel Newton tracking algorithm, where each node updates its local variable along a local Newton direction modified with neighboring and historical information. We investigate the connections between the proposed Newton tracking algorithm and several existing methods, including gradient tracking and second-order algorithms. Under the strong convexity assumption, we prove that it converges to the exact optimal solution at a linear rate. Numerical experiments demonstrate the efficacy of Newton tracking and validate the theoretical findings.