Network Newton-Part II: Convergence Rate and Implementation

TL;DR

This paper analyzes the convergence rate of Network Newton methods for decentralized optimization, introduces adaptive variants, and demonstrates through numerical experiments that these methods reduce iteration and communication costs compared to gradient descent.

Contribution

It provides a detailed convergence rate analysis, introduces adaptive NN-$K$ methods that converge to the original problem, and validates improvements via numerical experiments.

Findings

Network Newton methods exhibit at least linear convergence rate.

The quadratic convergence phase duration increases with the truncation parameter K.

Numerical results show fewer iterations and communication costs compared to distributed gradient descent.

Abstract

The use of network Newton methods for the decentralized optimization of a sum cost distributed through agents of a network is considered. Network Newton methods reinterpret distributed gradient descent as a penalty method, observe that the corresponding Hessian is sparse, and approximate the Newton step by truncating a Taylor expansion of the inverse Hessian. Truncating the series at $K$ terms yields the NN-$K$ that requires aggregating information from $K$ hops away. Network Newton is introduced and shown to converge to the solution of the penalized objective function at a rate that is at least linear in a companion paper [3]. The contributions of this work are: (i) To complement the convergence analysis by studying the methods' rate of convergence. (ii) To introduce adaptive formulations that converge to the optimal argument of the original objective. (iii) To perform numerical evaluations of NN-$K$ methods. The convergence analysis relates the behavior of NN-$K$ with the behavior of (regular) Newton's method and shows that the method goes through a quadratic convergence phase in a specific interval. The length of this quadratic phase grows with $K$ and can be made arbitrarily large. The numerical experiments corroborate reductions in the number of iterations and the communication cost that are necessary to achieve convergence relative to distributed gradient descent.