Newton-like method with diagonal correction for distributed optimization

TL;DR

This paper introduces a novel distributed Newton-like optimization method called DQN that efficiently approximates the Hessian inverse by splitting and weighting, achieving faster convergence in networked convex optimization problems.

Contribution

The paper proposes a new class of distributed quasi-Newton methods that approximate the Hessian inverse efficiently, overcoming the challenge of dense inverse Hessians in distributed settings.

Findings

DQN methods outperform traditional gradient methods in convergence speed.

Different variants of DQN balance communication and computation effectively.

Simulations confirm the effectiveness of the proposed methods.

Abstract

We consider distributed optimization problems where networked nodes cooperatively minimize the sum of their locally known convex costs. A popular class of methods to solve these problems are the distributed gradient methods, which are attractive due to their inexpensive iterations, but have a drawback of slow convergence rates. This motivates the incorporation of second-order information in the distributed methods, but this task is challenging: although the Hessians which arise in the algorithm design respect the sparsity of the network, their inverses are dense, hence rendering distributed implementations difficult. We overcome this challenge and propose a class of distributed Newton-like methods, which we refer to as Distributed Quasi Newton (DQN). The DQN family approximates the Hessian inverse by: 1) splitting the Hessian into its diagonal and off-diagonal part, 2) inverting the diagonal part, and 3) approximating the inverse of the off-diagonal part through a weighted linear function. The approximation is parameterized by the tuning variables which correspond to different splittings of the Hessian and by different weightings of the off-diagonal Hessian part. Specific choices of the tuning variables give rise to different variants of the proposed general DQN method -- dubbed DQN-0, DQN-1 and DQN-2 -- which mutually trade-off communication and computational costs for convergence. Simulations demonstrate the effectiveness of the proposed DQN methods.