A Decentralized Quasi-Newton Method for Dual Formulations of Consensus Optimization

TL;DR

This paper introduces a decentralized quasi-Newton method for consensus optimization that improves convergence in poorly conditioned problems by approximating curvature information without requiring second order data.

Contribution

It develops a dual D-BFGS algorithm that incorporates curvature correction in a fully decentralized manner for consensus optimization problems.

Findings

Convergence is formally proven for both synchronous and asynchronous settings.

Numerical experiments show performance advantages over existing decentralized algorithms.

The method effectively handles poorly conditioned problems where first order methods fail.

Abstract

This paper considers consensus optimization problems where each node of a network has access to a different summand of an aggregate cost function. Nodes try to minimize the aggregate cost function, while they exchange information only with their neighbors. We modify the dual decomposition method to incorporate a curvature correction inspired by the Broyden-Fletcher-Goldfarb-Shanno (BFGS) quasi-Newton method. The resulting dual D-BFGS method is a fully decentralized algorithm in which nodes approximate curvature information of themselves and their neighbors through the satisfaction of a secant condition. Dual D-BFGS is of interest in consensus optimization problems that are not well conditioned, making first order decentralized methods ineffective, and in which second order information is not readily available, making decentralized second order methods infeasible. Asynchronous implementation is discussed and convergence of D-BFGS is established formally for both synchronous and asynchronous implementations. Performance advantages relative to alternative decentralized algorithms are shown numerically.