Second-order Guarantees of Distributed Gradient Algorithms

TL;DR

This paper analyzes the behavior of distributed gradient algorithms in nonconvex optimization, showing that certain algorithms can avoid saddle points and converge to second-order stationary solutions, with new convergence rate results.

Contribution

It proves that gradient tracking algorithms can reach exact second-order solutions in distributed nonconvex optimization, improving understanding of their convergence properties.

Findings

Distributed gradient descent converges near second-order stationary points.

Gradient tracking algorithms can reach exact second-order solutions.

New convergence rates to first-order critical points are established.

Abstract

We consider distributed smooth nonconvex unconstrained optimization over networks, modeled as a connected graph. We examine the behavior of distributed gradient-based algorithms near strict saddle points. Specifically, we establish that (i) the renowned Distributed Gradient Descent (DGD) algorithm likely converges to a neighborhood of a Second-order Stationary (SoS) solution; and (ii) the more recent class of distributed algorithms based on gradient tracking--implementable also over digraphs--likely converges to exact SoS solutions, thus avoiding (strict) saddle-points. Furthermore, new convergence rate results to first-order critical points is established for the latter class of algorithms.