Gradient Primal-Dual Algorithm Converges to Second-Order Stationary Solutions for Nonconvex Distributed Optimization

TL;DR

This paper proves that certain primal-dual first-order algorithms can reliably find second-order stationary solutions in non-convex distributed optimization problems, extending the understanding of convergence properties in this domain.

Contribution

It demonstrates for the first time that primal-dual algorithms using only first-order information can converge to second-order stationary solutions in non-convex settings.

Findings

Algorithms converge to ss2 with probability one from random initialization.

First global convergence result to ss2 for distributed non-convex learning.

Extends primal-only algorithms to primal-dual methods for ss2 convergence.

Abstract

In this work, we study two first-order primal-dual based algorithms, the Gradient Primal-Dual Algorithm (GPDA) and the Gradient Alternating Direction Method of Multipliers (GADMM), for solving a class of linearly constrained non-convex optimization problems. We show that with random initialization of the primal and dual variables, both algorithms are able to compute second-order stationary solutions (ss2) with probability one. This is the first result showing that primal-dual algorithm is capable of finding ss2 when only using first-order information, it also extends the existing results for first-order, but primal-only algorithms. An important implication of our result is that it also gives rise to the first global convergence result to the ss2, for two classes of unconstrained distributed non-convex learning problems over multi-agent networks.