Gradient Primal-Dual Algorithm Converges to Second-Order Stationary Solution for Nonconvex Distributed Optimization Over Networks
Mingyi Hong, Meisam Razaviyayn, Jason Lee
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.
BibTeX
@InProceedings{pmlr-v80-hong18a,
title = {Gradient Primal-Dual Algorithm Converges to Second-Order Stationary Solution for Nonconvex Distributed Optimization Over Networks},
author = {Hong, Mingyi and Razaviyayn, Meisam and Lee, Jason},
booktitle = {Proceedings of the 35th International Conference on Machine Learning},
pages = {2009--2018},
year = {2018},
editor = {Dy, Jennifer and Krause, Andreas},
volume = {80},
series = {Proceedings of Machine Learning Research},
month = {10--15 Jul},
publisher = {PMLR},
pdf = {http://proceedings.mlr.press/v80/hong18a/hong18a.pdf},
url = {https://proceedings.mlr.press/v80/hong18a.html},
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.}
}