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

arXiv: https://arxiv.org/abs/1802.08941 · HTML: https://mlsanity.com/paper/1802.08941/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 25 Feb 2018.

## Built on by (most cited first, 1 of 1)

- [Delayed Projection Techniques for Linearly Constrained Problems: Convergence Rates, Acceleration, and Applications](https://mlsanity.com/paper/2101.01505/) (2021)

## Abstract (arXiv)

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.