SIAM Journal on Optimization, Volume 30, Issue 4, Page 3029-3068, January 2020.
We consider distributed smooth nonconvex unconstrained optimization over net- works, 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 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 for first-order critical points is established for the latter class of algorithms.

中文翻译：

---

分布梯度算法的二阶保证

SIAM优化杂志，第30卷，第4期，第3029-3068页，2020年1月。
我们考虑将网络上的分布式平滑无凸无约束优化建模为连通图。我们在严格的鞍点附近检查了基于分布式梯度算法的行为。具体来说，我们确定（i）著名的分布式梯度下降算法可能会收敛到二阶平稳（SoS）解决方案的邻域；（ii）基于梯度跟踪的最新一类分布式算法-也可以在图上实现-可能会收敛到精确的SoS解决方案，从而避免了（严格的）鞍点。此外，为后一类算法建立了针对一阶临界点的新收敛速度结果。