## On the Sublinear Convergence of Randomly Perturbed Alternating Gradient Descent to Second Order Stationary Solutions

2020-01-01
Lu, Songtao
Hong, Mingyi
Person
Wang, Zhengdao
Professor
##### Organizational Units
Organizational Unit
Electrical and Computer Engineering

The Department of Electrical and Computer Engineering (ECpE) contains two focuses. The focus on Electrical Engineering teaches students in the fields of control systems, electromagnetics and non-destructive evaluation, microelectronics, electric power & energy systems, and the like. The Computer Engineering focus teaches in the fields of software systems, embedded systems, networking, information security, computer architecture, etc.

History
The Department of Electrical Engineering was formed in 1909 from the division of the Department of Physics and Electrical Engineering. In 1985 its name changed to Department of Electrical Engineering and Computer Engineering. In 1995 it became the Department of Electrical and Computer Engineering.

Dates of Existence
1909-present

Historical Names

• Department of Electrical Engineering (1909-1985)
• Department of Electrical Engineering and Computer Engineering (1985-1995)

Related Units

##### Abstract

The alternating gradient descent (AGD) is a simple but popular algorithm which has been applied to problems in optimization, machine learning, data ming, and signal processing, etc. The algorithm updates two blocks of variables in an alternating manner, in which a gradient step is taken on one block, while keeping the remaining block fixed. When the objective function is nonconvex, it is well-known the AGD converges to the first-order stationary solution with a global sublinear rate.
In this paper, we show that a variant of AGD-type algorithms will not be trapped by "bad" stationary solutions such as saddle points and local maximum points. In particular, we consider a smooth unconstrained optimization problem, and propose a perturbed AGD (PA-GD) which converges (with high probability) to the set of second-order stationary solutions (SS2) with a global sublinear rate. To the best of our knowledge, this is the first alternating type algorithm which takes O(polylog(d)/ϵ7/3) iterations to achieve SS2 with high probability [where polylog(d) is polynomial of the logarithm of dimension d of the problem].

This is a pre-print of the article Lu, Songtao, Mingyi Hong, and Zhengdao Wang. "On the sublinear convergence of randomly perturbed alternating gradient descent to second order stationary solutions." arXiv preprint arXiv:1802.10418 (2018). Posted with permission.