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

## TL;DR

This paper introduces a perturbed alternating gradient descent algorithm that efficiently converges to second-order stationary solutions in nonconvex optimization, avoiding saddle points and local maxima with high probability.

## Contribution

It proposes a novel perturbed AGD variant that guarantees convergence to second-order stationary solutions with a sublinear rate, a first in alternating algorithms.

## Key findings

- Converges to second-order stationary solutions with high probability.
- Achieves a sublinear convergence rate of (polylog(d)/^{7/3}) iterations.
- Avoids saddle points and local maxima effectively.

## 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 $\mathcal{O}(\text{polylog}(d)/\epsilon^{7/3})$ iterations to achieve SS2 with high probability [where polylog$(d)$ is polynomial of the logarithm of dimension $d$ of the problem].