On the Convergence of Perturbed Distributed Asynchronous Stochastic Gradient Descent to Second Order Stationary Points in Non-convex Optimization

In this paper, the second order convergence of non-convex optimization in the asynchronous stochastic gradient descent (ASGD) algorithm is studied systematically. We investigate the behavior of ASGD near and away from saddle points. Different from the general stochastic gradient descent(SGD), we show that ASGD might return back even if it has escaped the saddle points, yet after staying near a strict saddle point for a long enough time ($O(T)$), ASGD will finally go away from strict saddle points. An inequality is given to describe the process of ASGD to escape saddle points. Using a novel Razumikhin-Lyapunov method, we show the exponential instability of the perturbed gradient dynamics near the strict saddle points and give a more detailed estimation about how the time delay parameter $T$ influences the speed to escape. In particular, we consider the optimization of smooth nonconvex functions, and propose a perturbed asynchronous stochastic gradient descent algorithm with guarantee of convergence to second order stationary points with high probability in $O(1/\epsilon^4)$ iterations. To the best of our knowledge, this is the first work on the second order convergence of asynchronous algorithm.