Accelerated Gradient Methods for Nonconvex Optimization: Escape Trajectories From Strict Saddle Points and Convergence to Local Minima

Abstract

This paper considers the problem of understanding the behavior of a general class of accelerated gradient methods on smooth nonconvex functions. Motivated by some recent works that have proposed effective algorithms, based on Polyak’s heavy ball method and the Nesterov accelerated gradient method, to achieve convergence to a local minimum of nonconvex functions, this work proposes a broad class of Nesterov-type accelerated methods and puts forth a rigorous study of these methods encompassing the escape from saddle points and convergence to local minima through both an asymptotic and a non-asymptotic analysis. In the asymptotic regime, this paper answers an open question of whether Nesterov’s accelerated gradient method (NAG) with variable momentum parameter avoids strict saddle points almost surely. This work also develops two metrics of asymptotic rates of convergence and divergence, and evaluates these two metrics for several popular standard accelerated methods such as the NAG and Nesterov’s accelerated gradient with constant momentum (NCM) near strict saddle points. In the non-asymptotic regime, this work provides an analysis that leads to the “linear” exit time estimates from strict saddle neighborhoods for trajectories of these accelerated methods as well the necessary conditions for the existence of such trajectories. Finally, this work studies a sub-class of accelerated methods that can converge in convex neighborhoods of nonconvex functions with a near optimal rate to a local minimum and at the same time this sub-class offers superior saddle-escape behavior compared to that of NAG.