Boundary Conditions for Linear Exit Time Gradient Trajectories Around Saddle Points: Analysis and Algorithm

TL;DR

This paper analyzes the behavior of gradient trajectories near saddle points in nonconvex optimization, deriving conditions for escape in linear time, and proposes a new gradient descent variant with proven convergence properties.

Contribution

It introduces the CCRGD algorithm, a simple gradient method with boundary conditions that guarantees escape from saddle neighborhoods in linear time, supported by convergence analysis.

Findings

Trajectories cannot re-enter saddle neighborhoods after escape.

The CCRGD algorithm effectively escapes saddle points in linear time.

Numerical experiments validate the algorithm's efficiency and convergence.

Abstract

Gradient-related first-order methods have become the workhorse of large-scale numerical optimization problems. Many of these problems involve nonconvex objective functions with multiple saddle points, which necessitates an understanding of the behavior of discrete trajectories of first-order methods within the geometrical landscape of these functions. This paper concerns convergence of first-order discrete methods to a local minimum of nonconvex optimization problems that comprise strict-saddle points within the geometrical landscape. To this end, it focuses on analysis of discrete gradient trajectories around saddle neighborhoods, derives sufficient conditions under which these trajectories can escape strict-saddle neighborhoods in linear time, explores the contractive and expansive dynamics of these trajectories in neighborhoods of strict-saddle points that are characterized by gradients of moderate magnitude, characterizes the non-curving nature of these trajectories, and highlights the inability of these trajectories to re-enter the neighborhoods around strict-saddle points after exiting them. Based on these insights and analyses, the paper then proposes a simple variant of the vanilla gradient descent algorithm, termed Curvature Conditioned Regularized Gradient Descent (CCRGD) algorithm, which utilizes a check for an initial boundary condition to ensure its trajectories can escape strict-saddle neighborhoods in linear time. Convergence analysis of the CCRGD algorithm, which includes its rate of convergence to a local minimum, is also presented in the paper. Numerical experiments are then provided on a test function as well as a low-rank matrix factorization problem to evaluate the efficacy of the proposed algorithm.