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Gradient-related first-order methods have become the workhorse of large-scalenumerical optimization problems. Many of these problems involve nonconvexobjective functions with multiple saddle points, which necessitates anunderstanding of the behavior of discrete trajectories of first-order methodswithin the geometrical landscape of these functions. This paper concernsconvergence of first-order discrete methods to a local minimum of nonconvexoptimization problems that comprise strict-saddle points within the geometricallandscape. To this end, it focuses on analysis of discrete gradienttrajectories around saddle neighborhoods, derives sufficient conditions underwhich these trajectories can escape strict-saddle neighborhoods in linear time,explores the contractive and expansive dynamics of these trajectories inneighborhoods of strict-saddle points that are characterized by gradients ofmoderate magnitude, characterizes the non-curving nature of these trajectories,and highlights the inability of these trajectories to re-enter theneighborhoods around strict-saddle points after exiting them. Based on theseinsights and analyses, the paper then proposes a simple variant of the vanillagradient descent algorithm, termed Curvature Conditioned Regularized GradientDescent (CCRGD) algorithm, which utilizes a check for an initial boundarycondition to ensure its trajectories can escape strict-saddle neighborhoods inlinear time. Convergence analysis of the CCRGD algorithm, which includes itsrate of convergence to a local minimum, is also presented in the paper.Numerical experiments are then provided on a test function as well as alow-rank matrix factorization problem to evaluate the efficacy of the proposedalgorithm.

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