  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.
