Boundary Conditions for Linear Exit Time Gradient Trajectories Around Saddle Points: Analysis and Algorithm
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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 within a geometrical landscape that has a maximum number of strict-saddle points, 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.
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