Exit Time Analysis for Approximations of Gradient Descent Trajectories Around Saddle Points

TL;DR

This paper provides a geometric analysis of gradient descent trajectories near saddle points, deriving explicit exit-time estimates that depend on problem geometry and initial conditions.

Contribution

It introduces a novel geometric framework using matrix perturbation theory to analyze and approximate gradient trajectories around saddle points, with explicit exit-time formulas.

Findings

Linear exit-time estimates for gradient descent near saddles

Dependence of escape time on problem dimension and conditioning

Approximate gradient trajectories based on initial conditions

Abstract

This paper considers the problem of understanding the exit time for trajectories of gradient-related first-order methods from saddle neighborhoods under some initial boundary conditions. Given the 'flat' geometry around saddle points, first-order methods can struggle to escape these regions in a fast manner due to the small magnitudes of gradients encountered. In particular, while it is known that gradient-related first-order methods escape strict-saddle neighborhoods, existing analytic techniques do not explicitly leverage the local geometry around saddle points in order to control behavior of gradient trajectories. It is in this context that this paper puts forth a rigorous geometric analysis of the gradient-descent method around strict-saddle neighborhoods using matrix perturbation theory. In doing so, it provides a key result that can be used to generate an approximate gradient trajectory for any given initial conditions. In addition, the analysis leads to a linear exit-time solution for gradient-descent method under certain necessary initial conditions, which explicitly bring out the dependence on problem dimension, conditioning of the saddle neighborhood, and more, for a class of strict-saddle functions.