Abstract. We examine the behavior of accelerated gradient methods in smooth nonconvex unconstrained optimization, focusing in particular on their behavior near strict saddle points. Accelerated methods are iterative methods that typically step along a direction that is a linear combination of the previous step and the gradient of the function evaluated at a point at or near the current iterate. (The previous step encodes gradient information from earlier stages in the iterative process.) We show by means of the stable manifold theorem that the heavy-ball method method is unlikely to converge to strict saddle points, which are points at which the gradient of the objective is zero but the Hessian has at least one negative eigenvalue. We then examine the behavior of the heavyball method and Nesterov's accelerated gradient method in the vicinity of a strict saddle point of a nonconvex quadratic function, showing that both methods can diverge from this point more rapidly than the steepest descent method.
The behavior of an accelerated gradient method near saddle points of unconstrained instances of (1) (i.e., with h = 0) is studied in REF .
Behavior of Accelerated Gradient Methods Near Critical Points of Nonconvex Functions
arXiv: Optimization and Control
Computer Science Mathematics