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We examine the behavior of accelerated gradient methods in smooth nonconvexunconstrained optimization, focusing in particular on their behavior nearstrict saddle points. Accelerated methods are iterative methods that typicallystep along a direction that is a linear combination of the previous step andthe gradient of the function evaluated at a point at or near the currentiterate. (The previous step encodes gradient information from earlier stages inthe iterative process.) We show by means of the stable manifold theorem thatthe 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 Hessianhas at least one negative eigenvalue. We then examine the behavior of theheavy-ball method and other accelerated gradient methods in the vicinity of astrict saddle point of a nonconvex quadratic function, showing that bothmethods can diverge from this point more rapidly than the steepest-descentmethod.

Behavior of Accelerated Gradient Methods Near Critical Points of Nonconvex Functions