First-order Methods Almost Always Avoid Saddle Points

TLDR: In this article, it was shown that first-order methods avoid saddle points for almost all initializations, and that neither access to second-order derivative information nor randomness beyond initialization is necessary to provably avoid saddle point.

Abstract: We establish that first-order methods avoid saddle points for almost all initializations. Our results apply to a wide variety of first-order methods, including gradient descent, block coordinate descent, mirror descent and variants thereof. The connecting thread is that such algorithms can be studied from a dynamical systems perspective in which appropriate instantiations of the Stable Manifold Theorem allow for a global stability analysis. Thus, neither access to second-order derivative information nor randomness beyond initialization is necessary to provably avoid saddle points.