First-order Methods Almost Always Avoid Saddle Points

Understand

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.