First-order Methods Almost Always Avoid Saddle Points

Jason D. Lee; Ioannis Panageas; Georgios Piliouras; Max Simchowitz; Michael I. Jordan; Benjamin Recht

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arXiv · 1710.07406

First-order Methods Almost Always Avoid Saddle Points

Jason D. Lee·Ioannis Panageas·Georgios Piliouras·Max Simchowitz·Michael I. Jordan·Benjamin Recht

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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.

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Jason D. Lee, Ioannis Panageas, Georgios Piliouras, Max Simchowitz, Michael I. Jordan, Benjamin Recht. 2017-10-20. First-order Methods Almost Always Avoid Saddle Points. https://arxiv.org/abs/1710.07406

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