Mathematical Programming

, Volume 176, Issue 1–2, pp 311–337 | Cite as

First-order methods almost always avoid strict saddle points

  • Jason D. LeeEmail author
  • Ioannis Panageas
  • Georgios Piliouras
  • Max Simchowitz
  • Michael I. Jordan
  • Benjamin Recht
Full Length Paper Series B


We establish that first-order methods avoid strict saddle points for almost all initializations. Our results apply to a wide variety of first-order methods, including (manifold) 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 strict saddle points.


Gradient descent Smooth optimization Saddle points Local minimum Dynamical systems 

Mathematics Subject Classification



Supplementary material

10107_2019_1374_MOESM1_ESM.pdf (649 kb)
Supplementary material 1 (pdf 649 KB)
10107_2019_1374_MOESM2_ESM.pdf (253 kb)
Supplementary material 2 (pdf 253 KB)


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society 2019

Authors and Affiliations

  1. 1.Data Sciences and OperationsUniversity of Southern CaliforniaLos AngelesUSA
  2. 2.Department of Information SystemsSingapore University of TechnologyTampinesSingapore
  3. 3.Engineering Systems and Design PillarSingapore University of Technology and DesignTampinesSingapore
  4. 4.Department of Electrical Engineering and Computer ScienceUC BerkeleyBerkeleyUSA

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