Stochastic Subgradient Method Converges on Tame Functions

  • Damek Davis
  • Dmitriy DrusvyatskiyEmail author
  • Sham Kakade
  • Jason D. Lee


This work considers the question: what convergence guarantees does the stochastic subgradient method have in the absence of smoothness and convexity? We prove that the stochastic subgradient method, on any semialgebraic locally Lipschitz function, produces limit points that are all first-order stationary. More generally, our result applies to any function with a Whitney stratifiable graph. In particular, this work endows the stochastic subgradient method, and its proximal extension, with rigorous convergence guarantees for a wide class of problems arising in data science—including all popular deep learning architectures.


Subgradient Proximal Stochastic subgradient method Differential inclusion Lyapunov function Semialgebraic Tame 

Mathematics Subject Classification

65K05 65K10 34A60 90C15 



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Copyright information

© SFoCM 2018

Authors and Affiliations

  • Damek Davis
    • 1
  • Dmitriy Drusvyatskiy
    • 2
    Email author
  • Sham Kakade
    • 3
  • Jason D. Lee
    • 4
  1. 1.School of Operations Research and Information EngineeringCornell UniversityIthacaUSA
  2. 2.Department of MathematicsUniversity of WashingtonSeattleUSA
  3. 3.Departments of Statistics and Computer ScienceUniversity of WashingtonSeattleUSA
  4. 4.Data Science and Operations Department, Marshall School of BusinessUniversity of Southern CaliforniaLos AngelesUSA

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