On the linear convergence of the stochastic gradient method with constant step-size

  • Volkan Cevher
  • Bằng Công VũEmail author
Original Paper


The strong growth condition (SGC) is known to be a sufficient condition for linear convergence of the stochastic gradient method using a constant step-size \(\gamma \) (SGM-CS). In this paper, we provide a necessary condition, for the linear convergence of SGM-CS, that is weaker than SGC. Moreover, when this necessary is violated up to a additive perturbation \(\sigma \), we show that both the projected stochastic gradient method using a constant step-size, under the restricted strong convexity assumption, and the proximal stochastic gradient method, under the strong convexity assumption, exhibit linear convergence to a noise dominated region, whose distance to the optimal solution is proportional to \({\gamma {\sigma }}\).


Stochastic gradient Linear convergence Strong growth condition 



The authors would like to thank Yen-Huan-Li, Ahmet Alacaoglu for useful discussions. We thank the referees for their suggestions and correction which helped to improve the first version of the manuscript. The work of B. Cong Vu and V. Cevher was supported by European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement no 725594 - time-data).


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Laboratory for Information and Inference Systems (LIONS)EPFLLausanneSwitzerland

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