Linear Convergence of Gradient and Proximal-Gradient Methods Under the Polyak-Łojasiewicz Condition

  • Hamed Karimi
  • Julie Nutini
  • Mark SchmidtEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9851)


In 1963, Polyak proposed a simple condition that is sufficient to show a global linear convergence rate for gradient descent. This condition is a special case of the Łojasiewicz inequality proposed in the same year, and it does not require strong convexity (or even convexity). In this work, we show that this much-older Polyak-Łojasiewicz (PL) inequality is actually weaker than the main conditions that have been explored to show linear convergence rates without strong convexity over the last 25 years. We also use the PL inequality to give new analyses of coordinate descent and stochastic gradient for many non-strongly-convex (and some non-convex) functions. We further propose a generalization that applies to proximal-gradient methods for non-smooth optimization, leading to simple proofs of linear convergence for support vector machines and L1-regularized least squares without additional assumptions.


Gradient descent Coordinate descent Stochastic gradient Variance-reduction Boosting Support vector machines L1-regularization 



We would like to thank Simon LaCoste-Julien and Martin Takáč for valuable discussions. This research was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC RGPIN-06068-2015). Julie Nutini is funded by a UBC Four Year Doctoral Fellowship (4YF) and Hamed Karimi is support by a Mathematics of Information Technology and Complex Systems (MITACS) Elevate Fellowship.

Supplementary material

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Supplementary material 1 (pdf 230 KB)


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Copyright information

© Springer International Publishing AG 2016

Authors and Affiliations

  1. 1.Department of Computer ScienceUniversity of British ColumbiaVancouverCanada

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