Calculus of the Exponent of Kurdyka–Łojasiewicz Inequality and Its Applications to Linear Convergence of First-Order Methods

Guoyin Li, Ting Kei Pong

Research output: Journal article publicationJournal articleAcademic researchpeer-review

125 Citations (Scopus)


Specifically, we develop various calculus rules to deduce the KL exponent of new (possibly nonconvex and nonsmooth) functions formed from functions with known KL exponents. In addition, we show that the well-studied Luo–Tseng error bound together with a mild assumption on the separation of stationary values implies that the KL exponent is (Formula presented.). The Luo–Tseng error bound is known to hold for a large class of concrete structured optimization problems, and thus we deduce the KL exponent of a large class of functions whose exponents were previously unknown. Building upon this and the calculus rules, we are then able to show that for many convex or nonconvex optimization models for applications such as sparse recovery, their objective function’s KL exponent is (Formula presented.). This includes the least squares problem with smoothly clipped absolute deviation regularization or minimax concave penalty regularization and the logistic regression problem with (Formula presented.) regularization. Since many existing local convergence rate analysis for first-order methods in the nonconvex scenario relies on the KL exponent, our results enable us to obtain explicit convergence rate for various first-order methods when they are applied to a large variety of practical optimization models. Finally, we further illustrate how our results can be applied to establishing local linear convergence of the proximal gradient algorithm and the inertial proximal algorithm with constant step sizes for some specific models that arise in sparse recovery.
Original languageEnglish
Pages (from-to)1199–1232
Number of pages34
JournalFoundations of Computational Mathematics
Publication statusPublished - 1 Oct 2018


  • Convergence rate
  • First-order methods
  • Kurdyka–Łojasiewicz inequality
  • Linear convergence
  • Luo–Tseng error bound
  • Sparse optimization

ASJC Scopus subject areas

  • Analysis
  • Computational Theory and Mathematics
  • Computational Mathematics
  • Applied Mathematics


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