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In this paper, we study the Kurdyka-{\L}ojasiewicz (KL) exponent, animportant quantity for analyzing the convergence rate of first-order methods.Specifically, we develop various calculus rules to deduce the KL exponent ofnew (possibly nonconvex and nonsmooth) functions formed from functions withknown KL exponents. In addition, we show that the well-studied Luo-Tseng errorbound together with a mild assumption on the separation of stationary valuesimplies that the KL exponent is $\frac12$. The Luo-Tseng error bound is knownto hold for a large class of concrete structured optimization problems, andthus we deduce the KL exponent of a large class of functions whose exponentswere previously unknown. Building upon this and the calculus rules, we are thenable to show that for many convex or nonconvex optimization models forapplications such as sparse recovery, their objective function's KL exponent is$\frac12$. This includes the least squares problem with smoothly clippedabsolute deviation (SCAD) regularization or minimax concave penalty (MCP)regularization and the logistic regression problem with $\ell_1$regularization. Since many existing local convergence rate analysis forfirst-order methods in the nonconvex scenario relies on the KL exponent, ourresults enable us to obtain explicit convergence rate for various first-ordermethods when they are applied to a large variety of practical optimizationmodels. Finally, we further illustrate how our results can be applied toestablishing local linear convergence of the proximal gradient algorithm andthe inertial proximal algorithm with constant step-sizes for some specificmodels that arise in sparse recovery.

Calculus of the exponent of Kurdyka-{\L}ojasiewicz inequality and its applications to linear convergence of first-order methods