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

TL;DR

This paper develops calculus rules for the Kurdyka-Łojasiewicz (KL) exponent, enabling explicit convergence rate analysis of first-order methods in nonconvex optimization problems like sparse recovery.

Contribution

It introduces calculus rules for the KL exponent, links it to error bounds, and applies these to establish local linear convergence of algorithms in practical models.

Findings

KL exponent is 1/2 for many structured problems

Explicit convergence rates derived for first-order methods

Local linear convergence shown for specific sparse recovery models

Abstract

In this paper, we study the Kurdyka-{\L}ojasiewicz (KL) exponent, an important quantity for analyzing the convergence rate of first-order methods. 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 $\frac12$. 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 $\frac12$. This includes the least squares problem with smoothly clipped absolute 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 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.