The exact modulus of the generalized concave Kurdyka-{\L}ojasiewicz property

TL;DR

This paper introduces a generalized concave KL property with an exact modulus that optimally characterizes the desingularizing function, leading to sharper bounds on algorithmic convergence.

Contribution

It defines the exact modulus of the generalized concave KL property, solving an open problem and enabling precise convergence analysis for optimization algorithms.

Findings

Exact modulus is the smallest among all concave desingularizing functions.

Provides the sharpest upper bound for the total length of PALM algorithm iterates.

Illustrates the property with concrete examples.

Abstract

We introduce a generalized version of the concave Kurdyka-\L ojasiewicz (KL) property by employing nonsmooth desingularizing functions. We also present the exact modulus of the generalized concave KL property, which provides an answer to the open question regarding the optimal concave desingularizing function. The exact modulus is designed to be the smallest among all possible concave desingularizing functions. Examples are given to illustrate this pleasant property. In turn, using the exact modulus we provide the sharpest upper bound for the total length of iterates generated by the celebrated Bolte-Sabach-Teboulle PALM algorithm.