Concrete convergence rates for common fixed point problems under Karamata regularity

TL;DR

This paper introduces Karamata regular operators to derive explicit convergence rates for fixed point algorithms, extending beyond H"olderian assumptions and applying to non-H"olderian problems involving exponential and logarithmic data.

Contribution

It develops a new framework using Karamata regularity and regularly varying functions to obtain concrete convergence rates in non-H"olderian settings, including intermediate rates and connections to o-minimal geometry.

Findings

Derived explicit convergence rates involving Lambert W function.

Extended convergence analysis beyond H"olderian assumptions.

Linked Karamata regularity to o-minimal definable operators.

Abstract

We introduce the notion of Karamata regular operators, which is a notion of regularity that is suitable for obtaining concrete convergence rates for common fixed point problems. This provides a broad framework that includes, but goes beyond, H\"olderian error bounds and H\"older regular operators. By concrete, we mean that the rates we obtain are explicitly expressed in terms of a function of the iteration number $k$ instead, of say, a function of the iterate $x^k$. While it is well-known that under H\"olderian-like assumptions many algorithms converge linearly/sublinearly (depending on the exponent), little it is known when the underlying problem data does not satisfy H\"olderian assumptions, which may happen if a problem involves exponentials and logarithms. Our main innovation is the usage of the theory of regularly varying functions which we showcase by obtaining concrete convergence rates for quasi-cylic algorithms in non-H\"olderian settings. This includes certain rates that are neither sublinear nor linear but sit somewhere in-between, including a case where the rate is expressed via the Lambert W function. Finally, we connect our discussion to o-minimal geometry and show that, under mild assumptions, definable operators in any o-minimal structure are always Karamata regular.