Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Concrete convergence rates for common fixed point problems under Karamata regularity

Tianxiang Liu, Bruno F. Lourenço

TL;DR

This work develops concrete convergence rates for common fixed point problems solved by quasi-cyclic algorithms under Karamata regularity. It introduces joint Karamata regularity (JKR) and provides two pathways to rates: (i) index-based rates from the regular variation index ρ, and (ii) tighter rates via arrow inverses and asymptotic equivalence, with explicit examples in non-Hölderian settings. The analysis unifies exotic error bounds, exponential-cone problems, and DR/AP methods, yielding rate formulas that can involve Lambert W functions and other non-Hölder phenomena. Moreover, it links definable operators in o-minimal structures to JK regularity, showing that in polynomially bounded settings rates are at least sublinear and often Hölder-like, broadening the applicability of rate results beyond semialgebraic data.

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, Hölderian error bounds and Hölder 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 Hölderian-like assumptions many algorithms converge linearly/sublinearly (depending on the exponent), little it is known when the underlying problem data does not satisfy Hölderian 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-Hölderian 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.

Concrete convergence rates for common fixed point problems under Karamata regularity

TL;DR

This work develops concrete convergence rates for common fixed point problems solved by quasi-cyclic algorithms under Karamata regularity. It introduces joint Karamata regularity (JKR) and provides two pathways to rates: (i) index-based rates from the regular variation index ρ, and (ii) tighter rates via arrow inverses and asymptotic equivalence, with explicit examples in non-Hölderian settings. The analysis unifies exotic error bounds, exponential-cone problems, and DR/AP methods, yielding rate formulas that can involve Lambert W functions and other non-Hölder phenomena. Moreover, it links definable operators in o-minimal structures to JK regularity, showing that in polynomially bounded settings rates are at least sublinear and often Hölder-like, broadening the applicability of rate results beyond semialgebraic data.

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, Hölderian error bounds and Hölder regular operators. By concrete, we mean that the rates we obtain are explicitly expressed in terms of a function of the iteration number instead, of say, a function of the iterate . While it is well-known that under Hölderian-like assumptions many algorithms converge linearly/sublinearly (depending on the exponent), little it is known when the underlying problem data does not satisfy Hölderian 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-Hölderian 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.
Paper Structure (24 sections, 29 theorems, 211 equations, 1 figure)

This paper contains 24 sections, 29 theorems, 211 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Our contributions
  3. Preliminaries and basic notions from Karamata theory
  4. Calculus rules
  5. Potter's bounds
  6. Generalized inverses
  7. Karamata regularity and convergence rates
  8. General convergence theory
  9. Quasi-cylic algorithms
  10. A general convergence rate result
  11. Rates based on the index of regular variation
  12. Tighter rates
  13. Applications and examples
  14. Exotic error bounds and convergence rate of alternating projections
  15. An example with $\rho \in (0,\,1)$: a Hölder-entropic error bound
  16. ...and 9 more sections

Key Result

Lemma 2.1

Let $T$ be an $\alpha$-averaged ($\alpha \in (0,1)$) operator on $\mathcal{E}$. Then it satisfies

Figures (1)

  • Figure 1: In the plot on the left, the nonlinear part of the boundary of $C_1$ is described by the graph of $W_0$, i.e., points of the form $(x, W_0(x))$. In the plot on the right, the sets are reflected and translated so that the nonlinear part of the boundary of $\widehat{C}_1$ is the described by $(x, \widehat{W}(x))$, where $\widehat{W}$ is $W_0^{-1}$ followed by a shift ensuring that $\widehat{W}(0) = 0$ holds.

Theorems & Definitions (66)

  • Lemma 2.1: $\alpha$-averaged operator
  • Definition 2.2: Regularly varying functions
  • Remark 2.3: About the function domain and image
  • Remark 2.4: Examples of regularly varying functions
  • Lemma 2.5
  • proof
  • Proposition 2.6
  • proof
  • Definition 2.7
  • Definition 3.1: Karamata regularity
  • ...and 56 more