Convergence analysis under consistent error bounds

TL;DR

This paper introduces a unified framework for various error bounds in convex feasibility problems, enabling explicit convergence rate analysis of projection algorithms using advanced mathematical tools.

Contribution

It develops the concept of consistent error bound functions, extending classical bounds and applying Karamata theory to analyze convergence rates in conic feasibility problems.

Findings

Convergence rates are explicitly expressed in terms of the error bound functions.

The framework encompasses Lipschitzian, H"olderian, logarithmic, and entropic error bounds.

Algorithms' convergence depends on the singularity degree of the problem.

Abstract

We introduce the notion of consistent error bound functions which provides a unifying framework for error bounds for multiple convex sets. This framework goes beyond the classical Lipschitzian and H\"olderian error bounds and includes logarithmic and entropic error bounds found in the exponential cone. It also includes the error bounds obtainable under the theory of amenable cones. Our main result is that the convergence rate of several projection algorithms for feasibility problems can be expressed explicitly in terms of the underlying consistent error bound function. Another feature is the usage of Karamata theory and functions of regular variations which allows us to reason about convergence rates while bypassing certain complicated expressions. Finally, applications to conic feasibility problems are given and we show that a number of algorithms have convergence rates depending explicitly on the singularity degree of the problem.