Tight Coefficients of Averaged Operators via Scaled Relative Graph

TL;DR

This paper demonstrates that the averagedness coefficients for certain composite operators in optimization are tight, using the scaled relative graph as a geometric analysis tool, which has implications for convergence rate guarantees.

Contribution

It establishes the tightness of averagedness coefficients for specific operator compositions using the scaled relative graph, advancing understanding of convergence properties.

Findings

Averagedness coefficients for operator compositions are tight.

The scaled relative graph effectively analyzes operator properties.

Results impact convergence rate analysis in optimization algorithms.

Abstract

Many iterative methods in optimization are fixed-point iterations with averaged operators. As such methods converge at an $\mathcal{O}(1/k)$ rate with the constant determined by the averagedness coefficient, establishing small averagedness coefficients for operators is of broad interest. In this paper, we show that the averagedness coefficients of the composition of averaged operators by Ogura and Yamada (Numer Func Anal Opt 32(1--2):113--137, 2002) and the three-operator splitting by Davis and Yin (Set-Valued Var Anal 25(4):829--858, 2017) are tight. The analysis relies on the scaled relative graph, a geometric tool recently proposed by Ryu, Hannah, and Yin (arXiv:1902.09788, 2019).