# Tight Coefficients of Averaged Operators via Scaled Relative Graph

arXiv: https://arxiv.org/abs/1912.01593 · HTML: https://mlsanity.com/paper/1912.01593/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 3 Dec 2019.

## Built on by (most cited first, 1 of 1)

- [Improved Analysis and Rates for Variance Reduction under Without-replacement Sampling Orders](https://mlsanity.com/paper/2104.12112/) (2021)

## Abstract (arXiv)

Many iterative methods in optimization are fixed-point iterations with averaged operators. As such methods converge at an 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).