Tight coefficients of averaged operators via scaled relative graph

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 (2002) [21] and the three-operator splitting by Davis and Yin (2017) [9] are tight. The analysis relies on the scaled relative graph, a geometric tool recently proposed by Ryu et al. (2019) [25].