The sharp step-size constant for one-call reflection splittings on monotone inclusions

Abstract

The forward-reflected-backward splitting of Malitsky and Tam converges weakly for step sizes $\lambda \in (0,\frac{1}{2L})$, where $B$ is monotone and $L$-Lipschitz, and the question whether this bound is tight has been recorded as open \cite{GS}. We show that it is: for the matched-skew instance $A=LJ$, $B=LJ$ ($J$ the counterclockwise rotation by $\pi /2$ in ${\mathbb{R}}^2$), the iterates fail to converge for every $\lambda \ge \frac{1}{2L}$ and diverge for $\lambda >\frac{1}{2L}$. More generally, for the skew--rotation family $A=\gamma J$, $B=J$, the exact stability threshold is ${\lambda }^{\star }(\gamma )=1/\sqrt{(1+\gamma )(3-\gamma )}$, which attains its minimum $1/2$ at the matched skew $\gamma =1$ and recovers the reflected gradient constant $1/\sqrt{3}$ at $\gamma =0$. The same instance is a counterexample for the reflected--forward--backward method of Cevher and Vũ, which coincides with forward-reflected-backward on linear operators.

arXiv comment5 pages, 2 figures