Forward-backward splitting with deviations for monotone inclusions

This is the authors' abstract. We don't add key points for this paper.

We propose and study a weakly convergent variant of the forward-backward algorithm for solving structured monotone inclusion problems.Our algorithm features a per-iteration deviation vector, providing additional degrees of freedom.The only requirement on the deviation vector to guarantee convergence is that its norm is bounded by a quantity that can be computed online.This approach offers great flexibility and paves the way for the design of new forward-backward-based algorithms, while still retaining global convergence guarantees.These guarantees include linear convergence under a metric subregularity assumption.Choosing suitable monotone operators enables the incorporation of deviations into other algorithms, such as the Chambolle-Pock method and Krasnosel'skiȋ-Mann iterations.We propose a novel inertial primal-dual algorithm by selecting the deviations along a momentum direction and deciding their size by using the norm condition.Numerical experiments validate our convergence claims and demonstrate that even this simple choice of a deviation vector can enhance the performance compared to, for instance, the standard Chambolle-Pock algorithm.