Preconditioning of a Generalized Forward-Backward Splitting and Application to Optimization on Graphs

TL;DR

This paper introduces a preconditioning method for a generalized forward-backward splitting algorithm, enhancing convergence and computational efficiency in large-scale graph-structured convex optimization problems.

Contribution

It proposes a novel preconditioning strategy that accelerates convergence and simplifies computations in monotone operator splitting methods, especially for graph-based optimization.

Findings

Preconditioning improves convergence speed in large-scale problems.

The method reduces storage and computational costs for auxiliary variables.

Application to graph-structured problems shows favorable performance.

Abstract

We present a preconditioning of a generalized forward-backward splitting algorithm for finding a zero of a sum of maximally monotone operators $\sum_{i=1}^{n} A_i + B$ with $B$ cocoercive, involving only the computation of $B$ and of the resolvent of each $A_i$ separately. This allows in particular to minimize functionals of the form $\sum_{i=1}^n g_i + f$ with $f$ smooth, using only the computation of the gradient of $f$ and of the proximity operator of each $g_i$ separately. By adapting the underlying metric, such preconditioning can serve two practical purposes: first, it might accelerate the convergence, or second, it might simplify the computation of the resolvent of $A_i$ for some $i$. In addition, in many cases of interest, our preconditioning strategy allows the economy of storage and computation concerning some auxiliary variables. In particular, we show how this approach can handle large-scale, nonsmooth, convex optimization problems structured on graphs, which arises in many image processing or learning applications, and that it compares favorably to alternatives in the literature.