# Variable Metric Forward-Backward Algorithm for Composite Minimization Problems

## TL;DR

This paper introduces a variable metric forward-backward algorithm for minimizing sums of differentiable and nonconvex nonsmooth functions, especially when the nonsmooth part is a composite of a concave and convex function, with proven convergence and practical advantages.

## Contribution

It proposes a novel C2FB algorithm that handles composite nonconvex functions without explicit proximity computation, extending reweighting methods with convergence guarantees.

## Key findings

- Requires fewer iterations to converge in image processing tasks.
- Achieves better critical points compared to traditional methods.
- Converges even with inexact proximity operator computations.

## Abstract

We present a forward-backward-based algorithm to minimize a sum of a differentiable function and a nonsmooth function, both being possibly nonconvex. The main contribution of this work is to consider the challenging case where the nonsmooth function corresponds to a sum of non-convex functions, resulting from composition between a strictly increasing, concave, differentiable function and a convex nonsmooth function. The proposed variable metric Composite Function Forward-Backward algorithm (C2FB) circumvents the explicit, and often challenging, computation of the proximity operator of the composite functions through a majorize-minimize approach. Precisely, each composite function is majorized using a linear approximation of the differentiable function, which allows one to apply the proximity step only to the sum of the nonsmooth functions. We prove the convergence of the algorithm iterates to a critical point of the objective function leveraging the Kurdyka-\L ojasiewicz inequality. The convergence is guaranteed even if the proximity operators are computed inexactly, considering relative errors. We show that the proposed approach is a generalization of reweighting methods, with convergence guarantees. In particular, applied to the log-sum function, our algorithm reduces to a generalized version of the celebrated reweighted $\ell_1$ method. Finally, we show through simulations on an image processing problem that the proposed C2FB algorithm necessitates less iterations to converge and leads to better critical points compared with traditional reweighting methods and classic forward-backward algorithms.