# Bregman Forward-Backward Operator Splitting

## TL;DR

This paper proves the convergence of a Bregman distance-based forward-backward splitting algorithm for monotone operators in Banach spaces, extending known results beyond minimization problems and providing sharper convergence rates.

## Contribution

It introduces a novel Bregman forward-backward splitting framework with variable distances and a new assumption on operators, broadening applicability and improving convergence rates.

## Key findings

- Convergence established for the algorithm in reflexive Banach spaces.
- Sharper convergence rates in the minimization setting.
- Framework accommodates iteration-varying Bregman distances.

## Abstract

We establish the convergence of the forward-backward splitting algorithm based on Bregman distances for the sum of two monotone operators in reflexive Banach spaces. Even in Euclidean spaces, the convergence of this algorithm has so far been proved only in the case of minimization problems. The proposed framework features Bregman distances that vary over the iterations and a novel assumption on the single-valued operator that captures various properties scattered in the literature. In the minimization setting, we obtain rates that are sharper than existing ones.