Bregman Circumcenters: Monotonicity and Forward Weak Convergence

TL;DR

This paper introduces the concept of forward Bregman monotonicity and demonstrates its application in optimization algorithms, establishing weak convergence of the associated iterative sequences.

Contribution

It generalizes Fejér monotonicity to Bregman settings, introduces Bregman circumcenter mappings, and proves their weak convergence to fixed points.

Findings

Forward Bregman monotonicity generalizes Fejér monotonicity.

Sequences of Bregman circumcenter mappings weakly converge to fixed points.

Provides conditions ensuring convergence in Bregman optimization methods.

Abstract

Recently, we systematically studied the basic theory of Bregman circumcenters in another paper. In this work, we aim to apply Bregman circumcenters to optimization algorithms. Here, we propose the forward Bregman monotonicity which is a generalization of the powerful Fej\'er monotonicity and show a weak convergence result of the forward Bregman monotone sequence. We also naturally introduce the Bregman circumcenter mappings associated with a finite set of operators. Then we provide sufficient conditions for the sequence of iterations of the forward Bregman circumcenter mapping to be forward Bregman monotone. Furthermore, we prove that the sequence of iterations of the forward Bregman circumcenter mapping weakly converges to a point in the intersection of the fixed point sets of relevant operators, which reduces to the known weak convergence result of the circumcentered method under the Euclidean distance. In addition, particular examples are provided to illustrate the Bregman isometry and Browder's demiclosedness principle, and our convergence result.