Bregman circumcenters: basic theory

TL;DR

This paper introduces Bregman circumcenters, providing a theoretical framework and explicit formulas, which could enhance the development of faster iterative optimization methods.

Contribution

It establishes the existence and formulas for Bregman pseudo-circumcenters, linking backward and forward versions through duality, and offers a foundational framework for acceleration in optimization.

Findings

Explicit formulas for Bregman pseudo-circumcenters

Connections between backward and forward Bregman circumcenters

Illustrative examples demonstrating the concepts

Abstract

Circumcenters play an important role in the design and analysis of accelerating various iterative methods in optimization. In this work, we propose Bregman (pseudo-)circumcenters associated with finite sets. We show the existence and give explicit formulae for the unique backward and forward Bregman pseudo-circumcenters of finite sets. Moreover, we use duality to establish connections between backward and forward Bregman (pseudo-)circumcenters. Various examples are presented to illustrate the backward and forward Bregman (pseudo-)circumcenters of finite sets. Our general framework for circumcenters paves the way for the development of accelerating iterative methods by Bregman circumcenters.