On the nature of Bregman functions

TL;DR

This paper characterizes Bregman functions as strictly convex functions on polytopal domains, providing new insights into convergence conditions for Bregman divergence-based algorithms.

Contribution

It proves that Bregman functions are exactly strictly convex functions on polytopal domains, clarifying convergence conditions for related algorithms.

Findings

Bregman functions are characterized as strictly convex on polytopes.

Convergence conditions can be expressed via explicit properties of h and C.

General convergence theory needs further refinement beyond Bregman's conditions.

Abstract

Let C be convex, compact, with nonempty interior and h be Legendre with domain C, continuous on C. We prove that h is Bregman if and only if it is strictly convex on C and C is a polytope. This provides insights on sequential convergence of many Bregman divergence based algorithm: abstract compatibility conditions between Bregman and Euclidean topology may equivalently be replaced by explicit conditions on h and C. This also emphasizes that a general convergence theory for these methods (beyond polyhedral domains) would require more refinements than Bregman's conditions.