Upper and lower bounds for the Bregman divergence

TL;DR

This paper investigates bounds on the Bregman divergence for convex functions, providing a simplified proof for specific cases and extending results to more general functions.

Contribution

It offers a simpler proof of existing inequalities for Bregman divergence and extends the bounds to broader classes of convex functions.

Findings

Simplified proof of inequalities for Bregman divergence when $\\mathcal{F}(x)=\|x\|^p$

Extension of bounds to more general convex functions

Applicable to convex analysis in normed spaces

Abstract

In this paper we study upper and lower bounds on the Bregman divergence $\Delta_{\mathcal{F}}^{\xi}(y,x):=\mathcal{F}(y)-\mathcal{F}(x)-\langle \xi, y-x\rangle $ for some convex functional $\mathcal{F}$ on a normed space $\mathcal{X}$, with subgradient $\xi\in\partial\mathcal{F}(x)$. We give a considerably simpler new proof of the inequalities by Xu and Roach for the special case $\mathcal{F}(x)=\left\| x\right\|^p, p>1$. The results can be transfered to more general functions as well.