Upper and lower bounds for the Bregman divergence

Abstract

In this paper we study upper and lower bounds on the Bregman divergence $Δ_{\mathcal{F}}^ξ(y,x):=\mathcal{F}(y)-\mathcal{F}(x)-\langle ξ, y-x\rangle $ for some convex functional $\mathcal{F}$ on a normed space $\mathcal{X}$, with subgradient $ξ\in\partial\mathcal{F}(x)$. 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.