Upper and lower bounds for the Bregman divergence

Abstract

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