Upper and lower bounds for the Bregman divergence

Abstract

In this paper we study upper and lower bounds on the Bregman divergence ΔFξ(y,x):=F(y)−F(x)−〈ξ,y−x〉 for some convex functional F on a normed space X, with subgradient ξ∈ ∂F(x). We give a considerably simpler new proof of the inequalities by Xu and Roach for the special case F(x) = ∥ x∥ p, p> 1. The results can be transferred to more general functions as well.