Upper and lower bounds for the Bregman divergence
Simplifies existing proofs for researchers working with Bregman divergences in optimization and functional analysis, but the contribution is incremental.
The paper provides simpler proofs of known bounds for the Bregman divergence of power norms and extends them to more general convex functionals. No new numerical results are reported.
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.