A note on the approximate symmetry of Bregman distances

Abstract

The Bregman distance $B_{{\xi }_x}(y,x)$, ${\xi }_x\in \partial J(y),$ associated to a convex sub-differentiable functional $J$ is known to be in general non-symmetric in its arguments $x$, $y$. In this note we address the question when Bregman distances can be bounded against each other when the arguments are switched, i.e., if some constant $C>0$ exists such that for all $x,y$ on a convex set $M$ it holds that $\frac{1}{C}{B}_{{\xi }_x}(y,x)\le B_{{\xi }_y}(x,y)\le C{B}_{{\xi }_x}(y,x).$ We state sufficient conditions for such an inequality and prove in particular that it holds for the $p$-powers of the ${\ell }_p$ and $L^p$-norms when $1<p<\infty$.