A Note on the Approximate Symmetry of Bregman Distances

The Bregman distance $\Breg{\xi_x}(y,x)$, $\xi_x \in \partial J(y),$ 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 $\frac{1}{C} \Breg{\xi_x}(y,x) \leq \Breg{\xi_y}(x,y) \leq C \Breg{\xi_x}(y,x).$ 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 $1 < p <\infty$.