Symmetric Bregman distance ($D_J^{\mathrm{sym}}(u,v)=⟨p_u-p_v,u-v⟩$)

For $p_u\in \partial J(u)$ and $p_v\in \partial J(v)$, add the two Bregman distances to get $D_J^{p_v}(u,v)+D_J^{p_u}(v,u)=⟨p_u-p_v,u-v⟩\ge 0$. The choice of subgradients matters. This is generally not a metric: it need not separate points or satisfy a triangle inequality.

Let $K:U\to V$ be a bounded linear operator from a real Banach space to a real Hilbert space, let $J$ be a proper lower-semicontinuous convex function, and let $\alpha >0$ and $\Vert f^{\delta }-K{u}^{†}\Vert \le \delta$. For an existing minimizer of $\Vert Ku-f^{\delta }{\Vert }^2/2+\alpha J(u)$ and a source condition in variational regularization $p^{†}=K^{\ast }w\in \partial J(u^{†})$, let $e=f^{\delta }-K{u}^{†}$ and $q=K(u_{\alpha }-u^{†})$. Its optimality subgradient is $p_{\alpha }=K^{\ast }(f^{\delta }-K{u}_{\alpha })/\alpha$. Then $\alpha D_J^{\mathrm{sym}}=⟨e-\alpha w,q⟩-\Vert q{\Vert }^2\le \Vert e-\alpha w{\Vert }^2/4$. The displayed estimate follows from $\Vert e-\alpha w{\Vert }^2\le 2{\delta }^2+2{\alpha }^2\Vert w{\Vert }^2$.