Suppose $\varphi : \mathbb{R} \to [0,\infty)$ is classification calibrated and convex. Then there exists a non-decreasing function $\psi : [0,\infty) \to [0,\infty)$ with $\psi(0) = 0$ such that for any measurable $h : \mathcal{X} \to \mathbb{R}$, \begin{align*} R(h) - R^* \leq \psi(R_\varphi(h) - R_\varphi^*). \end{align*}