Let $\mathcal L$ be a uniformly bounded excess loss class and suppose it satisfies a Bernstein condition with exponent $1$. Assume that a sub-root function $\psi$ controls the expected localized Rademacher complexities and has fixed point $r_n$. Then there are universal constants $C,c>0$ such that, for every $t>0$, with probability at least $1-e^{-t}$, every empirical risk minimizer $\hat f_n$ satisfies \begin{align*} R(\hat f_n)-R(f^*)\le C\left(r_n+\frac{t}{n}\right). \end{align*}