Log-sum-exp function
= Log-sum-exp function
{title2=$\operatorname{LSE}_\beta$}
{wiki}
For $z\in\mathbb R^m$ and $\beta>0$, the scaled log-sum-exp function is
$$
\operatorname{LSE}_\beta(z)
=\frac1\beta\log\sum_{i=1}^m e^{\beta z_i}.
$$
It is a smooth <convex function> satisfying
$$
\max_i z_i\leq\operatorname{LSE}_\beta(z)
\leq\max_i z_i+\frac{\log m}{\beta}.
$$