Log-sum-exp function (${\mathrm{LSE}}_{\beta }$)

For $z\in {\mathbb{R}}^m$ and $\beta >0$, the scaled log-sum-exp function isIt is a smooth convex function satisfying

Composing the log-sum-exp function with finitely many affine functions gives a differentiable approximation to their pointwise maximum. Increasing the inverse-temperature parameter $\beta$ reduces the uniform approximation error while increasing the Lipschitz gradient constant.