Let $f_i:\mathbb{R}^n\rightarrow [0,\infty)$ be $C^1$ for $i=1,...,n$. Under what (reasonable) conditions on the $f_i$ does the function $$ \sum_{i=1}^N \operatorname{softmax}(f_1(x),\dots,f_n(x))_if_i(x), $$ have Lipschitz gradients?
Where the $\operatorname{softmax}(z_1,\dots,z_n)=\sum_{i=1}^n \frac{e^{z_i}}{\sum_{i=1}^N e^{z_i}}$. I tried using the constants of this post but to no avail...