Consider being given a $n-$dimensional random vector with a distribution ${\cal D}$, vectors $a \in \mathbb{R}^k$, $\{ b_i \in \mathbb{R}^n \}_{i=1}^k$ and non-linear Lipschitz functions, $f_1,f_2 : \mathbb{R} \rightarrow \mathbb{R}$.
Now under what conditions can we exactly compute (or give tight bounds on)
$$ \mathbb{E} \Big [ x f_1 \Big ( \sum_{i=1}^k a_i f_2 (b_i^\top x) \Big ) \Big ] $$
as a function of the the $a$ and $b_i$ vectors ?
I can hardly find any examples of non-trivial $f_1$ and $f_2$ (and ${\cal D}$ being anything "reasonable" like log-concave etc.) for which this is computable!