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!

  • $\begingroup$ For an exact and non-trivial example, try a standard bivariate normal distribution with the function $f(x)=\max(-x,x/2)$. $\endgroup$ – Matt F. Mar 14 at 8:38
  • $\begingroup$ Could you kindly elaborate? I am actually quite surprised that this is the $f$ you chose. I had in mind initially a very similar $f$! $\endgroup$ – gradstudent Mar 15 at 1:16
  • $\begingroup$ Eg one part of that expectation could be wolframalpha.com/input/… $\endgroup$ – Matt F. Mar 15 at 3:47
  • $\begingroup$ Sorry! Could you kindly say some more details? Like the expression I have written above is a vector. But you are computing a scalar. Could you kindly check that? And what are your $a_i$s and the $b_i$s? $\endgroup$ – gradstudent Mar 15 at 4:58
  • $\begingroup$ That’s part of why I said “one part”. I don’t think these are research-level questions. $\endgroup$ – Matt F. Mar 15 at 10:50

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