Optimization over Poisson-binomial distributions

Question

I am studying the problem of how an expected utility maximizer should optimally form a portfolio of uncorrelated Bernoullis.

Fix an increasing sequence of $n$ numbers in $(0,1)$, $0<p_1<\dots<p_n<1$, to be interpreted as the parameters (probability of success) of a vector of independent Bernoulli random variables.

Given any $1\leq m\leq n$, we can define a random variable $S_m$ which counts the number of successes restricting to the subvector $p_m<\dots<p_n$. $S_m$ is supported in $\{0,\dots,n-m+1\}$ and is distributed according to a Poisson-Binomial $(p_i)_{i=m}^n$. In my application $S_m$ represents the ``payoff" of a risky portfolio.

Suppose that by choosing $m\in\{1,\dots,n\}$ the individual obtains a payoff of $$R_m=S_m+v(m-1)$$ where the term $v(m-1)$ represents the returns of a safe asset, for some $v\in (0,1)$. Consider the problem of maximizing the function

$$m\mapsto\mathbb{E}[u(R_m)]$$

for different choices of an increasing $u:[0,n]\to\mathbb{R}$. If $u$ is linear or convex (risk loving and risk neutrality) and $v<\frac{\sum_{i=1}^n p_i}{n}$, clearly the solution will be to choose $m=1$. Considering a concave $u$ is more interesting because, as $m$ decreases, the variables $S_m$ are such that both their mean and their variance increases. Hence, one has a trade off between enlarging the support of the portfolio and accepting more risk.

Q1 do you know any results about this problem?

The main problem working with the Poisson-Binomial to me is the combinatorics involved with its pmf and cdf. Hence, it would be nice to have a ``smooth" analog of the problem. In particular, replace the vector of $p$'s with a distribution over $[0,1]$ and let $F$ be its cdf.

Q2 How can we define an analog to the $S_m$'s random variables?

The idea would be that, by selecting a quantile of $F$, $q\in [0,1]$, one restricts to the Bernoulli's with parameter $p\geq q$. Hence, to each $q$, one should associate a random variable with support in $[0,1-F(q)]$. How to do this in a meaninful way? The problem of taking averages of the subfamily of Bernoulli's is that a law of large number kicks in (Law of large numbers for a continuum of Bernoullis) hence, the random variable is degenerate and the trade off above disappears.

Answer 1

Partial answer to Q2:

If the only reason to avoid Poisson-binomial is the combinatorics, its PDF and CDF can typically be well approximated in linear time with the saddlepoint approximation - the derivation for binomial (and hence Bernoulli) variables can be found in Liu & Quertermous: Approximating the Sum of Independent Non-Identical Binomial Random Variables - there's even an R implementation in the sinib package (though somewhat rough in my experience). See also the initial part of results section of Madson et al. 2017

The saddlepoint approximation is smooth, but requires you to numerically solve an equation for each evaluation, which is somewhat costly (though not combinatorially exploding). However, with some extra work, this still allows you to obtain gradients for optimization via the implicit function theorem. See see (Gaebler 2021) and (Margossian and Betancourt 2022) for description of the methods used in Stan to get those derivatives automatically.

I can provide some more thoughts in implementation, but since this is primarily math-oriented forum, I'll leave them out unless asked specifically :-)

I have no insight into Q1.