Say we have a function $f(x_1,x_2,x_3,\ldots,x_n): \mathbb R^n \mapsto \mathbb R$ and $n$ independent random variables $X_1, X_2, X_3, \ldots, X_n \in \mathbb R$.
Given a positive number $s \in \mathbb N$, what is the expected maximum of $f$ with $s$ samples?That is, let $x_{i1}, x_{i2}, \ldots, x_{is}$ be $s$ independent samples from $X_i$, for each $i \in [n]$, we want to compute$$\mathbb{E} \max_{j \in [s]} f(x_{1j}, x_{2j}, \ldots, x_{nj}).$$
Some special cases are:
- $s = 1 \Rightarrow \mathbb{E} \max_{j \in [s]} f(x_{1j}, x_{2j}, \ldots, x_{nj}) = \mathbb{E}_{x_1 \sim X_1, \ldots, X_n \sim X_n} f(x_{1}, x_{2}, \ldots, x_{n})$
- $s \to \infty \Rightarrow \mathbb{E} \max_{j \in [s]} f(x_{1j}, x_{2j}, \ldots, x_{nj}) \to \max_{x_1, \ldots, x_n} f(x_{1j}, x_{2j}, \ldots, x_{nj})$
General cases:
- I know that we can write$$\Pr[\max_{j \in [s]} f(x_{1j}, x_{2j}, \ldots, x_{nj}) \leq t] = (\Pr[f(x_{1}, x_{2}, \ldots, x_{n}) \leq t])^s,$$but this does not directly give us the expectation
- I am wondering whether we can compute (or estimate) the expectation, at least for some special forms of $f$ (e.g., linear functions)