Allocating Variance to Maximize Expectation

We design efficient approximation algorithms for maximizing the expectation of the supremum of families of Gaussian random variables. In particular, let $\mathrm{OPT}:=\max_{\sigma_1,\cdots,\sigma_n}\mathbb{E}\left[\sum_{j=1}^{m}\max_{i\in S_j} X_i\right]$, where $X_i$ are Gaussian, $S_j\subset[n]$ and $\sum_i\sigma_i^2=1$, then our theoretical results include:- We characterize the optimal variance allocation -- it concentrates on a small subset of variables as $|S_j|$ increases,- A polynomial time approximation scheme (PTAS) for computing $\mathrm{OPT}$ when $m=1$, and- An $O(\log n)$ approximation algorithm for computing $\mathrm{OPT}$ for general $m>1$.Such expectation maximization problems occur in diverse applications, ranging from utility maximization in auctions markets to learning mixture models in quantitative genetics.