Allocating Variance to Maximize Expectation

TL;DR

This paper develops efficient algorithms for allocating variance among Gaussian variables to maximize the expected supremum, with applications in auctions and genetics, including a PTAS for single sets and an approximation for multiple sets.

Contribution

It characterizes the optimal variance allocation and provides polynomial-time approximation schemes for maximizing the expectation of Gaussian suprema.

Findings

Optimal variance concentrates on small subsets as set size increases

PTAS for single-set expectation maximization

O(log n) approximation for multiple sets

Abstract

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.