Allocating Variance to Maximize Expectation

2025-02-25

AI Summary

This paper studies the optimal allocation of variances among Gaussian random variables under a fixed total variance constraint to maximize the expected supremum. The problem arises in auction mechanism design and mixed-model learning in quantitative genetics. The authors first characterize the structural properties of optimal solutions, establishing that variance should concentrate on a sparse subset of variables (“sparsity-concentration”). Leveraging this insight, they propose a polynomial-time approximation scheme (PTAS) for the single-group case—achieving (1−ε)-approximation in poly(n, 1/ε) time for any ε > 0—and an O(log n)-approximation algorithm for the multi-group case (m > 1). Their approach integrates tools from Gaussian process theory, probabilistic upper bound analysis, and combinatorial optimization. The theoretical guarantees strictly improve upon prior work, while the algorithms remain computationally tractable and practically implementable.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Game Theory and Economic Paradigms: Mechanism Design
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

We design efficient approximation algorithms for maximizing the expectation of the supremum of families of Gaussian random variables. In particular, let $mathrm{OPT}:=max_{sigma_1,cdots,sigma_n}mathbb{E}left[sum_{j=1}^{m}max_{iin S_j} X_i ight]$, where $X_i$ are Gaussian, $S_jsubset[n]$ and $sum_isigma_i^2=1$, 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.

Problem

• Optimize variance allocation for Gaussian variables
• Develop PTAS for expectation maximization
• Approximate $O(log n)$ algorithm for general cases

Innovation

• Gaussian random variables optimization
• Polynomial time approximation scheme
• Logarithmic approximation algorithm