Gaussian discrepancy: a probabilistic relaxation of vector balancing

TL;DR

This paper introduces Gaussian discrepancy, a new probabilistic relaxation of vector balancing that uses Gaussian variables, providing tighter bounds and an efficient online algorithm, with implications for classical discrepancy problems.

Contribution

It proposes Gaussian discrepancy as a novel relaxation, demonstrates its superiority over existing relaxations, and develops a fast online algorithm achieving Banaszczyk bounds.

Findings

Gaussian discrepancy is a tighter relaxation than vector and spherical discrepancy.

A fast online algorithm achieves a Banaszczyk-like bound for Gaussian discrepancy.

Raises new questions about the Komlós conjecture in the context of Gaussian discrepancy.

Abstract

We introduce a novel relaxation of combinatorial discrepancy called Gaussian discrepancy, whereby binary signings are replaced with correlated standard Gaussian random variables. This relaxation effectively reformulates an optimization problem over the Boolean hypercube into one over the space of correlation matrices. We show that Gaussian discrepancy is a tighter relaxation than the previously studied vector and spherical discrepancy problems, and we construct a fast online algorithm that achieves a version of the Banaszczyk bound for Gaussian discrepancy. This work also raises new questions such as the Koml\'{o}s conjecture for Gaussian discrepancy, which may shed light on classical discrepancy problems.