Gaussian discrepancy: A probabilistic relaxation of vector balancing

We introduce a novel relaxation of combinatorial discrepancy called Gaussian discrep-ancy, 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 Komlos conjecture for Gaussian discrepancy, which may shed light on classical discrepancy problems. (C) 2022 Elsevier B.V. All rights reserved.