Discrepancy Minimization via Regularization

TL;DR

This paper presents a novel regularization-based framework for discrepancy minimization, unifying and extending key results in the field and confirming conjectures in specific pseudorandom settings.

Contribution

It introduces a new algorithmic approach that reinterprets existing discrepancy bounds and proves conjectures in new regimes using regularization techniques.

Findings

Unified interpretation of discrepancy algorithms via regularization

Extended discrepancy bounds to pseudorandom instances

Confirmed Beck-Fiala and Komlos conjectures in new regimes

Abstract

We introduce a new algorithmic framework for discrepancy minimization based on regularization. We demonstrate how varying the regularizer allows us to re-interpret several breakthrough works in algorithmic discrepancy, ranging from Spencer's theorem [Spencer 1985, Bansal 2010] to Banaszczyk's bounds [Banaszczyk 1998, Bansal-Dadush-Garg 2016]. Using our techniques, we also show that the Beck-Fiala and Komlos conjectures are true in a new regime of pseudorandom instances.