The Gram-Schmidt Walk: A Cure for the Banaszczyk Blues

TL;DR

This paper introduces a randomized algorithm based on the Gram-Schmidt walk to efficiently find sign combinations of vectors that satisfy discrepancy bounds, resolving a longstanding open problem in discrepancy theory.

Contribution

It provides the first efficient randomized algorithm for Banaszczyk's discrepancy result, improving the constructive aspect of the theory.

Findings

Developed an efficient randomized algorithm for discrepancy problems.

Achieved bounds matching Banaszczyk's non-constructive result.

Enabled new practical algorithms for discrepancy minimization.

Abstract

An important result in discrepancy due to Banaszczyk states that for any set of $n$ vectors in $\mathbb{R}^m$ of $\ell_2$ norm at most $1$ and any convex body $K$ in $\mathbb{R}^m$ of Gaussian measure at least half, there exists a $\pm 1$ combination of these vectors which lies in $5K$. This result implies the best known bounds for several problems in discrepancy. Banaszczyk's proof of this result is non-constructive and a major open problem has been to give an efficient algorithm to find such a $\pm 1$ combination of the vectors. In this paper, we resolve this question and give an efficient randomized algorithm to find a $\pm 1$ combination of the vectors which lies in $cK$ for $c>0$ an absolute constant. This leads to new efficient algorithms for several problems in discrepancy theory.