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

An important result in discrepancy due to Banaszczyk states that for any set of n vectors in R^m of ℓ₂ norm at most 1 and any convex body K in R^m of Gaussian measure at least half, there exists a ±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 ±1 combination of the vectors.

In this paper, we resolve this question and give an efficient randomized algorithm to find a ±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.