# The Gram-Schmidt walk: a cure for the Banaszczyk blues

@article{Bansal2018TheGW,
title={The Gram-Schmidt walk: a cure for the Banaszczyk blues},
author={Nikhil Bansal and Daniel Dadush and Shashwat Garg and Shachar Lovett},
journal={Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing},
year={2018}
}
• Published 3 August 2017
• Mathematics
• Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing
An important result in discrepancy due to Banaszczyk states that for any set of n vectors in ℝm of ℓ2 norm at most 1 and any convex body K in ℝm 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 an open problem has been to give an efficient algorithm to find such a ± 1 combination of the vectors. In this paper…

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## References

SHOWING 1-10 OF 40 REFERENCES

### Efficient algorithms for discrepancy minimization in convex sets

• Mathematics, Computer Science
Random Struct. Algorithms
• 2018
A constructive version of the result of Gluskin and Giannopoulos, in which the coloring is attained via the optimization of a linear function is proved, which implies a linear programming based algorithm for combinatorial discrepancy obtaining the same result as Spencer.

### Constructive Discrepancy Minimization for Convex Sets

• T. Rothvoss
• Mathematics
2014 IEEE 55th Annual Symposium on Foundations of Computer Science
• 2014
It is shown that for any symmetric convex set K with measure at least e-n/500, the following algorithm finds a point y ∈ K ∩ [-1, 1]n with Ω(n) coordinates in ±1: (1) take a random Gaussian vector x, (2) compute the point y in K∩ [- 1, 1)n that is closest to x.

### Constructive Discrepancy Minimization by Walking on the Edges

• Mathematics
2012 IEEE 53rd Annual Symposium on Foundations of Computer Science
• 2012
A new randomized algorithm to find a coloring as in Spencer's result based on a restricted random walk which is “truly” constructive in that it does not appeal to the existential arguments, giving a new proof of Spencer's theorem and the partial coloring lemma.

### The Komlos Conjecture Holds for Vector Colorings

Here it is proved that if the columns of A are assigned unit real vectors rather than +/- 1 then the Komlos conjecture holds with K=1, which opens the way to proving tighter efficient (polynomial-time computable) upper bounds for the conjecture using semidefinite programming techniques.

### Six standard deviations suffice

Given n sets on n elements it is shown that there exists a two-coloring such that all sets have discrepancy at most Knl/2, K an absolute constant. This improves the basic probabilistic method with

### Algorithmic discrepancy beyond partial coloring

• Mathematics, Computer Science
STOC
• 2017
A new and general algorithmic framework is given that overcomes the limitations of the partial coloring method and can be applied in a black-box manner to various problems and gives new improved bounds and algorithms for several classic problems in discrepancy.

### Tighter Bounds for the Discrepancy of Boxes and Polytopes

New discrepancy upper bounds for geometrically defined set systems include sets induced by axis-aligned boxes, whose discrepancy is the subject of the well known Tusnady problem, are proved by extending the approach based on factorization norms previously used by the author and Matousek.

### Roth’s estimate of the discrepancy of integer sequences is nearly sharp

It is proved that R(N)=N1/4+o(1) thus showing that Roth’s original lower bound was essentially best possible, and the notion ofdiscrepancy of hypergraphs is introduced and derive an upper bound from which the above result follows.

### An Algorithm for Komlós Conjecture Matching Banaszczyk's Bound

• Computer Science, Mathematics
2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)
• 2016
An efficient algorithm is given that finds a coloring with discrepancy O((t log n)1/2), matching the best known non-constructive bound for the problem due to Banaszczyk, and gives an algorithmic O(log 1/2 n) bound.

### The geometry of differential privacy: the sparse and approximate cases

• Computer Science, Mathematics
STOC '13
• 2013
The connection between the hereditary discrepancy and the privacy mechanism enables the first polylogarithmic approximation to the hereditary discrepancies of a matrix A to be derived.