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}
}
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

TLDR
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
TLDR
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

TLDR
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

TLDR
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

TLDR
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

TLDR
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

TLDR
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

  • N. BansalD. DadushS. Garg
  • Computer Science, Mathematics
    2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)
  • 2016
TLDR
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

TLDR
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.