Tighter Bounds for the Discrepancy of Boxes and Polytopes

Nikolov, Aleksandar

doi:10.1112/s0025579317000250

Cited by 18 publications

(22 citation statements)

References 34 publications

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“…The last line follows by Bansal and Garg [3] showed disc(n, R d ) = O(log d n), and their proof provides a polynomial time algorithm. Nikolov [25] soon after showed that disc(n, R d ) = O(log d− 1 2 n) although this result does not describe how to efficiently construct the coloring. With these result we obtain the following.…”

Section: Gaussian Kernel Coresets Bounded Using Rectangle Discrepancymentioning

confidence: 99%

Improved Coresets for Kernel Density Estimates

Phillips

Tai

2018

Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms

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We study the construction of coresets for kernel density estimates. That is we show how to approximate the kernel density estimate described by a large point set with another kernel density estimate with a much smaller point set. For characteristic kernels (including Gaussian and Laplace kernels), our approximation preserves the L ∞ error between kernel density estimates within error ε, with coreset size 4/ε 2 , but no other aspects of the data, including the dimension, the diameter of the point set, or the bandwidth of the kernel common to other approximations. When the dimension is unrestricted, we show this bound is tight for these kernels as well as a much broader set.This work provides a careful analysis of the iterative Frank-Wolfe algorithm adapted to this context, an algorithm called kernel herding. This analysis unites a broad line of work that spans statistics, machine learning, and geometry.When the dimension d is constant, we demonstrate much tighter bounds on the size of the coreset specifically for Gaussian kernels, showing that it is bounded by the size of the coreset for axis-aligned rectangles. Currently the best known constructive bound is O( 1 ε log d 1 ε ), and non-constructively, this can be improved by log 1 ε . This improves the best constant dimension bounds polynomially for d ≥ 3.

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Section: Gaussian Kernel Coresets Bounded Using Rectangle Discrepancymentioning

confidence: 99%

Improved Coresets for Kernel Density Estimates

Phillips

Tai

2018

Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms

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“…Theorem 1.1 was also used in a very interesting non-black box way in a later work by Banaszczyk [Ban12] to show improved bounds for several variants of the Steinitz problem. Recently, [Nik17] used this to obtain improved bounds for the Tusnady's problem.…”

Section: Introductionmentioning

confidence: 99%

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

Bansal

Dadush

Garg

et al. 2018

Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing

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An important result in discrepancy due to Banaszczyk states that for any set of n vectors in R m of ℓ 2 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.

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“…Despite this success, Banaszczyk's more recent results on signed series [3], and the last author's upper bounds on the discrepancy of axis-aligned boxes [27], both proved using the techniques used in [2], remain out of reach algorithmically. Extending the techniques of this paper and of [6] to give constructive proofs of these results is an interesting open problem.…”

Section: Conclusion and Follow-up Workmentioning

confidence: 99%

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Dadush¹,

Garg²,

Lovett³

et al. 2019

Theory of Comput.

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An important theorem of Banaszczyk (Random Structures & Algorithms 1998) states that for any sequence of vectors of 2 norm at most 1/5 and any convex body K of Gaussian measure 1/2 in R n , there exists a signed combination of these vectors which lands inside K. A major open problem is to devise a constructive version of Banaszczyk's vector balancing theorem, i. e., to find an efficient algorithm which constructs the signed combination. We make progress towards this goal along several fronts. As our first contribution, we show an equivalence between Banaszczyk's theorem and the existence of O(1)-subgaussian distributions over signed combinations. For the case of symmetric convex bodies, our equivalence implies the existence of a universal signing algorithm (i. e., independent of the An extended abstract of this paper appeared in the Proceedings of the 20th International Workshop on Randomization and Computation, 2016. * Supported by the NWO Veni grant 639.071.510. † Supported by the Netherlands Organisation for Scientific Research (NWO) under project no. 022.005.025. ‡ Supported by an NSF CAREER award 1614023. § Supported by an NSERC Discovery grant, application number RGPIN-2016-06333.

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Tighter Bounds for the Discrepancy of Boxes and Polytopes

Cited by 18 publications

References 34 publications

Improved Coresets for Kernel Density Estimates

Improved Coresets for Kernel Density Estimates

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

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