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

Bansal, Nikhil; Dadush, Daniel; Garg, Swati; Lovett, Shachar

doi:10.1145/3188745.3188850

Cited by 39 publications

(61 citation statements)

References 36 publications

(42 reference statements)

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“…We will apply these properties shortly. A recent result by Matousek et al [33] shows the following property about connecting discrepancy to γ 2 , which was recently made constructive in polynomial time [5]. Let the size of S be m = O(n O(d) ), and define an m × n matrix G so its rows are indexed by x ∈ S and columns indexed by p ∈ P , and G x,p = K(p, x).…”

Section: Upper Bound For Kde Coresetmentioning

confidence: 99%

Near-Optimal Coresets of Kernel Density Estimates

Phillips

Tai

2019

Discrete Comput Geom

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We construct near-optimal coresets for kernel density estimates for points in R d when the kernel is positive definite. Specifically we show a polynomial time construction for a coreset of size O( √ d/ε · log 1/ε), and we show a near-matching lower bound of size Ω(min{ √ d/ε, 1/ε 2 }). When d ≥ 1/ε 2 , it is known that the size of coreset can be O(1/ε 2 ). The upper bound is a polynomial-in-(1/ε) improvement when d ∈ [3, 1/ε 2 ) and the lower bound is the first known lower bound to depend on d for this problem. Moreover, the upper bound restriction that the kernel is positive definite is significant in that it applies to a wide-variety of kernels, specifically those most important for machine learning. This includes kernels for information distances and the sinc kernel which can be negative.

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Section: Upper Bound For Kde Coresetmentioning

confidence: 99%

Near-Optimal Coresets of Kernel Density Estimates

Phillips

Tai

2019

Discrete Comput Geom

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“…Plugging back into (17), we get E[e λy (Xτ −x) ] ≤ e τ λ 2 ε 2 0 n 2 /ε 1 . The standard exponential moment argument then implies (16).…”

Section: We Finish This Section With the Proof Of Lemma 13mentioning

confidence: 97%

Sticky Brownian Rounding and its Applications to Constraint Satisfaction Problems

Abbasi-Zadeh¹,

Bansal²,

Guruganesh³

et al. 2020

Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms

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Semidefinite programming is a powerful tool in the design and analysis of approximation algorithms for combinatorial optimization problems. In particular, the random hyperplane rounding method of Goemans and Williamson [31] has been extensively studied for more than two decades, resulting in various extensions to the original technique and beautiful algorithms for a wide range of applications. Despite the fact that this approach yields tight approximation guarantees for some problems, e.g., Max-Cut, for many others, e.g., Max-SAT and Max-DiCut, the tight approximation ratio is still unknown. One of the main reasons for this is the fact that very few techniques for rounding semi-definite relaxations are known.In this work, we present a new general and simple method for rounding semi-definite programs, based on Brownian motion. Our approach is inspired by recent results in algorithmic discrepancy theory. We develop and present tools for analyzing our new rounding algorithms, utilizing mathematical machinery from the theory of Brownian motion, complex analysis, and partial differential equations. Focusing on constraint satisfaction problems, we apply our method to several classical problems, including Max-Cut, Max-2SAT, and Max-DiCut, and derive new algorithms that are competitive with the best known results. To illustrate the versatility and general applicability of our approach, we give new approximation algorithms for the Max-Cut problem with side constraints that crucially utilizes measure concentration results for the Sticky Brownian Motion, a feature missing from hyperplane rounding and its generalizations.In the presence of a single side constraint, the problem is closely related to the Max-Bisection problem [12,50], and, more generally to Max-Cut with a cardinality constraint. While our methods use the stronger semi-definite programs considered in [50] and [12], the main new technical ingredient is showing that the Sticky Brownian Motion possesses concentration of measure properties that allow us to approximately satisfy multiple constraints. By contrast, the hyperplane rounding and its generalizations that have been applied previously to the Max-Cut and Max-Bisection problems do not seem to allow for such strong concentration bounds. For this reason, the rounding and analysis used in [50] only give an O(n poly(k/ε) ) time algorithm for the Max-Cut-SC problem, which is trivial for k = Ω(n), whereas our algorithm has non-trivial quasi-polynomial running time even in this regime. We expect that this concentration of measure property will find further applications, in particular to constraint satisfaction problems with global constraints.Remark. We can achieve better results using Sticky Brownian Motion with slowdown. In particular, in time O(n poly(log(k)/ε) ) we can get a (0.858 − ε, ε)-approximation with high probability for any satisfiable instance. However, we focus on the basic Sticky Brownian Motion algorithm to simplify exposition. Note that due to the recent work by Austrin and Stanković [11], we know ...

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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: 97%

“…Subsequently to the publication of an extended abstract of this paper [14], the first three named authors, jointly with Bansal, showed the existence of a polynomial-time sampling algorithm for an O(1)-subgaussian distribution on signed sums of vectors with 2 norm at most 1 [6]. Together with the results in this paper (specifically Corollary 6.6), this sampling algorithm implies a new constructive proof of Banaszczyk's theorem in its full generality.…”

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

Cited by 39 publications

References 36 publications

Near-Optimal Coresets of Kernel Density Estimates

Near-Optimal Coresets of Kernel Density Estimates

Sticky Brownian Rounding and its Applications to Constraint Satisfaction Problems

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