Towards a Constructive Version of Banaszczyk's Vector Balancing Theorem
Daniel Dadush, Shashwat Garg, Shachar Lovett, Aleksandar Nikolov

TL;DR
This paper advances the quest for a constructive version of Banaszczyk's vector balancing theorem by establishing equivalences with subgaussian distributions and providing efficient algorithms for specific vector lengths.
Contribution
It introduces an equivalence between Banaszczyk's theorem and subgaussian distributions, and develops algorithms for vector lengths of O(1/√log n) using random walks and stochastic gradient ascent.
Findings
Established equivalence with O(1)-subgaussian distributions
Designed a universal signing algorithm for symmetric bodies
Implemented efficient algorithms for vectors of length O(1/√log n)
Abstract
An important theorem of Banaszczyk (Random Structures & Algorithms `98) states that for any sequence of vectors of norm at most and any convex body of Gaussian measure in , there exists a signed combination of these vectors which lands inside . 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 -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 body), which simply samples from the subgaussian sign distribution and checks to…
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