Towards a Constructive Version of Banaszczyk's Vector Balancing Theorem

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 $\ell_2$ norm at most $1/5$ and any convex body $K$ of Gaussian measure $1/2$ in $\mathbb{R}^n$, 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 body), which simply samples from the subgaussian sign distribution and checks to see if the associated combination lands inside the body. For asymmetric convex bodies, we provide a novel recentering procedure, which allows us to reduce to the case where the body is symmetric. As our second main contribution, we show that the above framework can be efficiently implemented when the vectors have length $O(1/\sqrt{\log n})$, recovering Banaszczyk's results under this stronger assumption. More precisely, we use random walk techniques to produce the required $O(1)$-subgaussian signing distributions when the vectors have length $O(1/\sqrt{\log n})$, and use a stochastic gradient ascent method to implement the recentering procedure for asymmetric bodies.