Revised: October 8, 2019

Published: December 13, 2019

**Keywords:**discrepancy, vector balancing, convex geometry

**Categories:**algorithms, randomized, discrepancy, vector balancing, convex geometry, RANDOM, APPROX-RANDOM 2016 special issue

**ACM Classification:**F.2.2

**AMS Classification:**68W20

**Abstract:**
[Plain Text Version]

An important theorem of Banaszczyk (Random Structures & Algorithms 1998) 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.

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An extended abstract of this paper appeared in the Proceedings of the 20th International Workshop on Randomization and Computation, RANDOM 2016.