Vector Balancing in Lebesgue Spaces

1 Introduction

The celebrated Spencer’s Theorem in discrepancy theory [Spencer1985SixSD]

shows that "six standard deviations suffice" for balancing vectors in the

ℓ_{\infty}

-norm: for any

a_{1}, \dots, a_{n} \in [- 1, 1]^{n}

, there exist signs

x \in {- 1, 1}^{n}

such that

∥ \sum_{i = 1}^{n} x_{i} a_{i} ∥_{\infty} \leq 6 \sqrt{n}

. More generally, Spencer showed that for vectors in

[- 1, 1]^{m}

with

n \leq m

one can achieve a bound of

O (\sqrt{n log (2 m / n)})

. While his proof used a nonconstructive form of the partial coloring lemma based on the pigeonhole principle, in the past decade several approaches starting with the breakthrough work of Bansal [DiscrepancyMinimization-Bansal-FOCS2010] did succeed in computing such signs in polynomial time [DiscrepancyMinimization-LovettMekaFOCS12, ConstructiveDiscrepancy-Rothvoss-FOCS2014, DBLP:journals/corr/LevyRR16, DBLP:journals/rsa/EldanS18].

As for balancing vectors of bounded $ℓ_{2}$ -norm, the situation has been more delicate. In the same paper, Spencer [Spencer1985SixSD] showed a nonconstructive bound of $O (log n)$ for the $ℓ_{\infty}$ discrepancy of vectors $a_{1}, \dots, a_{n} \in B_{2}^{m}$ and also stated a conjecture of Komlós that this may be improved to $O (1)$ . This was improved to $O (\sqrt{log n})$ by Banaszczyk [BalancingVectors-Banaszczyk98] who showed that in fact for any set of $n$ vectors of $ℓ_{2}$ -norm at most 1 and any convex body $K \subseteq R^{m}$ of Gaussian measure at least $1 / 2$ , some $\pm 1$ combination of such vectors lies in $5 \cdot K$ . For the more general setting of $ℓ_{q}$ discrepancy, the work of Barthe, Guédon, Mendelson and Naor [GeometryOfLpBall-BartheGuedonMendelsonNaor2005] shows that, for $q \geq 2$ , a $O (\sqrt{q} \cdot n^{1 / q})$ scaling of $n$ -dimensional slices of the $ℓ_{q}$ ball in $R^{m}$ does have Gaussian measure at least $1 / 2$ , thus implying a corresponding $O (\sqrt{q} \cdot n^{1 / q})$ upper bound for balancing vectors from $ℓ_{2}$ to $ℓ_{q}$ . For $q = log n$ , this matches the $ℓ_{2}$ to $ℓ_{\infty}$ bound of $O (\sqrt{log n})$ . Banaszczyk’s proof was nonconstructive and the first polynomial time algorithm in the general convex body setting was found only recently by Bansal, Dadush, Garg and Lovett [GramSchmidtWalk-BansalDGL-STOC18], while the Komlós conjecture remains an open problem. The work of [GramSchmidtWalk-BansalDGL-STOC18] actually shows that for any vectors $a_{1}, \dots, a_{n} \in B_{2}^{m}$ there exists an efficiently computable distribution over signs $x \in {- 1, 1}^{n}$ so that the sum $\sum_{i = 1}^{n} x_{i} a_{i}$ is $O (1)$ -subgaussian and will be in $O (1) \cdot K$

with good probability. Interestingly, this means their algorithm is

oblivious to the body

K

, which is a striking difference to the regime of

γ_{n} (K) = e^{- Θ (n)}

where any algorithm needs to be dependent on

K

. The connection between Banaszczyk’s theorem and subgaussianity is due to Dadush et al. [DBLP:conf/approx/DadushGLN16].

For the general setting of balancing vectors from $ℓ_{p}$ to $ℓ_{q}$ norms, not much was known beyond Spencer’s theorem ( $p = \infty$ ) or what can be deduced from Banaszczyk’s theorem as above: any vector in $B_{p}^{m}$ also belongs to $m^{max (0, 1 / 2 - 1 / p)} \cdot B_{2}^{m}$ , thus implying a discrepancy bound of $O (\sqrt{q}) \cdot m^{max (0, 1 / 2 - 1 / p)} \cdot n^{1 / q}$ . Even in the square case $m = n$ , it has been an open problem to remove the dependency on $\sqrt{q}$ [8555088]. The goal of this paper is to provide a unified approach for balancing from $ℓ_{p}$ to $ℓ_{q}$ via optimal constructive fractional partial colorings, which yield optimal bounds for most of the range $1 \leq p \leq q \leq \infty$ . We obtain such fractional partial colorings by proving a new measure lower bound on the relevant linear preimages of $ℓ_{q}$ balls (Section 3) and an improved algorithm which works for sets of Gaussian measure $e^{- δ n}$ for any $δ > 0$ (Section 4), as opposed to previous work ([ConstructiveDiscrepancy-Rothvoss-FOCS2014, DBLP:journals/rsa/EldanS18]) which required measure $e^{- δ n}$ for sufficiently small $δ > 0$ .

As an application of our results, we show a slight improvement to the bounds for the well-known Beck-Fiala conjecture [BECK19811], a discrete version of Komlós. It asks for a $O (\sqrt{t})$ bound on the $ℓ_{\infty}$ discrepancy of any $a_{1}, \dots, a_{n} \in {0, 1}^{m}$ , each with at most $t$ ones. We establish the conjecture for $t \geq n$ and show slightly improved bounds when $t$ is close to $n$ (Corollary 4).

Notation. Let $B_{p}^{m} := {x \in R^{m} : ∥ x ∥_{p} \leq 1}$ denote the unit ball in the $ℓ_{p}$ -norm. The Gaussian measure of a measurable set $K \subseteq R^{n}$ is given by $γ_{n} (K) := {Pr}_{x \sim N (0, I_{n})} [x \in K]$ . We denote the mean width of a convex set as $w (K) := E_{θ \in S^{n - 1}} [{sup}_{x \in K} ⟨ θ, x ⟩]$ . The Euclidean distance to a set $S \subseteq R^{n}$ is denoted by $d (x, S) := min {∥ x - y ∥_{2} : y \in S}$ . If $A \in R^{m \times n}$ is a matrix, we denote its rows by $A_{1}, \dots, A_{m} \in R^{n}$ and its columns by $a_{1}, \dots, a_{n} \in R^{m}$ . Naturally, a matrix can also be interpreted as a (not necessarily invertible) linear map. Then for any set $K \subseteq R^{m}$ , we use the notation $A^{- 1} (K) := {x \in R^{n} : A x \in K}$ .

1.1 Our contribution

Our main contribution is a tight bound on partial colorings for balancing from $ℓ_{p}$ to $ℓ_{q}$ :

Theorem 1.

Let $n \leq m$ and $1 \leq p \leq q \leq \infty$ . Then for any $a_{1}, \dots, a_{n} \in B_{p}^{m}$ , there exists a polynomial-time computable partial coloring $x \in [- 1, 1]^{n}$ with $| {i : x_{i}^{2} = 1} | \geq n / 2$ so that

∥ ∥ n \sum i = 1 x_{i} a_{i} {∥ ∥}_{q} \leq C \sqrt{min (p, log (\frac{2 m}{n}))} \cdot n^{max (0, 1 / 2 - 1 / p) + 1 / q},