Constructive l2-Discrepancy Minimization with Additive Deviations

TL;DR

This paper presents a polynomial-time randomized algorithm that achieves near-optimal bounds for the signed series problem in -norm, improving previous constructive bounds and matching conjectured bounds for high-dimensional cases.

Contribution

The authors develop a new algorithm that improves constructive bounds for -discrepancy minimization, incorporating novel spectral orthogonality constraints and a Freedman-like concentration inequality.

Findings

Achieves -discrepancy bounds of O(^{1/2} + \u221a{ }log n)

Matches Banaszczyk's bounds when d ; log^2 n for high dimensions

Provides a constructive approach to the Steinitz problem in -norm

Abstract

The \emph{signed series} problem in the $\ell_2$ norm asks, given set of vectors $v_1,\ldots,v_n\in \mathbf{R}^d$ having at most unit $\ell_2$ norm, does there always exist a series $(\varepsilon_i)_{i\in [n]}$ of $\pm 1$ signs such that for all $i\in [n]$, $\max_{i\in [n]} \|\sum_{j=1}^i \varepsilon_i v_i\|_2 = O(\sqrt{d})$. A result of Banaszczyk [2012, \emph{Rand. Struct. Alg.}] states that there exist signs $\varepsilon_i\in \{-1,1\},\; i\in [n]$ such that $\max_{i\in [n]} \|\sum_{j=1}^i \varepsilon_i v_i\|_2 = O(\sqrt{d+\log n})$. The best constructive bound known so far is of $O(\sqrt{d\log n})$, by Bansal and Garg [2017, \emph{STOC.}, 2019, \emph{SIAM J. Comput.}]. We give a polynomial-time randomized algorithm to find signs $x(i) \in \{-1,1\},\; i\in [n]$ such that \[ \max_{i\in [n]} \|\sum_{j=1}^i x(i)v_i\|_2 = O(\sqrt{d + \log^2 n}) = O(\sqrt{d}+\log n).\] By the constructive reduction of Harvey and Samadi [\emph{COLT}, 2014], this also yields a constructive bound of $O(\sqrt{d}+\log n)$ for the Steinitz problem in the $\ell_2$-norm. Thus, we algorithmically achieve Banaszczyk's bounds for both problems when $d \geq \log^2n$, which also matches the conjectured bounds. Our algorithm is based on the framework on Bansal and Garg, together with a new analysis involving $(i)$ additional linear and spectral orthogonality constraints during the construction of the covariance matrix of the random walk steps, which allow us to control the quadratic variation in the linear as well as the quadratic components of the discrepancy increment vector, alongwith $(ii)$ a ``Freedman-like" version of the Hanson-Wright concentration inequality, for filtration-dependent sums of subgaussian chaoses.