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.

翻译：$\ell_2$ 范数下的\emph{符号级数}问题是指：给定一组向量 $v_1,\ldots,v_n\in \mathbf{R}^d$，其 $\ell_2$ 范数至多为 1，是否总是存在一个由 $\pm 1$ 符号组成的序列 $(\varepsilon_i)_{i\in [n]}$，使得对于所有 $i\in [n]$，都有 $\max_{i\in [n]} \|\sum_{j=1}^i \varepsilon_i v_i\|_2 = O(\sqrt{d})$。Banaszczyk [2012, \emph{Rand. Struct. Alg.}] 的一个结果表明，存在符号 $\varepsilon_i\in \{-1,1\},\; i\in [n]$，使得 $\max_{i\in [n]} \|\sum_{j=1}^i \varepsilon_i v_i\|_2 = O(\sqrt{d+\log n})$。目前已知的最佳构造性界是 $O(\sqrt{d\log n})$，由 Bansal 和 Garg [2017, \emph{STOC.}, 2019, \emph{SIAM J. Comput.}] 给出。我们提出了一种多项式时间随机算法来寻找符号 $x(i) \in \{-1,1\},\; i\in [n]$，使得 \[ \max_{i\in [n]} \|\sum_{j=1}^i x(i)v_i\|_2 = O(\sqrt{d + \log^2 n}) = O(\sqrt{d}+\log n).\] 根据 Harvey 和 Samadi [\emph{COLT}, 2014] 的构造性归约，这也为 $\ell_2$ 范数下的 Steinitz 问题产生了一个 $O(\sqrt{d}+\log n)$ 的构造性界。因此，当 $d \geq \log^2n$ 时，我们通过算法实现了 Banaszczyk 对这两个问题的界，这也与猜想界相符。我们的算法基于 Bansal 和 Garg 的框架，并结合了一种新的分析，该分析涉及 $(i)$ 在构造随机游走步骤的协方差矩阵时，附加的线性和谱正交性约束，这使我们能够控制离散度增量向量的线性分量和二次分量的二次变差，以及 $(ii)$ 一个“类 Freedman”版本的 Hanson-Wright 集中不等式，用于处理依赖于滤子的亚高斯混沌和。