We study discrepancy minimization for vectors in $\mathbb{R}^n$ under various settings. The main result is the analysis of a new simple random process in multiple dimensions through a comparison argument. As corollaries, we obtain bounds which are tight up to logarithmic factors for several problems in online vector balancing posed by Bansal, Jiang, Singla, and Sinha (STOC 2020), as well as linear time algorithms for logarithmic bounds for the Koml\'{o}s conjecture.

中文翻译：

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通过自平衡行走最小化差异

我们研究了 $\mathbb{R}^n$ 在各种设置下向量的差异最小化。主要结果是通过比较论证在多个维度上分析了一个新的简单随机过程。作为推论，对于 Bansal、Jiang、Singla 和 Sinha (STOC 2020) 提出的在线向量平衡中的几个问题，我们获得了与对数因子紧密相关的边界，以及 Koml 对数边界的线性时间算法\'{ o} 的猜想。