We study the online discrepancy minimization problem for vectors in $\mathbb{R}^d$ in the oblivious setting where an adversary is allowed fix the vectors $x_1, x_2, \ldots, x_n$ in arbitrary order ahead of time. We give an algorithm that maintains $O(\sqrt{\log(nd/\delta)})$ discrepancy with probability $1-\delta$, matching the lower bound given in [Bansal et al. 2020] up to an $O(\sqrt{\log \log n})$ factor in the high-probability regime. We also provide results for the weighted and multi-color versions of the problem.

翻译：我们用 $mathbb{R ⁇ d$ 来研究对矢量的在线差异最小化问题, 以 $mathbb{R ⁇ d$ 为单位, 允许对手在高概率制度中将矢量修正为$_ 1, x_2,\ldots, x_n$ 任意顺序。 我们给出的算法以 $(\ sqrt}log(nd/\delta)} 为单位, 以 $- delta$ 为单位, 与 [Bansal 等人 2020] 给出的较低约束值匹配, 最多为 $(\ sqrt\log\log n} 。 我们还给出了问题加权和多色版本的结果 。