Online Discrepancy Minimization via Persistent SelfBalancing Walks
Abstract
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 highprobability regime. We also provide results for the weighted and multicolor versions of the problem.
 Publication:

arXiv eprints
 Pub Date:
 February 2021
 arXiv:
 arXiv:2102.02765
 Bibcode:
 2021arXiv210202765A
 Keywords:

 Computer Science  Data Structures and Algorithms;
 Computer Science  Discrete Mathematics;
 Mathematics  Combinatorics
 EPrint:
 The proof of Lemma 7 is incorrect. There is a serious issue that we don't know how to fix at the moment. We thank Yang, Nikhil and collaborators for bringing it to our attention