Online Discrepancy Minimization via Persistent Self-Balancing Walks

TL;DR

This paper introduces an online discrepancy minimization algorithm for vectors in high-dimensional space, achieving near-optimal bounds with high probability, and extends results to weighted and multi-color variants.

Contribution

It presents a novel algorithm that maintains low discrepancy in an online setting, matching known lower bounds up to a logarithmic factor, and addresses weighted and multi-color cases.

Findings

Achieves $O(\sqrt{\log(nd/\delta)})$ discrepancy with high probability.

Matches the lower bound up to an $O(\sqrt{\log \log n})$ factor.

Extends results to weighted and multi-color discrepancy problems.

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 high-probability regime. We also provide results for the weighted and multi-color versions of the problem.