Online Vector Balancing and Geometric Discrepancy

TL;DR

This paper introduces a new framework for online vector balancing that achieves near-optimal bounds under arbitrary distributions, significantly improving previous results and extending to various geometric discrepancy problems.

Contribution

The authors develop a novel framework for online vector balancing with dependencies, achieving polylogarithmic bounds for general distributions and extending to multiple geometric discrepancy problems.

Findings

Achieved poly(n, log T) bound for online vector balancing with arbitrary input distributions.

Established poly(log T) bound for online interval discrepancy.

Extended the framework to online d-dimensional Tusnady's problem with poly(log^d T) bound.

Abstract

We consider an online vector balancing question where $T$ vectors, chosen from an arbitrary distribution over $[-1,1]^n$, arrive one-by-one and must be immediately given a $\pm$ sign. The goal is to keep the discrepancy small as possible. A concrete example is the online interval discrepancy problem where T points are sampled uniformly in [0,1], and the goal is to immediately color them $\pm$ such that every sub-interval remains nearly balanced. As random coloring incurs $\Omega(T^{1/2})$ discrepancy, while the offline bounds are $\Theta(\sqrt{n \log (T/n)})$ for vector balancing and $1$ for interval balancing, a natural question is whether one can (nearly) match the offline bounds in the online setting for these problems. One must utilize the stochasticity as in the worst-case scenario it is known that discrepancy is $\Omega(T^{1/2})$ for any online algorithm. Bansal and Spencer recently show an $O(\sqrt{n}\log T)$ bound when each coordinate is independent. When there are dependencies among the coordinates, the problem becomes much more challenging, as evidenced by a recent work of Jiang, Kulkarni, and Singla that gives a non-trivial $O(T^{1/\log\log T})$ bound for online interval discrepancy. Although this beats random coloring, it is still far from the offline bound. In this work, we introduce a new framework for online vector balancing when the input distribution has dependencies across coordinates. This lets us obtain a $poly(n, \log T)$ bound for online vector balancing under arbitrary input distributions, and a $poly(\log T)$ bound for online interval discrepancy. Our framework is powerful enough to capture other well-studied geometric discrepancy problems; e.g., a $poly(\log^d (T))$ bound for the online $d$-dimensional Tusn\'ady's problem. A key new technical ingredient is an {anti-concentration} inequality for sums of pairwise uncorrelated random variables.