Online Vector Balancing and Geometric Discrepancy

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 ± sign. The goal is to keep the discrepancy—the ℓ∞-norm of any signed prefix-sum—as small as possible. A concrete example of this question is the online interval discrepancy problem where T points are sampled one-by-one uniformly in the unit interval [0,1], and the goal is to immediately color them ± such that every sub-interval remains always nearly balanced. As random coloring incurs Ω(T1/2) discrepancy, while the worst-case offline bounds are Θ(√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 Ω(T1/2) for any online algorithm. In a special case of online vector balancing, Bansal and Spencer [BS19] recently show an O(√nlogT) bound when each coordinate is independently chosen. When there are dependencies among the coordinates, as in the interval discrepancy problem, the problem becomes much more challenging, as evidenced by a recent work of Jiang, Kulkarni, and Singla [JKS19] that gives a non-trivial O(T1/loglogT) 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 that allows us to handle online vector balancing even when the input distribution has dependencies across coordinates. In particular, this lets us obtain a poly(n, logT) bound for online vector balancing under arbitrary input distributions, and a polylog (T) bound for online interval discrepancy. Our framework is powerful enough to capture other well-studied geometric discrepancy problems; e.g., we obtain a poly(logd (T)) bound for the online d-dimensional Tusnády’s problem. All our bounds are tight up to polynomial factors. A key new technical ingredient in our work is an anti-concentration inequality for sums of pairwise uncorrelated random variables, which might also be of independent interest.