Online vector balancing and geometric discrepancy

Abstract

We consider an online vector balancing question where T vectors, chosen from an arbitrary distribution over [−1, 1] n , arrive one-byone 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(T 1/2 ) 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(T 1/2 ) for any online algorithm.In a special case of online vector balancing, Bansal and Spencer [BS19] recently show an O( √ n logT ) 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(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 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(log d (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 anticoncentration inequality for sums of pairwise uncorrelated random variables, which might also be of independent interest.

Discrepancy Minimization via a Self-Balancing Walk

“…We also note that this theorem is sharp up to logarithmic factors in n and t due to a lower bound of Ω( log t/ log log t) given in [8]. It seems possible to the authors that a variant of Algorithm 1 can maintain an ℓ ∞ bound of O( √ log nt) instead of O(log nt).…”

“…Algorithm 1 works against oblivious adversaries. Therefore, Theorem 1.1 implies tight bounds up to logarithmic factors in n and t for all of Questions 1-5 in [8]. We state Questions 4 and 5, which are about oblivious adversaries, as these generalize the stochastic and prophet models discussed in the other questions raised in [8].…”

“…The best previous results in this direction instead required that v i were sampled in an iid manner from a distribution p which is known beforehand. In this direction Bansal and Spencer [9] showed an ℓ ∞ guarantee of O( √ n log t) when the vectors v i were sampled uniformly from [−1, 1] n and Bansal, Jiang, Singla, and Sinha [8] achieved a ℓ ∞ guarantee of O(n 2 log(nt)) when the vectors v i were sampled from a known distribution p supported on [−1, 1] n .…”

Sequential Fair Allocation: Achieving the Optimal Envy-Efficiency Tradeoff Curve

“…Our model instead imposes stochastic assumptions on the number of arriving individuals and characterizes probabilistic instead of sample-path fairness criteria. Online Resources: One line of work considers the resource (here to be thought of as the units of food, processing power, etc) are online and the agents are fixed [13,4,36,35,2,9,10,16]. In [51] they study the tradeoffs between fairness and efficiency when items arrive under several adversarial models.…”

A Gaussian fixed point random walk

“…In the more general setting where p is a general distribution supported on [−1, 1] n , Aru, Narayanan, Scott, and Venkatesan [3] achieved a bound of O n ( √ log t) (where the implicit dependence on n is super-exponential) and Bansal, Jiang, Meka, Singla, and Sinha [9] (building on work of Bansal, Jiang, Singla, and Sinha [10]) achieved an ℓ ∞ guarantee of O( √ n log(nt) 4 ).…”