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.
We study the problem of estimating the number of edges in a graph with access to only an independent set oracle. Independent set queries draw motivation from group testing and have applications to the complexity of decision versus counting problems. We give two algorithms to estimate the number of edges in an n-vertex graph: one that uses only polylog(n) bipartite independent set queries, and another one that uses n 2/3 • polylog(n) independent set queries.
We show that a black-box construction of a pseudorandom generator from a one-way function needs to make Ω( n log(n) ) calls to the underlying one-way function. The bound even holds if the one-way function is guaranteed to be regular. In this case it matches the best known construction due to Goldreich, Krawczyk, and Luby (SIAM J. Comp. 22, 1993), which uses O( n log(n) ) calls.
In the stochastic online vector balancing problem, vectors v 1 , v 2 , . . . , v T chosen independently from an arbitrary distribution in R n arrive one-by-one and must be immediately given a ± sign. The goal is to keep the norm of the discrepancy vector, i.e., the signed prefix-sum, as small as possible for a given target norm.We consider some of the most well-known problems in discrepancy theory in the above online stochastic setting, and give algorithms that match the known offline bounds up to polylog(nT ) factors. This substantially generalizes and improves upon the previous results of Bansal, Jiang, Singla, and Sinha (STOC' 20). In particular, for the Komlós problem where v t 2 ≤ 1 for each t, our algorithm achieves O(1) discrepancy with high probability, improving upon the previous O(n 3/2 ) bound. For Tusnády's problem of minimizing the discrepancy of axisaligned boxes, we obtain an O(log d+4 T ) bound for arbitrary distribution over points. Previous techniques only worked for product distributions and gave a weaker O(log 2d+1 T ) bound. We also consider the Banaszczyk setting, where given a symmetric convex body K with Gaussian measure at least 1/2, our algorithm achieves O(1) discrepancy with respect to the norm given by K for input distributions with sub-exponential tails.Our results are based on a new potential function approach.
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