
Online Discrepancy Minimization for Stochastic Arrivals
In the stochastic online vector balancing problem, vectors v_1,v_2,…,v_T...
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Online Geometric Discrepancy for Stochastic Arrivals with Applications to Envy Minimization
Consider a unit interval [0,1] in which n points arrive onebyone indep...
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OnLine Balancing of Random Inputs
We consider an online vector balancing game where vectors v_t, chosen un...
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Efficient Splitting of Measures and Necklaces
We provide approximation algorithms for two problems, known as NECKLACE ...
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Discrepancy Minimization via a SelfBalancing Walk
We study discrepancy minimization for vectors in ℝ^n under various setti...
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Gaussian discrepancy: a probabilistic relaxation of vector balancing
We introduce a novel relaxation of combinatorial discrepancy called Gaus...
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Online Discrepancy Minimization via Persistent SelfBalancing Walks
We study the online discrepancy minimization problem for vectors in ℝ^d ...
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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 onebyone and must be immediately given a ± 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 ± such that every subinterval remains nearly balanced. As random coloring incurs Ω(T^1/2) discrepancy, while the offline bounds are Θ((nlog T)^1/2) 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 worstcase scenario it is known that discrepancy is Ω(T^1/2) for any online algorithm. Bansal and Spencer recently show an O(√(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 nontrivial O(T^1/loglog 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 wellstudied geometric discrepancy problems; e.g., a poly(log^d (T)) bound for the online ddimensional Tusńady's problem. A key new technical ingredient is an anticoncentration inequality for sums of pairwise uncorrelated random variables.
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