DeepAI

# 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 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 sub-interval 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 worst-case 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 non-trivial 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 well-studied geometric discrepancy problems; e.g., a poly(log^d (T)) bound for the online d-dimensional Tusńady's problem. A key new technical ingredient is an anti-concentration inequality for sums of pairwise uncorrelated random variables.

• 27 publications
• 21 publications
• 43 publications
• 11 publications
07/21/2020

### Online Discrepancy Minimization for Stochastic Arrivals

In the stochastic online vector balancing problem, vectors v_1,v_2,…,v_T...
10/02/2019

### Online Geometric Discrepancy for Stochastic Arrivals with Applications to Envy Minimization

Consider a unit interval [0,1] in which n points arrive one-by-one indep...
03/16/2019

### On-Line Balancing of Random Inputs

We consider an online vector balancing game where vectors v_t, chosen un...
11/11/2021

### Online Discrepancy with Recourse for Vectors and Graphs

The vector-balancing problem is a fundamental problem in discrepancy the...
06/30/2020

### Efficient Splitting of Measures and Necklaces

We provide approximation algorithms for two problems, known as NECKLACE ...
09/17/2021

### Gaussian discrepancy: a probabilistic relaxation of vector balancing

We introduce a novel relaxation of combinatorial discrepancy called Gaus...
10/29/2022

### The Vector Balancing Constant for Zonotopes

The vector balancing constant vb(K,Q) of two symmetric convex bodies K,Q...