Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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 $\pm$ 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 $\pm$ such that every sub-interval remains nearly balanced. As random
coloring incurs $\Omega(T^{1/2})$ discrepancy, while the offline bounds are
$\Theta(\sqrt{n \log (T/n)})$ 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
$\Omega(T^{1/2})$ for any online algorithm.
  Bansal and Spencer recently show an $O(\sqrt{n}\log T)$ 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 $O(T^{1/\log\log T})$
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 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 Tusn\'ady's problem. A key new technical ingredient is an
{anti-concentration} inequality for sums of pairwise uncorrelated random
variables.

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The categories of this paper are: bayes-an
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The categories of this paper are: q-alg
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The categories of this paper are: gr-qc
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The categories of this paper are: math
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The categories of this paper are: chao-dyn
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The categories of this paper are: hep-th
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The categories of this paper are: dg-ga
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The categories of this paper are: solv-int
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The categories of this paper are: astro-ph
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The categories of this paper are: mtrl-th
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The categories of this paper are: quant-ph
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The categories of this paper are: math-ph
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The categories of this paper are: econ
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The categories of this paper are: supr-con
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The categories of this paper are: nucl-ex
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The categories of this paper are: hep-lat