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Discrepancy and $\epsilon$-Approximations for Bounded VC-Dimensions

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Emo Welzl (joint work with Jiri Matousek, Lorenz Wernisch)
Freie Universität Berlin
Institut für Informatik
Email: welzl@inf.fu-berlin.de
Report B 91-06
April 1991

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File: pub/reports/tr-b-91-06.ps.gz

Let $(X,\RR)$ be a set system on an $n$-point set $X$.
We investigate colorings $\chi:X\to\{-1,+1\}$, such that the sum
of colors in each set $R \in \RR$ does not deviate too much
from $0$. The largest value $| \sum_{x \in R} \chi(x)|$ over all
$R \in \RR$ is the {\em discrepancy} of $\chi$ on ${\cal R}$.
We also consider {$\eps$-ap\-prox\-i\-ma\-tions}: For $0 < \eps < 1$,
a subset $A$ of $X$ is an {\em $\eps$-ap\-prox\-i\-ma\-tion} for
$(X,\RR)$, if
$|\, |A \cap R| / |A| - |R| / n \,| < \eps $ for all $R \in \RR$.
Let $d$ be a fixed integer such that for any $Y\subseteq X$,
the number of distinct sets of the form $R\cap Y$, $R\in\RR$
is bounded by $O(|Y|^d)$ (i.e., $(X,\RR)$ is an $n$-point range space
with the {\em primal shatter function} $\pi_{\cal R}(m)$ bounded by
$O(m^d)$). Then we prove that there is always a coloring with discrepancy
bounded by $O(\pdiscub)$. We show that this implies that, for any $r$,
there exists a {\em $(1/r)$-ap\-prox\-i\-ma\-tion} for $(X,{\cal R})$ of size
$O(\paproub)$. This improves on a previous bound of $O(r^2\log r)$ due to
Vapnik and Chervonenkis.
If any subcollection of $m$ sets of $\RR$ partitions the points into
at most $O(m^d)$ classes (i.e., the {\em dual shatter function} is
bounded by $O(m^d)$), then we get a bound of $O(\ddiscub)$ for
discrepancy and of $O(\daproub)$ for $(1/r)$-ap\-prox\-i\-ma\-tions. These
bounds via the dual shatter function can be realized by deterministic
polynomial time algorithms.
All the bounds are tight upto polylogarithmic factors in the worst
case. Our results allow to generalize several results of Beck
bounding the discrepancy in certain geometric settings to the case
when the discrepancy is taken relative to an arbitrary measure.