A threshold for online balancing of sparse i.i.d. vectors

Abstract

Consider the task of \textit{online} vector balancing for stochastic arrivals $(X_i{)}_{i\in [T]}$, where the time horizon satisfies $T=\Theta (n)$, and the $X_i$ are i.i.d uniform $d$--sparse $n$--dimensional binary vectors, with $2\le d\le (\log \log n{)}^2/\log \log \log n$. We show that for this range of parameters, every online algorithm incurs discrepancy at least $\Omega (\log \log n)$, and there is an efficient algorithm which achieves a matching discrepancy bound of $O(\log \log n)$ w.h.p. This establishes an asymptotic gap, both existential and algorithmic, between the online and offline versions of the average--case Beck--Fiala problem. Strikingly, the optimal online discrepancy in the considered setting is order $\log \log n$, independent of $d$ and the norms of the vectors $(X_i{)}_i$. Our assumptions on $d$ are nearly optimal, as this independence ceases when $d=\omega ((\log \log n{)}^2)$.

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