Online Discrepancy of Sparse Random Vectors over Long Horizons

Source abstract

We study online vector balancing for independent sparse random vectors: each arrival is supported on a uniformly random set of $s$ of the $n$ coordinates, with nonzero entries equal to one (binary model) or independent random signs (signed model), and an online algorithm must sign each vector upon arrival. The prefix discrepancy is the largest $\ell_\infty$-norm of the signed sums up to the horizon $T$, the terminal discrepancy that of the sum at time $T$. For binary inputs with $2\le s\le n/2$ and $T=Θ(n)$, Altschuler and Tikhomirov showed that both are of order $\sqrt{s}+\log\log n$; they asked how the prefix discrepancy depends on the horizon and whether a matching terminal bound holds at every prescribed time. Over long horizons, the two differ. For every fixed $s\ge2$ and $c>0$, every online algorithm, even a randomized one that knows the horizon, incurs with high probability prefix discrepancy $Ω(\log\log n+(\log T/\log n)^{α_s})$ for $n\le T\le e^{cn}$, where $α_s=(2^{s-1}-1)/(2^s-1)$. For $s=2$, quadratic Greedy matches this bound; for fixed $s\ge3$, we give an online algorithm with prefix discrepancy $O(\log\log n+(\log T/\log n)^{1/2})$. In contrast, a deterministic algorithm that knows neither the horizon nor the failure probability keeps the terminal discrepancy $O_k(\sqrt{s}+\log\log n)$ at each prescribed time, with failure probability $n^{-k}$ for every fixed $k$, for $2\le s\le n/2$ (binary model) and $2\le s\le n$ (signed model); for $s=2$, quadratic Greedy itself does so. This answers their terminal question, and the same order is optimal for prefixes at all polynomial horizons $T\ge n$. The lower bound uses a finite safety game realized within the input law; the upper bounds use moments of elementary symmetric polynomials and, at terminal times, a $\cosh$ potential or a translated gauge with a Foster inequality at every state.