发表机构
The University of Texas at Austin; Carnegie Mellon University（德克萨斯大学奥斯汀分校; 卡内基梅隆大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该论文研究了随机到达的稀疏二元向量的在线平衡问题，确定了最优前缀偏差的量级为 $\Theta(\max\{\sqrt d,\log\log n\})$，并给出了达到上界的有效在线算法，揭示了稀疏度影响在线偏差的阈值。
AI 中文摘要
考虑随机到达的在线向量平衡任务，其中 $X_1,\ldots,{X_T}$ 是 $\{0,1\}^n$ 中独立的均匀随机 $d$ 稀疏二元向量。这是在线 Beck--Fiala 问题的随机类比。我们证明，对于 $2\le d\le n/2$ 和 $T = \Theta(n)$，最优在线前缀偏差 $\max\limits_{t\leq T}\left\\|\sum_{i=1}^t\sigma_i X_i\right\\|_\infty$ 的量级为 $\Theta\big(\max\{\sqrt d,\log\log n\}\big)$。上界由一种高效的在线算法实现。因此，对于 $d\le(\log\log n)^2$，最优偏差为 $\Theta(\log\log n)$，且在常数因子内与稀疏度无关；而当 $d$ 超过此尺度时，最优偏差为 $\Theta(\sqrt d)$，与离线偏差的量级一致。这确定了稀疏度开始主导随机 Beck--Fiala 模型在线偏差的阈值。
英文摘要
Consider the task of online vector balancing for stochastic arrivals $X_1,\ldots,{X_T}$, where the $X_i$ are independent uniformly random $d$--sparse binary vectors in $\{0,1\}^n$. This is a random analogue of the online Beck--Fiala problem. We show that uniformly for $2\le d\le n/2$ and $T = Θ(n)$, the optimal online prefix discrepancy $\max\limits_{t\leq T}\left\|\sum_{i=1}^tσ_i X_i\right\|_\infty$ is of order \[ Θ\big(\max\{\sqrt d,\log\log n\}\big). \] The upper bound is achieved by an efficient online algorithm. Thus, for $d\le(\log\log n)^2$, the optimal discrepancy is $Θ(\log\log n)$ and is independent of the sparsity up to constant factors, whereas above this scale it is $Θ(\sqrt d)$, matching the order of the offline discrepancy. This identifies the threshold at which sparsity begins to govern the online discrepancy of the random Beck--Fiala model.
CommentsSupersedes and replaces arxiv.org/abs/2509.02432, which has the same lower bound, but only has an upper bound for the ultra-sparse regime
