发表机构
Georgia Institute of Technology; The University of Texas at Austin（佐治亚理工学院; 德克萨斯大学奥斯汀分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明在线向量平衡问题的最优值在随机输入下具有分布无关的普适极限，并据此构造了达到该极限的随机在线算法，同时应用于对称二元感知器的在线阈值普适性。
AI 中文摘要
我们研究在线向量平衡问题，其中$\mathbb{R}^M$中的$N$个随机向量按顺序揭示，每个向量在到达时必须被赋予不可撤销的符号。目标是最终符号和的期望$\ell^\infty$范数最小化。对于均值为零、方差为一且具有有限四阶矩的独立同分布条目，我们证明，当$M/N\to\alpha\in(0,\infty)$时，最优值除以$\sqrt N$收敛到一个与条目分布无关的极限$R_\alpha$。该极限是Fiedler、Jackson、Lacker和Niles-Weed为高斯输入确定的随机控制值。特别地，它决定了Rademacher输入的精确渐近最优值。对于每个$\kappa>R_\alpha$，我们构造一个随机在线算法，其最终符号和的$\ell^\infty$范数以高概率至多为$\kappa\sqrt N$；对于$\kappa<R_\alpha$，每个在线算法的成功概率趋于零。因此，对称二元感知器的在线阈值在每个正余量下都是普适的。主要步骤是一种耦合，将布朗控制转移到非高斯输入，而截断控制罕见的较大条目。
英文摘要
We study online vector balancing with $N$ random vectors in $\mathbb{R}^M$ revealed sequentially, where each vector must be assigned an irrevocable sign upon arrival. The goal is to minimize the expected $\ell^\infty$ norm of the final signed sum. For i.i.d. entries with mean zero, variance one, and a finite fourth moment, we prove that, as $M/N\toα\in(0,\infty)$, the optimal value divided by $\sqrt N$ converges to a limit $R_α$ independent of the entry distribution. This limit is the stochastic control value identified for Gaussian inputs by Fiedler, Jackson, Lacker, and Niles-Weed. In particular, it determines the exact asymptotic optimum for Rademacher inputs. For every $κ>R_α$, we construct a randomized online algorithm whose final signed sum has $\ell^\infty$ norm at most $κ\sqrt N$ with high probability; for $κ<R_α$, every online algorithm has vanishing success probability. Consequently, the online threshold of the symmetric binary perceptron is universal at every positive margin. The main step is a coupling that transfers Brownian controls to non-Gaussian inputs, while truncation controls rare large entries.
Comments40 pages
