AI 中文总结
该研究针对亚高斯输入的在线偏差最小化问题，提出结合ℓ_∞-范数正则化与自适应坐标限制的多项式时间算法，得到不依赖时间T的偏差界，推广了相关定理并接近最优。
AI 中文摘要
我们研究在线偏差最小化问题：向量v₁,…,v_T∈ℝⁿ依次到达，每个向量必须立即被分配一个符号x_t∈{±1}，目标是最小化‖∑_{t=1}^T x_t v_t‖_∞。我们提出一种基于多项式时间势函数的算法，该算法结合了ℝⁿ上的ℓ_∞-范数正则化与对自适应选择的坐标集的限制。对于满足以下条件的独立同分布输入：坐标为独立、对称、中心化、单位方差的亚高斯变量，且亚高斯范数至多为σ，该算法实现的最终偏差为O(σ⁸√n)，失败概率至少为1−exp(−Ω(σ³√n))。若坐标被均值为k/n的伯努利变量独立掩蔽，其中k≳(log n)²，则偏差界改进为O(σ⁸√k)，失败概率为exp(−Ω(σ³√k))。这两种保证均对任意给定的有限时间 horizon T 成立，且不依赖于T。该稠密输入结果大幅推广了Bansal与Spencer（2020）针对拉德马赫输入的定理，并为高斯输入提供了高效的O(√n)偏差界，这一结果是Gamarnik等人（2022）提出的猜想。当T比n大一个多项式因子时，该偏差界在条件上接近最优：根据标准近似格问题的最坏情况困难性假设，Vafa与Vaikuntanathan（2025）证明，即使是离线的多项式时间算法，也无法将√n的尺度在T/n的固定多项式因子下进一步改进。
英文摘要
We study online discrepancy minimization: vectors $v_1,\ldots,v_T\in\mathbb{R}^n$ arrive sequentially, and each must immediately be assigned a sign $x_t\in\{\pm1\}$, with the aim of minimizing $\|\sum_{t=1}^T x_t v_t\|_\infty$. We give a polynomial-time potential-based algorithm combining a regularization of the $\ell_\infty$-norm with restriction to an adaptively chosen coordinate set. For i.i.d. inputs with independent, symmetric, centered, unit-variance sub-Gaussian coordinates of sub-Gaussian norm at most $σ$, the algorithm achieves terminal discrepancy $O(σ^8\sqrt{n})$ with probability at least $1-\exp(-Ω(σ^3\sqrt{n}))$. If the coordinates are independently masked by Bernoulli variables with mean $k/n$, where $k\gtrsim(\log n)^2$, the bound improves to $O(σ^8\sqrt{k})$, with failure probability $\exp(-Ω(σ^3\sqrt{k}))$. Both guarantees hold for every prescribed finite horizon $T$, with no dependence on $T$. The dense result substantially generalizes a theorem of Bansal and Spencer (2020) for Rademacher inputs and gives an efficient $O(\sqrt{n})$ bound for Gaussian inputs, as conjectured by Gamarnik et al. (2022). When $T$ is polynomially larger than $n$, this is conditionally close to optimal: under worst-case hardness assumptions for standard approximate lattice problems, Vafa and Vaikuntanathan (2025) showed that no polynomial-time algorithm, even offline, can improve the $\sqrt{n}$ scale by a fixed polynomial factor in $T/n$.
