带小坐标向量的在线平衡
Online balancing of vectors with small coordinates
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中文总结 AI 辅助
该研究针对带小坐标的向量序列，提出随机在线符号函数方法，证明其前缀偏差概率界，给出非均匀版本与下界，扩展至一般对称目标体，解决在线向量平衡问题。
中文摘要 AI 辅助
设 $v_1,\ldots,v_T\in B_2^m$ 预先固定并依次揭示，对所有 $1\leqslant t\leqslant T$，存在 $d\geqslant1$ 使得 $\\|v_t\\|_\infty\leqslant d^{-1/2}$。存在绝对常数 $L,C,c>0$ 和随机在线符号函数，使得 $\mathbb{P}\left\{\max_{k\leqslant T}\left\\|\sum_{t=1}^k\varepsilon_t v_t\right\\|_\infty>6L\right\} \leqslant CT\exp\left(-\frac{cd}{\ln^2(ed)}\right)$。因此，当 $d$ 至少为 $C\ln\frac{3T}{\varepsilon}\left[\ln\left(e+\ln\frac{3T}{\varepsilon}\right)\right]^2$ 时，前缀偏差为常数的概率至少为 $1-\varepsilon$。特别地，每个至多有 $d$ 个非零坐标的固定向量序列 $a_t\in[-1,1]^m$ 都存在在线符号函数，其前缀偏差为 $O(\sqrt{d})$，失败概率至多为 $CT\exp[-cd/\ln^2(ed)]$。我们还证明了非均匀版本，其中失败概率依赖于个体参数 $d_t=\\|v_t\\|_\infty^{-2}$，并给出下界：当 $d=o(\ln T)$ 时，通用常数前缀偏差不可能存在。我们确定了紧致势方法对应的 $\ln^2 d$ 障碍，并将论证扩展到具有二次光滑性估计的一般对称目标体。
英文摘要
Let $v_1,\ldots,v_T\in B_2^m$ be fixed in advance and revealed sequentially, and assume that $\|v_t\|_\infty\leqslant d^{-1/2}$ for some $d\geqslant 1$ and every $1\leqslant t\leqslant T$. There are absolute constants $L,C,c>0$ and a randomized online signing such that $$\mathbb{P}\left\{\max_{k\leqslant T}\left\|\sum_{t=1}^k\varepsilon_t v_t\right\|_\infty>6L\right\} \leqslant CT\exp\left(-\frac{cd}{\ln^2(ed)}\right).$$ Consequently, constant prefix discrepancy holds with probability at least $1-\varepsilon$ once $d$ is at least $C\ln\frac{3T}{\varepsilon}\left[\ln\left(e+\ln\frac{3T}{\varepsilon}\right)\right]^2$. In particular, every fixed sequence of vectors $a_t\in[-1,1]^m$ with at most $d$ nonzero coordinates admits an online signing with prefix discrepancy $O(\sqrt d)$ and failure probability at most $CT\exp[-cd/\ln^2(ed)]$. We also prove a nonuniform version in which the failure probability depends on the individual parameters $d_t=\|v_t\|_\infty^{-2}$, and a lower bound showing that a universal constant prefix discrepancy is impossible when $d=o(\ln T)$. We identify the corresponding $\ln^2 d$ barrier for the compact-potential method and extend the argument to general symmetric target bodies admitting a quadratic smoothness estimate.
