发表机构
Northeastern University; Beijing Jiaotong University; Xiangtan University（东北大学; 北京交通大学; 湘潭大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文解决分段-Lipschitz函数在线优化中的一维局部根反集中问题，建立区间命中常数的尖锐无维表征，给出图学习应用的遗憾界，消除此前损失并验证了相关系数律的判据。
AI 中文摘要
本文针对分段-Lipschitz函数在线优化背景下Balcan、Pegden和Sharma提出的一维局部根反集中问题给出解答。对于齐次特征曲线及系数（其相对于对称凸体K上均匀律的密度被A界定），本文证明最坏区间命中常数等于A乘以截面平均投影关联速度。对于立方体支撑系数，该速度在通用常数范围内等价于投影Lipschitz常数，由此得到尖锐的无维表征并消除了此前的√N损失。对于首项系数为1的d次多项式及任意系数律，本文证明区间命中常数有限当且仅当有序实根律具有有界密度，且存在尖锐的d倍比较因子。条件与联合系数空间面积公式结合双图表准则，使该判据可用于相关及奇异系数律的验证。本文还给出两个图学习应用以完成遗憾转换链：代价敏感高斯-RBF调和分类器利用投影关联定理，达到期望遗憾Õ((An²D e^{BD}/ℓ+1)√T)；共偏移多项式核模型利用有序根的刚性平移，达到Õ((qn²κ+1)√T)的遗憾，即使诱导系数律在环境系数空间中是奇异的。
英文摘要
This paper answers the one-dimensional local root anti-concentration questions posed by Balcan, Pegden, and Sharma in the context of online optimization of piecewise-Lipschitz functions. For a homogeneous feature curve and coefficients whose density relative to the uniform law on a symmetric convex body $K$ is bounded by $A$, we show that the worst-case interval-hitting constant equals $A$ times a section-averaged projective incidence speed. For cube-supported coefficients, this speed is equivalent, up to universal constants, to the projective Lipschitz constant. This yields a sharp, dimension-free characterization and removes the previous $\sqrt N$ loss. For monic degree-$d$ polynomials under arbitrary coefficient laws, we prove that the interval-hitting constant is finite if and only if the ordered real-root laws have bounded densities, with a factor-$d$ comparison that is sharp. Conditional and joint coefficient-space area formulas, together with a two-chart certificate, make this criterion verifiable for dependent and singular coefficient laws. We also give two graph-learning applications that complete the transition-to-regret chain. A cost-sensitive Gaussian-RBF harmonic classifier uses the projective incidence theorem and achieves expected regret $\widetilde O((An^2D e^{BD}/\ell+1)\sqrt T)$. A common-offset polynomial-kernel model uses rigid translation of the ordered roots and achieves $\widetilde O((qn^2κ+1)\sqrt T)$ regret, even when the induced coefficient law is singular in the ambient coefficient space.
Comments27 pages, 3 figures
