发表机构
School of Physical and Mathematical Sciences, Nanyang Technological University; John Hopcroft Center for Computer Science, Shanghai Jiao Tong University（南洋理工大学物理与数学科学学院; 上海交通大学约翰·霍普克罗夫特计算机科学中心）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对一般p≥1下未知链接函数的单索引模型主动ℓₚ回归问题，提出非自适应采样算法并建立几乎紧下界，缩小了该领域的理论差距。
AI 中文摘要
本文研究一般ℓₚ损失下未知1-利普希茨链接函数f的单索引模型主动回归问题，建模为min_{f,x} ||f(Ax)-b||ₚᵖ，其中可完全访问矩阵A，仅能对向量b进行坐标查询。现有工作针对已知链接函数对所有p≥1的情况建立了上界，针对未知链接函数仅在p=2的情况建立了上界，同时针对p≤2的情况建立了下界。本文解决了更具挑战性的未知链接函数及一般p≥1的情况，提出了一种非自适应采样算法，使用O(d^{p/2∨1}/ε^{p∨2}poly log(n/ε))次查询实现(1+ε)近似，还针对p>2的情况建立了几乎紧的下界，这些结果缩小了单索引模型主动ℓₚ回归中剩余的大量差距。
英文摘要
This paper studies active regression for single-index models under general $\ell_p$-loss with an unknown $1$-Lipschitz link function $f$, formulated as $\min_{f,x} \|f(Ax)-b\|_p^p$ with full access to $A$ but coordinate-query access to $b$. Prior work established upper bounds for known link functions for all $p\geq 1$ and for unknown link functions only in the $p=2$ case, together with lower bounds for $p\leq 2$. This work addresses the more challenging setting of unknown link functions and general $p \geq 1$. A non-adaptive sampling algorithm is presented that achieves a $(1+ε)$-approximation using $O(d^{p/2\vee 1}/ε^{p\vee 2}\operatorname{poly}\log(n/ε))$ queries. Nearly tight lower bounds are also established for $p>2$. These results close much of the remaining gap in active $\ell_p$-regression for single-index models.
CommentsEarlier version accepted to ICML 2026; the lower bound has been extended to adaptive queries
