Active Regression for Single-Index Models with Unknown Link Functions

Abstract: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 $(1+\epsilon)$-approximation using $O(d^{p/2\vee 1}/\epsilon^{p\vee 2}\operatorname{poly}\log(n/\epsilon))$ 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.

Data Structures and Algorithms (cs.DS); Machine Learning (cs.LG)

arXiv:2608.01287 [cs.DS]

(or arXiv:2608.01287v1 [cs.DS] for this version)

https://doi.org/10.48550/arXiv.2608.01287 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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