Active Regression for Single-Index Models with Unknown Link Functions

2026-08-02

AI Summary

This work addresses the problem of active regression for single-index models under an unknown 1-Lipschitz link function, where the response vector can only be accessed via coordinate queries. The goal is to jointly estimate both the link function and the underlying parameter vector under ℓ_p loss for any p ≥ 1. The paper proposes the first computationally efficient algorithm applicable to arbitrary p ≥ 1, combining non-adaptive sampling, ℓ_p regression, and function approximation techniques. It achieves a (1+ε)-approximate solution with query complexity O(d^{p/2 ∨ 1}/ε^{p ∨ 2} · polylog(n/ε)). Notably, for p > 2, the authors establish a nearly tight information-theoretic lower bound, thereby closing a significant theoretical gap in this setting.

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 $(1+ε)$-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.

Problem

• active regression
• single-index models
• unknown link function
• ℓ_p-loss

Innovation

• active regression
• single-index models
• unknown link function
• ℓ_p-loss
• non-adaptive sampling