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arXiv · 2502.18213

Near-optimal Active Regression of Single-Index Models

Yi Li·Wai Ming Tai
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Abstract

The active regression problem of the single-index model is to solve $\min_x \lVert f(Ax)-b\rVert_p$, where $A$ is fully accessible and $b$ can only be accessed via entry queries, with the goal of minimizing the number of queries to the entries of $b$. When $f$ is Lipschitz, previous results only obtain constant-factor approximations. This work presents the first algorithm that provides a $(1+\varepsilon)$-approximation solution by querying $\tilde{O}(d^{\frac{p}{2}\vee 1}/\varepsilon^{p\vee 2})$ entries of $b$. This query complexity is also shown to be optimal up to logarithmic factors for $p\in [1,2]$ and the $\varepsilon$-dependence of $1/\varepsilon^p$ is shown to be optimal for $p>2$.

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Yi Li, Wai Ming Tai. 2025-02-25. Near-optimal Active Regression of Single-Index Models. https://arxiv.org/abs/2502.18213

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