Near-optimal Active Regression of Single-Index Models
Yi Li Wai Ming Tai
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 $p>2$.
Lewis weights, Active regression, Query complexity
We provide a near-optimal query complexity for the active regression of single-index models.