  We study active learning methods for single index models of the form
$F({\mathbf x}) = f(\langle {\mathbf w}, {\mathbf x}\rangle)$, where
$f:\mathbb{R} \to \mathbb{R}$ and ${\mathbf x,\mathbf w} \in \mathbb{R}^d$. In
addition to their theoretical interest as simple examples of non-linear neural
networks, single index models have received significant recent attention due to
applications in scientific machine learning like surrogate modeling for partial
differential equations (PDEs). Such applications require sample-efficient
active learning methods that are robust to adversarial noise. I.e., that work
even in the challenging agnostic learning setting.
  We provide two main results on agnostic active learning of single index
models. First, when $f$ is known and Lipschitz, we show that $\tilde{O}(d)$
samples collected via {statistical leverage score sampling} are sufficient to
learn a near-optimal single index model. Leverage score sampling is simple to
implement, efficient, and already widely used for actively learning linear
models. Our result requires no assumptions on the data distribution, is optimal
up to log factors, and improves quadratically on a recent ${O}(d^{2})$ bound of
\cite{gajjar2023active}. Second, we show that $\tilde{O}(d)$ samples suffice
even in the more difficult setting when $f$ is \emph{unknown}. Our results
leverage tools from high dimensional probability, including Dudley's inequality
and dual Sudakov minoration, as well as a novel, distribution-aware
discretization of the class of Lipschitz functions.
