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We study active learning methods for single index models of the form $$F({\bm x}) = f(\langle {\bm w}, {\bm x}\rangle)$$, where $$f:\mathbb{R} \to \mathbb{R}$$ and $${\bx,\bm w} \in \mathbb{R}^d$$. 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. 

A radiative vapor condenser sheds heat in the form of infrared radiation and cools itself to below the ambient air temperature to produce liquid water from vapor. This effect has been known for centuries, and is exploited by some insects to survive in dry deserts. Humans have also been using radiative condensation for dew collection. However, all existing radiative vapor condensers must operate during the nighttime. Here, we develop daytime radiative condensers that continue to operate 24 h a day. These daytime radiative condensers can produce water from vapor under direct sunlight, without active consumption of energy. Combined with traditional passive cooling via convection and conduction, radiative cooling can substantially increase the performance of passive vapor condensation, which can be used for passive water extraction and purification technologies.