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We show how to obtain improved active learning methods in the agnostic(adversarial noise) setting by combining marginal leverage score sampling withnon-independent sampling strategies that promote spatial coverage. Inparticular, we propose an easily implemented method based on the pivotalsampling algorithm, which we test on problems motivated by learning-basedmethods for parametric PDEs and uncertainty quantification. In comparison toindependent sampling, our method reduces the number of samples needed to reacha given target accuracy by up to $50\%$. We support our findings with twotheoretical results. First, we show that any non-independent leverage scoresampling method that obeys a weak one-sided $\ell_{\infty}$ independencecondition (which includes pivotal sampling) can actively learn $d$ dimensionallinear functions with $O(d\log d)$ samples, matching independent sampling. Thisresult extends recent work on matrix Chernoff bounds under $\ell_{\infty}$independence, and may be of interest for analyzing other sampling strategiesbeyond pivotal sampling. Second, we show that, for the important case ofpolynomial regression, our pivotal method obtains an improved bound of $O(d)$samples.

Improved Active Learning via Dependent Leverage Score Sampling