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Active Local Learning

In this work we consider active local learning: given a query point $x$, and active access to an unlabeled training set $S$, output the prediction $h(x)$ of a near-optimal $h \in H$ using significantly fewer labels than would be needed to actually learn $h$ fully. In particular, the number of label queries should be independent of the complexity of $H$, and the function $h$ should be well-defined, independent of $x$. This immediately also implies an algorithm for distance estimation: estimating the value $opt(H)$ from many fewer labels than needed to actually learn a near-optimal $h \in H$, by running local learning on a few random query points and computing the average error. For the hypothesis class consisting of functions supported on the interval $[0,1]$ with Lipschitz constant bounded by $L$, we present an algorithm that makes $O(({1 / ε^6}) \log(1/ε))$ label queries from an unlabeled pool of $O(({L / ε^4})\log(1/ε))$ samples. It estimates the distance to the best hypothesis in the class to an additive error of $ε$ for an arbitrary underlying distribution. We further generalize our algorithm to more than one dimensions. We emphasize that the number of labels used is independen