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Active Tolerant Testing

In this work, we give the first algorithms for tolerant testing of nontrivial classes in the active model: estimating the distance of a target function to a hypothesis class C with respect to some arbitrary distribution D, using only a small number of label queries to a polynomial-sized pool of unlabeled examples drawn from D. Specifically, we show that for the class D of unions of d intervals on the line, we can estimate the error rate of the best hypothesis in the class to an additive error epsilon from only $O(\frac{1}{\epsilon^6}\log \frac{1}{\epsilon})$ label queries to an unlabeled pool of size $O(\frac{d}{\epsilon^2}\log \frac{1}{\epsilon})$. The key point here is the number of labels needed is independent of the VC-dimension of the class. This extends the work of Balcan et al. [2012] who solved the non-tolerant testing problem for this class (distinguishing the zero-error case from the case that the best hypothesis in the class has error greater than epsilon). We also consider the related problem of estimating the performance of a given learning algorithm A in this setting. That is, given a large pool of unlabeled examples drawn from distribution D, can we, from only a few label queries, estimate how well A would perform if the entire dataset were labeled? We focus on k-Nearest Neighbor style algorithms, and also show how our results can be applied to the problem of hyperparameter tuning (selecting the best value of k for the given learning problem).

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