  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 $O(({1 / \epsilon^6}) \log(1/\epsilon))$ label queries from an unlabeled
pool of $O(({L / \epsilon^4})\log(1/\epsilon))$ samples. It estimates the
distance to the best hypothesis in the class to an additive error of $\epsilon$
for an arbitrary underlying distribution. We further generalize our algorithm
to more than one dimensions. We emphasize that the number of labels used is
independent of the complexity of the hypothesis class which depends on $L$.
Furthermore, we give an algorithm to locally estimate the values of a
near-optimal function at a few query points of interest with number of labels
independent of $L$.
  We also consider the related problem of approximating the minimum error that
can be achieved by the Nadaraya-Watson estimator under a linear diagonal
transformation with eigenvalues coming from a small range. For a
$d$-dimensional pointset of size $N$, our algorithm achieves an additive
approximation of $\epsilon$, makes $\tilde{O}({d}/{\epsilon^2})$ queries and
runs in $\tilde{O}({d^2}/{\epsilon^{d+4}}+{dN}/{\epsilon^2})$ time.
