  Active learning seeks to build the best possible model with a budget of
labelled data by sequentially selecting the next point to label. However the
training set is no longer \textit{iid}, violating the conditions required by
existing consistency results. Inspired by the success of Stone's Theorem we aim
to regain consistency for weighted averaging estimators under active learning.
Based on ideas in \citet{dasgupta2012consistency}, our approach is to enforce a
small amount of random sampling by running an augmented version of the
underlying active learning algorithm. We generalize Stone's Theorem in the
noise free setting, proving consistency for well known classifiers such as
$k$-NN, histogram and kernel estimators under conditions which mirror classical
results. However in the presence of noise we can no longer deal with these
estimators in a unified manner; for some satisfying this condition also
guarantees sufficiency in the noisy case, while for others we can achieve near
perfect inconsistency while this condition holds. Finally we provide conditions
for consistency in the presence of noise, which give insight into why these
estimators can behave so differently under the combination of noise and active
learning.
