  $k$-nearest neighbour ($k$-NN) is one of the simplest and most widely-used
methods for supervised classification, that predicts a query's label by taking
weighted ratio of observed labels of $k$ objects nearest to the query. The
weights and the parameter $k \in \mathbb{N}$ regulate its bias-variance
trade-off, and the trade-off implicitly affects the convergence rate of the
excess risk for the $k$-NN classifier; several existing studies considered
selecting optimal $k$ and weights to obtain faster convergence rate. Whereas
$k$-NN with non-negative weights has been developed widely, it was also proved
that negative weights are essential for eradicating the bias terms and
attaining optimal convergence rate. In this paper, we propose a novel
multiscale $k$-NN (MS-$k$-NN), that extrapolates unweighted $k$-NN estimators
from several $k \ge 1$ values to $k=0$, thus giving an imaginary 0-NN
estimator. Our method implicitly computes optimal real-valued weights that are
adaptive to the query and its neighbour points. We theoretically prove that the
MS-$k$-NN attains the improved rate, which coincides with the existing optimal
rate under some conditions.
