We derive a new asymptotic expansion for the global excess risk of a
local-$k$-nearest neighbour classifier, where the choice of $k$ may depend upon
the test point. This expansion elucidates conditions under which the dominant
contribution to the excess risk comes from the decision boundary of the optimal
Bayes classifier, but we also show that if these conditions are not satisfied,
then the dominant contribution may arise from the tails of the marginal
distribution of the features. Moreover, we prove that, provided the
$d$-dimensional marginal distribution of the features has a finite $\rho$th
moment for some $\rho > 4$ (as well as other regularity conditions), a local
choice of $k$ can yield a rate of convergence of the excess risk of
$O(n^{-4/(d+4)})$, where $n$ is the sample size, whereas for the standard
$k$-nearest neighbour classifier, our theory would require $d \geq 5$ and $\rho
> 4d/(d-4)$ finite moments to achieve this rate. These results motivate a new
$k$-nearest neighbour classifier for semi-supervised learning problems, where
the unlabelled data are used to obtain an estimate of the marginal feature
density, and fewer neighbours are used for classification when this density
estimate is small. Our worst-case rates are complemented by a minimax lower
bound, which reveals that the local, semi-supervised $k$-nearest neighbour
classifier attains the minimax optimal rate over our classes for the excess
risk, up to a subpolynomial factor in $n$. These theoretical improvements over
the standard $k$-nearest neighbour classifier are also illustrated through a
simulation study.

研究使用局部 -$k$- 最近邻分类器的全局超额风险的渐近展开式，通过此理论发现半监督学习问题中的局部选择 $k$ 能够实现额外风险的收敛速率，同时通过模拟研究验证了该理论。