Many modern machine learning models are trained to achieve zero or near-zero
training error in order to obtain near-optimal (but non-zero) test error. This
phenomenon of strong generalization performance for "overfitted" / interpolated
classifiers appears to be ubiquitous in high-dimensional data, having been
observed in deep networks, kernel machines, boosting and random forests. Their
performance is consistently robust even when the data contain large amounts of
label noise.
Very little theory is available to explain these observations. The vast
majority of theoretical analyses of generalization allows for interpolation
only when there is little or no label noise. This paper takes a step toward a
theoretical foundation for interpolated classifiers by analyzing local
interpolating schemes, including geometric simplicial interpolation algorithm
and singularly weighted $k$-nearest neighbor schemes. Consistency or
near-consistency is proved for these schemes in classification and regression
problems. Moreover, the nearest neighbor schemes exhibit optimal rates under
some standard statistical assumptions.
Finally, this paper suggests a way to explain the phenomenon of adversarial
examples, which are seemingly ubiquitous in modern machine learning, and also
discusses some connections to kernel machines and random forests in the
interpolated regime.

本文分析局部插值方案，包括几何单纯插值算法和单一加权 k 近邻算法，在分类和回归问题中证明了这些方案的一致性或近一致性，并提出了一种解释对抗性示例的方法，同时讨论了与核机器和随机森林的一些联系。