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Many modern machine learning models are trained to achieve zero or near-zerotraining error in order to obtain near-optimal (but non-zero) test error. Thisphenomenon of strong generalization performance for "overfitted" / interpolatedclassifiers appears to be ubiquitous in high-dimensional data, having beenobserved in deep networks, kernel machines, boosting and random forests. Theirperformance is consistently robust even when the data contain large amounts oflabel noise.  Very little theory is available to explain these observations. The vastmajority of theoretical analyses of generalization allows for interpolationonly when there is little or no label noise. This paper takes a step toward atheoretical foundation for interpolated classifiers by analyzing localinterpolating schemes, including geometric simplicial interpolation algorithmand singularly weighted $k$-nearest neighbor schemes. Consistency ornear-consistency is proved for these schemes in classification and regressionproblems. Moreover, the nearest neighbor schemes exhibit optimal rates undersome standard statistical assumptions.  Finally, this paper suggests a way to explain the phenomenon of adversarialexamples, which are seemingly ubiquitous in modern machine learning, and alsodiscusses some connections to kernel machines and random forests in theinterpolated regime.

Overfitting or perfect fitting? Risk bounds for classification and regression rules that interpolate