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

TL;DR

This paper develops theoretical risk bounds for interpolating classifiers and regressors, explaining their strong generalization in high-dimensional noisy data and connecting to phenomena like adversarial examples.

Contribution

It introduces and analyzes local interpolating schemes, providing consistency proofs and optimal rates, and offers insights into adversarial examples and connections to other models.

Findings

Interpolating classifiers can achieve consistency in noisy settings.

Nearest neighbor schemes attain optimal rates under standard assumptions.

Theoretical insights explain the robustness of overfitted models in practice.

Abstract

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.