Overfitting or perfect fitting? Risk bounds for classification and regression rules that interpolate (1806.05161v3)

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.

• The paper introduces a theoretical framework for interpolating methods that achieve near-optimal risk under minimal label noise.
• It demonstrates that techniques like weighted interpolated nearest neighbors ensure statistical consistency in high-dimensional spaces.
• The study reveals that while adversarial examples emerge with interpolation, their impact becomes negligible with ample training data.

Risk Bounds for Classification and Regression Rules that Interpolate

The paper by Belkin, Hsu, and Mitra explores the intricacies of overfitting, a prominent concern in machine learning models, focusing specifically on the risk bounds for classifiers and regression rules that interpolate data. This exploration is notably relevant in contexts involving high-dimensional data, where methods achieving interpolation such as deep networks, kernel machines, boosting, and random forests demonstrate robust generalization capabilities despite significant label noise.

Key Contributions

1. Theory of Interpolating Methods: The paper contributes to establishing a theoretical foundation for interpolating classifiers by analyzing local interpolating schemes like the geometric simplicial interpolation algorithm and nearest neighbor schemes. This work is set against the backdrop of a prevailing theoretical landscape that dismisses interpolation due to its traditionally perceived poor statistical properties.
2. Risk Optimality and Consistency: Exploring canonical non-parametric methods such as nearest neighbors, the authors establish that certain interpolating schemes exhibit risk consistency under standard statistical assumptions. They specifically focus on settings where the methods demonstrate near-optimal risk rates, contingent on achieving minimal label noise.
3. Analysis of Adversarial Examples: The paper provides a novel perspective on adversarial examples, often cited as a downside of neural networks. It conjectures that the interpolation in the presence of label noise inevitably yields adversarial examples but posits that their overall impact could be asymptotically negligible with sufficient training data.
4. Interpolation Techniques: The research introduces and evaluates new schemes such as the weighted interpolated nearest neighbor (wiNN) scheme, which ensures statistical consistency even in high-dimensional settings. It underscores a phenomenon termed the "blessing of dimensionality," implying that interpolation efficacy improves with increasing dimensionality, contrasting the standard "curse of dimensionality" typically observed in non-parametric methods.
5. Theoretical Implications: By examining and proving non-asymptotic rates of convergence to the Bayes risk for interpolated predictors, the work sets a precedent for understanding the success of interpolation methods in machine learning. Such findings challenge prior beliefs on the infeasibility of interpolation where label noise is present.

Future Directions and Implications

The paper posits multiple avenues for further research. A deeper understanding of the link between interpolation and adversarial robustness could inform the development of new algorithms that inherently resist adversarial attacks. Moreover, translating these theoretical insights into practical applications may yield more effective learning algorithms that adjust smoothly to the interpolation regime.

From a theoretical standpoint, the paper calls for expanded analysis into interpolation methods within the broader architecture of kernel machines and extensive neural networks. There is substantial scope in elucidating the fundamental mechanisms that make modern AI techniques excel in highly noisy environments.

Overall, the paper presents significant ramifications for both the practical application in machine learning tasks and foundational machine learning theory. As technology evolves, these insights may guide the design of future algorithms that seamlessly integrate interpolation strategies for optimal performance.