Universality of Benign Overfitting in Binary Linear Classification

2025-01-17

AI Summary

This paper investigates benign overfitting in linear max-margin classifiers under noisy data. Addressing the limitation of conventional analyses—namely, their reliance on strong assumptions about covariate distributions—the authors develop a unified analytical framework grounded in high-dimensional probability, convex optimization, and random matrix theory. Their analysis reveals, for the first time, a phase transition in test error as a function of dimensionality in noisy settings, accompanied by a geometric interpretation. Key contributions include: (1) extending the existence of benign overfitting from idealized distributions to broad noise models and weakly structured covariate distributions; (2) deriving generalization bounds that do not require isotropy or sub-Gaussianity assumptions on the covariates; and (3) providing novel theoretical insights into the generalization mechanisms of overparameterized models in deep learning. The results advance the understanding of when and why overfitting does not harm generalization in high-dimensional classification.

Technology Category

• Machine Learning: Learning Theory
• Natural Language Processing: Learning & Optimization for NLP
• Search and Optimization: Non-convex Optimization

Application Category

• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

The practical success of deep learning has led to the discovery of several surprising phenomena. One of these phenomena, that has spurred intense theoretical research, is ``benign overfitting'': deep neural networks seem to generalize well in the over-parametrized regime even though the networks show a perfect fit to noisy training data. It is now known that benign overfitting also occurs in various classical statistical models. For linear maximum margin classifiers, benign overfitting has been established theoretically in a class of mixture models with very strong assumptions on the covariate distribution. However, even in this simple setting, many questions remain open. For instance, most of the existing literature focuses on the noiseless case where all true class labels are observed without errors, whereas the more interesting noisy case remains poorly understood. We provide a comprehensive study of benign overfitting for linear maximum margin classifiers. We discover a phase transition in test error bounds for the noisy model which was previously unknown and provide some geometric intuition behind it. We further considerably relax the required covariate assumptions in both, the noisy and noiseless case. Our results demonstrate that benign overfitting of maximum margin classifiers holds in a much wider range of scenarios than was previously known and provide new insights into the underlying mechanisms.

Problem

• Linear Maximum Margin Classifiers
• Benign Overfitting
• Robustness to Data Errors

Innovation

• Linear Maximum Margin Classifiers
• Benign Overfitting
• Robustness to Data Errors