Benign Overfitting and the Geometry of the Ridge Regression Solution in Binary Classification

2025-03-11

AI Summary

This work studies the generalization behavior of overparameterized ridge regression for high-dimensional binary classification under label-flip noise and anisotropic cluster-structured data. Leveraging high-dimensional probability theory and anisotropic random matrix theory, we characterize the classification error’s joint dependence on cluster-mean magnitude and the effective rank of the covariance tail. We establish, for the first time, precise necessary and sufficient conditions for “benign overfitting” in binary classification via ridge regression—and prove their equivalence to those in linear regression. Furthermore, we show that while label noise shifts the geometry of the minimum-norm interpolating solution, it preserves its qualitative generalization behavior. Our theoretical framework quantitatively disentangles the bias mechanism induced by noise and delineates sharp generalization boundaries. This provides a novel perspective on the robustness of linear classifiers operating on structured, high-dimensional data.

Technology Category

• Machine Learning: Classification and Regression
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

In this work, we investigate the behavior of ridge regression in an overparameterized binary classification task. We assume examples are drawn from (anisotropic) class-conditional cluster distributions with opposing means and we allow for the training labels to have a constant level of label-flipping noise. We characterize the classification error achieved by ridge regression under the assumption that the covariance matrix of the cluster distribution has a high effective rank in the tail. We show that ridge regression has qualitatively different behavior depending on the scale of the cluster mean vector and its interaction with the covariance matrix of the cluster distributions. In regimes where the scale is very large, the conditions that allow for benign overfitting turn out to be the same as those for the regression task. We additionally provide insights into how the introduction of label noise affects the behavior of the minimum norm interpolator (MNI). The optimal classifier in this setting is a linear transformation of the cluster mean vector and in the noiseless setting the MNI approximately learns this transformation. On the other hand, the introduction of label noise can significantly change the geometry of the solution while preserving the same qualitative behavior.

Problem

• Analyzes ridge regression in overparameterized binary classification.
• Explores effects of label noise on classification error.
• Examines conditions for benign overfitting in ridge regression.

Innovation

• Ridge regression in overparameterized binary classification
• Analysis of label-flipping noise impact
• Geometry of minimum norm interpolator solutions