A Classical View on Benign Overfitting: The Role of Sample Size

2025-05-16

AI Summary

This paper investigates the phenomenon of “nearly benign overfitting,” wherein models achieve near-Bayes-optimal generalization despite arbitrarily small (yet nonzero) training error. Method: Within the classical statistical learning framework—where sample size and model complexity coevolve—the authors analyze gradient flow dynamics for two-layer ReLU networks without assuming specific forms for the regression function or noise distribution (only boundedness is required). They introduce a novel analytical paradigm combining excess risk decomposition with implicit regularization analysis to circumvent limitations of uniform convergence. Their approach integrates kernel ridge regression modeling, dynamical systems analysis, and error decoupling techniques. Contribution/Results: The work establishes universal generalization upper bounds for both kernel methods and shallow neural networks. It demonstrates that sufficiently large models, when appropriately scaled to the sample size, simultaneously attain low training error and near-optimal test performance—thereby challenging the conventional bias–variance trade-off paradigm.

Technology Category

• Machine Learning: Learning Theory
• Natural Language Processing: Learning & Optimization for NLP
• Search and Optimization: Learning to Search

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for 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

Benign overfitting is a phenomenon in machine learning where a model perfectly fits (interpolates) the training data, including noisy examples, yet still generalizes well to unseen data. Understanding this phenomenon has attracted considerable attention in recent years. In this work, we introduce a conceptual shift, by focusing on almost benign overfitting, where models simultaneously achieve both arbitrarily small training and test errors. This behavior is characteristic of neural networks, which often achieve low (but non-zero) training error while still generalizing well. We hypothesize that this almost benign overfitting can emerge even in classical regimes, by analyzing how the interaction between sample size and model complexity enables larger models to achieve both good training fit but still approach Bayes-optimal generalization. We substantiate this hypothesis with theoretical evidence from two case studies: (i) kernel ridge regression, and (ii) least-squares regression using a two-layer fully connected ReLU neural network trained via gradient flow. In both cases, we overcome the strong assumptions often required in prior work on benign overfitting. Our results on neural networks also provide the first generalization result in this setting that does not rely on any assumptions about the underlying regression function or noise, beyond boundedness. Our analysis introduces a novel proof technique based on decomposing the excess risk into estimation and approximation errors, interpreting gradient flow as an implicit regularizer, that helps avoid uniform convergence traps. This analysis idea could be of independent interest.

Problem

• Explores almost benign overfitting in classical regimes
• Analyzes sample size and model complexity interaction
• Provides generalization results without strong assumptions

Innovation

• Analyzing almost benign overfitting in classical regimes
• Using kernel ridge regression and ReLU networks
• Novel proof technique decomposing excess risk