Spectral-Transport Stability and Benign Overfitting in Interpolating Learning

TL;DR

This paper introduces a spectral-transport stability framework to understand why overparameterized models can generalize well despite interpolating data, linking stability, spectral geometry, and noise structure.

Contribution

It develops a unified theoretical framework connecting stability, spectral geometry, and noise to characterize benign overfitting and derive risk bounds for interpolating estimators.

Findings

Finite-sample risk bounds established.

Benign-overfitting criterion via spectral-transport index.

Explicit phase-transition rates under polynomial spectral decay.

Abstract

We develop a theoretical framework for generalization in the interpolating regime of statistical learning. The central question is why highly overparameterized estimators can attain zero empirical risk while still achieving nontrivial predictive accuracy, and how to characterize the boundary between benign and destructive overfitting. We introduce a spectral-transport stability framework in which excess risk is controlled jointly by the spectral geometry of the data distribution, the sensitivity of the learning rule under single-sample replacement, and the alignment structure of label noise. This leads to a scale-dependent Fredriksson index that combines effective dimension, transport stability, and noise alignment into a single complexity parameter for interpolating estimators. We prove finite-sample risk bounds, establish a sharp benign-overfitting criterion through the vanishing of the index along admissible spectral scales, and derive explicit phase-transition rates under polynomial spectral decay. For a model-specific specialization, we obtain an explicit theorem for polynomial-spectrum linear interpolation, together with a proof of the resulting rate. The framework also clarifies implicit regularization by showing how optimization dynamics can select interpolating solutions of minimal spectral-transport energy. These results connect algorithmic stability, double descent, benign overfitting, operator-theoretic learning theory, and implicit bias within a unified structural account of modern interpolation.