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This preprint is withdrawn due to an error in Lemma 4.2 regarding the replacement symmetrization proof for interpolating estimators, halting its contribution to current theoretical discourse on benign overfitting and spectral-transport stability. The withdrawal underscores the rigorous verification required before publishing novel mathematical frameworks that claim to resolve fundamental questions about generalization boundaries in highly overparameterized models.
Spectral-Transport Stability and Benign Overfitting in Interpolating Learning
arXiv:2604.08625v2 Announce Type: replace-cross 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.
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.
The authors are withdrawing this preprint due to a technical error discovered in the proof of Lemma 4.2 (Replacement Symmetrization) in Appendix B.2. Specifically, the expansion of the ghost-averaged estimator in the covariance operator's eigenbasis contains a dependency step that invalidates the finite-sample master bound in Theorem 4.3
Machine Learning (stat.ML); Machine Learning (cs.LG); Statistics Theory (math.ST)
arXiv:2604.08625 [stat.ML]
(or arXiv:2604.08625v2 [stat.ML] for this version)
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