Loop Corrections to the Training Error and Generalization Gap of Random Feature Models

TL;DR

This paper develops a theoretical framework to analyze how finite-width effects, modeled as loop corrections, influence the training and test errors of random feature models beyond the mean kernel approximation.

Contribution

It introduces a field-theoretic approach to derive loop corrections for errors in random feature models, capturing finite-width effects and their scaling laws.

Findings

Derived explicit formulas for loop corrections to errors.

Validated theoretical predictions with experimental results.

Showed finite-width effects significantly impact generalization performance.

Abstract

We investigate random feature models in which neural networks sampled from a prescribed initialization ensemble are frozen and used as random features, with only the readout weights optimized. Adopting a statistical-physics viewpoint, we study the training error, test error, and generalization gap beyond the mean kernel approximation. Since the predictor is a nonlinear functional of the induced random kernel, the ensemble-averaged errors depend not only on the mean kernel but also on higher-order fluctuation statistics. Within an effective field-theoretic framework, these finite-width contributions naturally appear as loop corrections. We derive loop corrections to the training error, test error, and generalization gap, obtain their scaling laws, and support the theory with experimental verification.