Scaling and renormalization in high-dimensional regression

TL;DR

This paper offers a unified analytical framework for understanding high-dimensional ridge regression, revealing how statistical fluctuations and weight structures influence model performance and scaling laws.

Contribution

It introduces a deterministic equivalence approach using free probability to derive explicit formulas for errors and scaling behaviors in ridge regression and related models.

Findings

Statistical fluctuations can be absorbed into a renormalized ridge parameter.

Derived explicit formulas for training and generalization errors.

Identified a variance-limited scaling regime in random feature models.

Abstract

From benign overfitting in overparameterized models to rich power-law scalings in performance, simple ridge regression displays surprising behaviors sometimes thought to be limited to deep neural networks. This balance of phenomenological richness with analytical tractability makes ridge regression the model system of choice in high-dimensional machine learning. In this paper, we present a unifying perspective on recent results on ridge regression using the basic tools of random matrix theory and free probability, aimed at readers with backgrounds in physics and deep learning. We highlight the fact that statistical fluctuations in empirical covariance matrices can be absorbed into a renormalization of the ridge parameter. This `deterministic equivalence' allows us to obtain analytic formulas for the training and generalization errors in a few lines of algebra by leveraging the properties of the $S$-transform of free probability. From these precise asymptotics, we can easily identify sources of power-law scaling in model performance. In all models, the $S$-transform corresponds to the train-test generalization gap, and yields an analogue of the generalized-cross-validation estimator. Using these techniques, we derive fine-grained bias-variance decompositions for a very general class of random feature models with structured covariates. This allows us to discover a scaling regime for random feature models where the variance due to the features limits performance in the overparameterized setting. We also demonstrate how anisotropic weight structure in random feature models can limit performance and lead to nontrivial exponents for finite-width corrections in the overparameterized setting. Our results extend and provide a unifying perspective on earlier models of neural scaling laws.