High-Dimensional Asymptotics of Prediction: Ridge Regression and Classification

2015/07/10 by Edgar Dobriban, Stefan Wager, Dobriban, Edgar +1 · 30 citations
Mathematics · Physics and Astronomy · #FOS: Computer and information sciences #FOS: Mathematics #Machine Learning (stat.ML) #Markov Chains and Monte Carlo Methods #Random Matrices and Applications #Statistics Theory (math.ST) #Theoretical and Computational Physics #math.ST #stat.ML #stat.TH

paper · pdf · doi:10.48550/arxiv.1507.03003

Added a section on prediction versus estimation for ridge regression. Rewrote introduction. Other results unchanged

openalex publication_date 2015/07/10 · arxiv created 2015/11/04 · arxiv updated 2015/11/05 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28

Abstract

We provide a unified analysis of the predictive risk of ridge regression and regularized discriminant analysis in a dense random effects model. We work in a high-dimensional asymptotic regime where p, n → ∞ and p/n → γ∈ (0, ∞), and allow for arbitrary covariance among the features. For both methods, we provide an explicit and efficiently computable expression for the limiting predictive risk, which depends only on the spectrum of the feature-covariance matrix, the signal strength, and the aspect ratio γ. Especially in the case of regularized discriminant analysis, we find that predictive accuracy has a nuanced dependence on the eigenvalue distribution of the covariance matrix, suggesting that analyses based on the operator norm of the covariance matrix may not be sharp. Our results also uncover several qualitative insights about both methods: for example, with ridge regression, there is an exact inverse relation between the limiting predictive risk and the limiting estimation risk given a fixed signal strength. Our analysis builds on recent advances in random matrix theory.

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