High-Dimensional Asymptotics of Prediction: Ridge Regression and\n Classification

2015/07/10 by Edgar Dobriban, Stefan Wager, Dobriban, Edgar +1 · 18 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

paper · pdf · doi:10.48550/arxiv.1507.03003

openalex publication_date 2015/07/10 · 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\nregularized discriminant analysis in a dense random effects model. We work in a\nhigh-dimensional asymptotic regime where p, n \→ \∞ and p/n \→ \γ\n\∈ (0, , \∞), and allow for arbitrary covariance among the features. For\nboth methods, we provide an explicit and efficiently computable expression for\nthe limiting predictive risk, which depends only on the spectrum of the\nfeature-covariance matrix, the signal strength, and the aspect ratio \γ.\nEspecially in the case of regularized discriminant analysis, we find that\npredictive accuracy has a nuanced dependence on the eigenvalue distribution of\nthe covariance matrix, suggesting that analyses based on the operator norm of\nthe covariance matrix may not be sharp. Our results also uncover several\nqualitative insights about both methods: for example, with ridge regression,\nthere is an exact inverse relation between the limiting predictive risk and the\nlimiting estimation risk given a fixed signal strength. Our analysis builds on\nrecent advances in random matrix theory.\n

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