High-Dimensional Asymptotics of Prediction: Ridge Regression and Classification (1507.03003v2)

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 $p, n \to \infty$ and $p/n \to \gamma \in (0, \, \infty)$, 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 $\gamma$. 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.

• The paper derives explicit formulas for the limiting predictive risk in ridge regression and RDA using high-dimensional asymptotic analysis.
• It reveals a phase transition in predictive accuracy at γ = 1 for ridge regression and highlights the role of covariance eigenvalue distribution in model performance.
• The study offers practical insights for tuning regularization parameters in high-dimensional settings, enhancing predictions across applications like genetics and image classification.

High-Dimensional Asymptotics of Prediction: Ridge Regression and Classification

The paper by Dobriban and Wager explores the high-dimensional predictive performance of two classical statistical methods—ridge regression and regularized discriminant analysis (RDA). These methods are analyzed within a high-dimensional asymptotic framework where the number of variables $p$ and the sample size $n$ tend towards infinity, with their ratio $\gamma = p/n$ converging to a constant. This setting is crucial to understanding the performance of these methods in modern applications where the dimensionality often rivals or surpasses the sample size.

Key Contributions

This research provides explicit, efficiently computable formulas for the predictive risk associated with both ridge regression and RDA under a dense random effects model. The results suggest a nuanced dependence of the predictive accuracy on the eigenvalue distribution of the feature covariance matrix, which deviates from simpler characterizations based on the operator norm. The key contributions include:

1. Ridge Regression: The paper yields a formula for the limiting predictive risk, demonstrating an inverse relation between limiting predictive and estimation risk under fixed signal strength. This inference is substantiated through advanced random matrix theory, offering a refined understanding compared to traditional operator norm analyses. The work underlines a phase transition in predictive accuracy that occurs at $\gamma = 1$ for high signal-to-noise ratios, signifying the difficulty of prediction and estimation.
2. Regularized Discriminant Analysis (RDA): The analysis shows that predictive performance in RDA can be better described through the Stieltjes transform of the empirical spectral distribution of the sample covariance matrix. Notably, the work highlights how correlated features influence RDA's efficacy, indicating classical condition number analyses might lack precision.

Theoretical Insights

The formulas derived enable researchers to gain qualitative insights into the performance of ridge regression and RDA under various covariance structures:

• For ridge regression, the risk analysis reveals that when signaled, i.e., $\alpha^2 \gg 1$, the transition at $\gamma = 1$ plays a pivotal role, marking a switch in complexity regimes.
• RDA's performance significantly hinges on its regularization parameter $\lambda$, with the research explicating its large-signal limiting behavior dependent on the spectral properties of the covariance matrix $\Sigma$.

Practical Implications and Future Directions

The findings have substantial implications for high-dimensional predictions across various domains—ranging from genetics to image classification—where these analytic tools could guide model selection and regularization parameter tuning to enhance predictive performance. Practitioners can exploit these results for a principled approach to tuning ridge regression and RDA under specific data distributions and sample sizes.

Future research might be oriented towards extending these high-dimensional asymptotic results to broader classes of models and more diverse data distributions. Another promising direction could involve exploring non-Gaussian settings and assessing robustness against data anomalies or misspecifications, which are prevalent in real-world scenarios.

In conclusion, Dobriban and Wager's paper significantly advances the understanding of ridge regression and RDA in high-dimensional contexts, offering both theoretical and practical tools for predictive model optimization in an array of scientific and engineering applications.