Prediction Errors for Penalized Regressions based on Generalized Approximate Message Passing

Abstract

We discuss the prediction accuracy of assumed statistical models in terms of prediction errors for the generalized linear model and penalized maximum likelihood methods. We derive the forms of estimators for the prediction errors, such as $C_p$ criterion, information criteria, and leave-one-out cross validation (LOOCV) error, using the generalized approximate message passing (GAMP) algorithm and replica method. These estimators coincide with each other when the number of model parameters is sufficiently small; however, there is a discrepancy between them in particular in the parameter region where the number of model parameters is larger than the data dimension. In this paper, we review the prediction errors and corresponding estimators, and discuss their differences. In the framework of GAMP, we show that the information criteria can be expressed by using the variance of the estimates. Further, we demonstrate how to approach LOOCV error from the information criteria by utilizing the expression provided by GAMP.

73 pages, 13 figures, accepted Journal of Physics A

References in corpus (7)

• First M87 Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole
• On the "degrees of freedom" of the lasso
• Coordinate descent algorithms for nonconvex penalized regression, with applications to biological feature selection
• Probabilistic Reconstruction in Compressed Sensing: Algorithms, Phase Diagrams, and Threshold Achieving Matrices
• Double Trouble in Double Descent : Bias and Variance(s) in the Lazy Regime
• Universality of empirical risk minimization
• Posterior covariance information criterion for general loss functions

Cited by in corpus (1)

• Replica Analysis for Ensemble Techniques in Variable Selection