Cross validation in LASSO and its acceleration

TL;DR

This paper develops efficient formulas for leave-one-out cross validation in LASSO using message passing and replica methods, enabling faster penalty parameter selection and demonstrating their accuracy through simulations and real data application.

Contribution

It introduces simple, computationally efficient formulas for CV errors in LASSO based on message passing and replica analysis, linking CV errors to residual sums of squares.

Findings

Formulas accurately estimate CV errors with reduced computational cost.

Analytical evaluation of CV errors for large random matrices confirms the formulas.

Application to supernovae data demonstrates practical utility.

Abstract

We investigate leave-one-out cross validation (CV) as a determinator of the weight of the penalty term in the least absolute shrinkage and selection operator (LASSO). First, on the basis of the message passing algorithm and a perturbative discussion assuming that the number of observations is sufficiently large, we provide simple formulas for approximately assessing two types of CV errors, which enable us to significantly reduce the necessary cost of computation. These formulas also provide a simple connection of the CV errors to the residual sums of squares between the reconstructed and the given measurements. Second, on the basis of this finding, we analytically evaluate the CV errors when the design matrix is given as a simple random matrix in the large size limit by using the replica method. Finally, these results are compared with those of numerical simulations on finite-size systems and are confirmed to be correct. We also apply the simple formulas of the first type of CV error to an actual dataset of the supernovae.