The out-of-sample prediction error of the square-root-LASSO and related estimators

TL;DR

This paper analyzes the out-of-sample prediction error of the square-root LASSO and similar estimators, providing new theoretical insights, distributionally robust interpretations, and practical guidelines for regularization and model comparison.

Contribution

It introduces conditions linking these estimators to distributionally robust optimization, offers finite-sample and asymptotic analysis, and proposes methods for regularization tuning and estimator ranking without sparsity assumptions.

Findings

Linear predictors minimize worst-case prediction error over Wasserstein-like distributional neighborhoods.

Provides finite-sample and asymptotic bounds for distributionally robust prediction error.

Offers practical procedures for regularization parameter selection and estimator comparison.

Abstract

We study the classical problem of predicting an outcome variable, $Y$, using a linear combination of a $d$-dimensional covariate vector, $\mathbf{X}$. We are interested in linear predictors whose coefficients solve: % \begin{align*} \inf_{\boldsymbol{\beta} \in \mathbb{R}^d} \left( \mathbb{E}_{\mathbb{P}_n} \left[ \left(Y-\mathbf{X}^{\top}\beta \right)^r \right] \right)^{1/r} +\delta \, \rho\left(\boldsymbol{\beta}\right), \end{align*} where $\delta>0$ is a regularization parameter, $\rho:\mathbb{R}^d\to \mathbb{R}_+$ is a convex penalty function, $\mathbb{P}_n$ is the empirical distribution of the data, and $r\geq 1$. We present three sets of new results. First, we provide conditions under which linear predictors based on these estimators % solve a \emph{distributionally robust optimization} problem: they minimize the worst-case prediction error over distributions that are close to each other in a type of \emph{max-sliced Wasserstein metric}. Second, we provide a detailed finite-sample and asymptotic analysis of the statistical properties of the balls of distributions over which the worst-case prediction error is analyzed. Third, we use the distributionally robust optimality and our statistical analysis to present i) an oracle recommendation for the choice of regularization parameter, $\delta$, that guarantees good out-of-sample prediction error; and ii) a test-statistic to rank the out-of-sample performance of two different linear estimators. None of our results rely on sparsity assumptions about the true data generating process; thus, they broaden the scope of use of the square-root lasso and related estimators in prediction problems.