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We study the classical problem of predicting an outcome variable, $Y$, usinga linear combination of a $d$-dimensional covariate vector, $\mathbf{X}$. Weare 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 aregularization parameter, $\rho:\mathbb{R}^d\to \mathbb{R}_+$ is a convexpenalty 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 conditionsunder which linear predictors based on these estimators % solve a\emph{distributionally robust optimization} problem: they minimize theworst-case prediction error over distributions that are close to each other ina type of \emph{max-sliced Wasserstein metric}. Second, we provide a detailedfinite-sample and asymptotic analysis of the statistical properties of theballs of distributions over which the worst-case prediction error is analyzed.Third, we use the distributionally robust optimality and our statisticalanalysis to present i) an oracle recommendation for the choice ofregularization parameter, $\delta$, that guarantees good out-of-sampleprediction error; and ii) a test-statistic to rank the out-of-sampleperformance of two different linear estimators. None of our results rely onsparsity assumptions about the true data generating process; thus, they broadenthe scope of use of the square-root lasso and related estimators in predictionproblems.

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