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.

翻译：我们研究使用d维协变量向量X的线性组合预测结果变量Y的经典问题。我们关注系数通过求解以下问题的线性预测器：inf_{β∈R^d} (E_{P_n}[(Y-X^T β)^r])^{1/r} + δ ρ(β)，其中δ>0是正则化参数，ρ:R^d→R_+是凸惩罚函数，P_n是数据的经验分布，且r≥1。我们提出三组新结果。首先，我们给出条件，使基于这些估计量的线性预测器能够解决分布鲁棒优化问题：它们最小化在一种最大切片Wasserstein度量下彼此接近的分布上的最坏情况预测误差。其次，我们对分析最坏情况预测误差的分布球的统计性质进行了详细的有限样本和渐近分析。第三，利用分布鲁棒最优性和我们的统计分析，我们提出：i) 一种关于正则化参数δ的Oracle推荐，确保良好的样本外预测误差；ii) 一种用于排序两个不同线性估计量样本外性能的检验统计量。我们的结果均不依赖于真实数据生成过程的稀疏性假设；因此，它们拓宽了平方根LASSO及相关估计量在预测问题中的适用范围。