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THE DISTRIBUTIONALLY ROBUST PREDICTION ERROR OF THE √ LASSO AND RELATED ESTIMATORS
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DOI：10.1214/25-AOS2599.png)
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Abstract
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We study the classical problem of predicting an outcome variable, Y, using a linear combination of a d-dimensional covariate vector, X. We are interested in linear predictors whose coefficients solve inf(BeRd) (Ep(n)[Y-X-T beta (R)])(1/r) + delta rho(beta). where delta > 0 is a regularization parameter, p: R-d -> R is a convex penalty function, P is the empirical distribution of the data and r >= 1 Our main contribution is a new bound on the out-of-distribution prediction error of such estimators. The new bound is obtained by combining three new sets of results. First, we provide conditions under which linear predictors based on these estimators solve a distributionally robust optimization problem: they minimize the worst-case prediction error over distributions that are close to each other in a type of 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 present an oracle recommendation for the choice of regularization param-eter, delta, that guarantees good out-of-distribution prediction error.
Keywords:
istributional robust optimization
max-sliced Wasserstein
square-root LASSO
out-of-distribution prediction error
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