Aggregated hold out for sparse linear regression with a robust loss function

Guillaume Maillard (1, 2, 3)

1 Uni.lu - Université du Luxembourg
2 Université Paris-Saclay
3 CELESTE - Statistique mathématique et apprentissage

Sparse linear regression methods generally have a free hyperparameter which controls the amount of sparsity, and is subject to a bias-variance tradeoff. This article considers the use of Aggregated hold-out to aggregate over values of this hyperparameter, in the context of linear regression with the Huber loss function. Aggregated hold-out (Agghoo) is a procedure which averages estimators selected by hold-out (cross-validation with a single split). In the theoretical part of the article, it is proved that Agghoo satisfies a non-asymptotic oracle inequality when it is applied to sparse estimators which are parametrized by their zero-norm. In particular , this includes a variant of the Lasso introduced by Zou, Hastié and Tibshirani. Simulations are used to compare Agghoo with cross-validation. They show that Agghoo performs better than CV when the intrinsic dimension is high and when there are confounders correlated with the predictive covariates.

Mots clés: Hyperparameter selection; Sparse regression; Cross-validation; Robust regression; Lasso; Aggregation; Model selection

Domaines: Statistiques [math.ST]

Guillaume Maillard. Aggregated hold out for sparse linear regression with a robust loss function. Electronic Journal of Statistics , 2022, 16 (1), pp.935 -- 997. ⟨10.1214/21-EJS1952⟩. ⟨hal-02485694v3⟩