Information criteria bias correction for group selection

The main contribution of this paper lies in the extension towards group lasso of a Mallows’ Cp-like information criterion used in finetuning the lasso selection in a high-dimensional, sparse regression model. The optimisation of an information criterion paired with an $$\ell _1$$ ℓ 1 -norm regularisation method of the lasso leads to an overestimation of the model size. This is because the shrinkage following from the $$\ell _1$$ ℓ 1 regularisation is too permissive towards false positives, since shrinkage reduces the effects of false positives. The problem does not arise with $$\ell _0$$ ℓ 0 -norm regularisation but this is a combinatorial problem, which is computationally unfeasible in the high-dimensional setting. The strategy adopted in this paper is to select the non-zero variables with $$\ell _1$$ ℓ 1 method and estimate their values with the $$\ell _0$$ ℓ 0 , meaning that lasso is used for selection, followed by an orthogonal projection, i.e., debiasing after selection. This approach necessitates the information criterion to be adapted, in particular, by including what is called a “mirror correction”, leading to smaller models. A second contribution of the paper is situated at the methodological level, more precisely in the development of the corrected information criterion using random hard thresholds as a model for the selection process.