The distribution of the Lasso: Uniform control over sparse balls and adaptive parameter tuning

Abstract

The Lasso is a popular regression method for high-dimensional problems in which the number of parameters ${\mathit{\theta }}_1,\dots ,{\mathit{\theta }}_{\mathit{N}}$, is larger than the number n of samples: $\mathit{N}>\mathit{n}$. A useful heuristics relates the statistical properties of the Lasso estimator to that of a simple soft-thresholding denoiser, in a denoising problem in which the parameters ${({\mathit{\theta }}_{\mathit{i}})}_{\mathit{i}\le \mathit{N}}$ are observed in Gaussian noise, with a carefully tuned variance. Earlier work confirmed this picture in the limit $\mathit{n},\mathit{N}\to \infty$, pointwise in the parameters θ and in the value of the regularization parameter.

Here, we consider a standard random design model and prove exponential concentration of its empirical distribution around the prediction provided by the Gaussian denoising model. Crucially, our results are uniform with respect to θ belonging to ${\ell }_{\mathit{q}}$ balls, $\mathit{q}\in [0,1]$, and with respect to the regularization parameter. This allows us to derive sharp results for the performances of various data-driven procedures to tune the regularization.

Our proofs make use of Gaussian comparison inequalities, and in particular of a version of Gordon’s minimax theorem developed by Thrampoulidis, Oymak and Hassibi, which controls the optimum value of the Lasso optimization problem. Crucially, we prove a stability property of the minimizer in Wasserstein distance that allows one to characterize properties of the minimizer itself.