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

The Lasso is a popular regression method for high-dimensional problems in which the number of parameters θ1,…,θN, is larger than the number n of samples: N>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 (θi)i≤N are observed in Gaussian noise, with a carefully tuned variance. Earlier work confirmed this picture in the limit n,N→∞, 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 ℓq balls, q∈[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.