The distribution of the Lasso: Uniform control over sparse balls and adaptive parameter tuning
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The Lasso is a popular regression method for high-dimensional problems in which the number of parameters θ_1,...,θ_N, 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 (θ_i)_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 ℓ_q balls, q∈ [0,1], and with respect to the regularization parameter. This allows 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 to characterize properties of the minimizer itself.
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