Monte Carlo Simulation for Lasso-Type Problems by Estimator Augmentation
Regularized linear regression under the ℓ1 penalty, such as the Lasso, has been shown to be effective in variable selection and sparse modeling. The sampling distribution of an ℓ1-penalized estimator is hard to determine as the estimator is defined by an optimization problem that in general can only be solved numerically and many of its components may be exactly zero. Let S be the subgradient of the ℓ1 norm of the coefficient vector β evaluated at . We find that the joint sampling distribution of and S, together called an augmented estimator, is much more tractable and has a closed-form density under a normal error distribution in both low-dimensional (p ⩽ n) and high-dimensional (p > n) settings. Given β and the error variance σ2, one may employ standard Monte Carlo methods, such as Markov chain Monte Carlo (MCMC) and importance sampling (IS), to draw samples from the distribution of the augmented estimator and calculate expectations with respect to the sampling distribution of . We develop a few concrete Monte Carlo algorithms and demonstrate with numerical examples that our approach may offer huge advantages and great flexibility in studying sampling distributions in ℓ1-penalized linear regression. We also establish nonasymptotic bounds on the difference between the true sampling distribution of and its estimator obtained by plugging in estimated parameters, which justifies the validity of Monte Carlo simulation from an estimated sampling distribution even when p ≫ n → ∞.