  We establish statistical properties of random-weighting methods in LASSO
regression under different regularization parameters $\lambda_n$ and suitable
regularity conditions. The random-weighting methods in view concern repeated
optimization of a randomized objective function, motivated by the need for
computational approximations to Bayesian posterior sampling. In the context of
LASSO regression, we repeatedly assign analyst-drawn random weights to terms in
the objective function (including the penalty terms), and optimize to obtain a
sample of random-weighting estimators. We show that existing approaches have
conditional model selection consistency and conditional asymptotic normality at
different growth rates of $\lambda_n$ as $n \to \infty$. We propose an
extension to the available random-weighting methods and establish that the
resulting samples attain conditional sparse normality and conditional
consistency in a growing-dimension setting. We find that random-weighting has
both approximate-Bayesian and sampling-theory interpretations. Finally, we
illustrate the proposed methodology via extensive simulation studies and a
benchmark data example.
