  We propose a residual randomization procedure designed for robust Lasso-based
inference in the high-dimensional setting. Compared to earlier work that
focuses on sub-Gaussian errors, the proposed procedure is designed to work
robustly in settings that also include heavy-tailed covariates and errors.
Moreover, our procedure can be valid under clustered errors, which is important
in practice, but has been largely overlooked by earlier work. Through extensive
simulations, we illustrate our method's wider range of applicability as
suggested by theory. In particular, we show that our method outperforms
state-of-art methods in challenging, yet more realistic, settings where the
distribution of covariates is heavy-tailed or the sample size is small, while
it remains competitive in standard, "well behaved" settings previously studied
in the literature.
