  We develop a new method for selecting the penalty parameter for
$\ell_1$-penalized M-estimators in high dimensions, which we refer to as
bootstrapping after cross-validation. We derive rates of convergence for the
corresponding $\ell_1$-penalized M-estimator and also for the
post-$\ell_1$-penalized M-estimator, which refits the non-zero parameters of
the former estimator without penalty in the criterion function. We demonstrate
via simulations that our method is not dominated by cross-validation in terms
of estimation errors and outperforms cross-validation in terms of inference. As
an illustration, we revisit Fryer Jr (2019), who investigated racial
differences in police use of force, and confirm his findings.
