We develop two new methods for selecting the penalty parameter for the $\ell^1$-penalized high-dimensional M-estimator, which we refer to as the analytic and bootstrap-after-cross-validation methods. For both methods, we derive nonasymptotic error bounds for the corresponding $\ell^1$-penalized M-estimator and show that the bounds converge to zero under mild conditions, thus providing a theoretical justification for these methods. We demonstrate via simulations that the finite-sample performance of our methods is much better than that of previously available and theoretically justified methods.

翻译：我们开发了两种新的方法来选择 $\ ell\ $1$ 被惩罚的高维M 估计值的处罚参数,我们称之为分析和靴套件交叉校验方法。对于这两种方法,我们得出相应的 $\ $1$ 美元 被惩罚的 M 估计值的非预防性误差界限,并显示在温和条件下,界限会趋近为零,从而为这些方法提供了理论上的理由。我们通过模拟来证明,我们方法的有限性能比以前可用和理论上合理的方法要好得多。