Sorted L-One Penalized Estimation (SLOPE) has shown the nice theoretical property as well as empirical behavior recently on the false discovery rate (FDR) control of high-dimensional feature selection by adaptively imposing the non-increasing sequence of tuning parameters on the sorted $\ell_1$ penalties. This paper goes beyond the previous concern limited to the FDR control by considering the stepdown-based SLOPE to control the probability of $k$ or more false rejections ($k$-FWER) and the false discovery proportion (FDP). Two new SLOPEs, called $k$-SLOPE and F-SLOPE, are proposed to realize $k$-FWER and FDP control respectively, where the stepdown procedure is injected into the SLOPE scheme. For the proposed stepdown SLOPEs, we establish their theoretical guarantees on controlling $k$-FWER and FDP under the orthogonal design setting, and also provide an intuitive guideline for the choice of regularization parameter sequence in much general setting. Empirical evaluations on simulated data validate the effectiveness of our approaches on controlled feature selection and support our theoretical findings.

翻译：排序L1惩罚估计（SLOPE）通过自适应地对排序后的$\ell_1$惩罚施加非递增调参序列，近期在高维特征选择的错误发现率（FDR）控制方面展现出良好的理论性质与实证表现。本文突破以往仅关注FDR控制的局限，提出了基于逐步回归的SLOPE方法，用于控制$k$次及以上错误拒绝的概率（$k$-FWER）以及错误发现比例（FDP）。为实现$k$-FWER和FDP控制，我们分别提出了两种新的SLOPE方法——$k$-SLOPE和F-SLOPE，其中将逐步回归过程融入SLOPE框架。针对所提出的逐步SLOPE方法，我们在正交设计条件下建立了其控制$k$-FWER和FDP的理论保证，并提供了在更一般情境下选择正则化参数序列的直观指导。模拟数据的实证评估验证了所提方法在受控特征选择中的有效性，并支持了我们的理论发现。