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:L-One 惩罚性估计(SLOPE) 已经分类了 L- One 惩罚性估计(SLOPE), 显示了最近对高维特征选择的虚假发现率(FDR) 控制方面良好的理论属性和经验行为, 通过对分类的 $\ ell_ 1美元 罚款 施以不增加的调制参数序列, 对高维特征选择进行了适应性调整 。 本文超越了此前对FDR 控制的关注范围, 通过考虑以逐步降低为基础的 SLOPE 控制 控制 $k$- FWER 或更多假拒绝的概率 (k$-FWER) 和假发现比例 (FDP ) 。 提出了两个新的 SLOPE, 名为 $k$- SLOPE 和 F- SLOPE, 分别实现 $k$-FWER 和 FDP 控制 。 在 SLOPE 计划 中注入了逐步递减程序 。 对于拟议的逐步削减 SLOPE, 我们为控制特性选择方法的有效性建立了理论保证, 我们为控制性选择了控制性参数选择 提供了直观的理论支持。