  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.
