On the Asymptotic Properties of SLOPE

Abstract

AbstractSorted L-One Penalized Estimator (SLOPE) is a relatively new convex optimization procedure for selecting predictors in high dimensional regression analyses. SLOPE extends LASSO by replacing theL1penalty norm with a SortedL1norm, based on the non-increasing sequence of tuning parameters. This allows SLOPE to adapt to unknown sparsity and achieve an asymptotic minimax convergency rate under a wide range of high dimensional generalized linear models. Additionally, in the case when the design matrix is orthogonal, SLOPE with the sequence of tuning parametersλBHcorresponding to the sequence of decaying thresholds for the Benjamini-Hochberg multiple testing correction provably controls the False Discovery Rate (FDR) in the multiple regression model. In this article we provide new asymptotic results on the properties of SLOPE when the elements of the design matrix are iid random variables from the Gaussian distribution. Specifically, we provide conditions under which the asymptotic FDR of SLOPE based on the sequenceλBHconverges to zero and the power converges to 1. We illustrate our theoretical asymptotic results with an extensive simulation study. We also provide precise formulas describing FDR of SLOPE under different loss functions, which sets the stage for future investigation on the model selection properties of SLOPE and its extensions.