On the asymptotic properties of SLOPE

TL;DR

This paper investigates the asymptotic behavior of SLOPE, a convex estimator for predictor selection, showing conditions under which it controls false discovery rate and achieves high power in high-dimensional Gaussian models.

Contribution

The paper provides new asymptotic results for SLOPE with Gaussian design matrices, including conditions for FDR control and power convergence, and offers formulas for FDR under various loss functions.

Findings

Asymptotic FDR of SLOPE with $ ext{lambda}^{BH}$ converges to zero.

Power of SLOPE converges to one under certain conditions.

Simulation studies support theoretical results.

Abstract

Sorted L-One Penalized Estimator (SLOPE) is a relatively new convex optimization procedure for selecting predictors in large data bases. Contrary to LASSO, SLOPE has been proved to be asymptotically minimax in the context of sparse high-dimensional generalized linear models. Additionally, in case when the design matrix is orthogonal, SLOPE with the sequence of tuning parameters $\lambda^{BH}$, corresponding to the sequence of decaying thresholds for the Benjamini-Hochberg multiple testing correction, provably controls False Discovery Rate 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 the conditions, under which the asymptotic FDR of SLOPE based on the sequence $\lambda^{BH}$ converges to zero and the power converges to 1. We illustrate our theoretical asymptotic results with extensive simulation study. We also provide precise formulas describing FDR of SLOPE under different loss functions, which sets the stage for future results on the model selection properties of SLOPE and its extensions.