Beyond False Discovery Rate: A Stepdown Group SLOPE Approach for Grouped Variable Selection

TL;DR

This paper introduces the Group Stepdown SLOPE, a new method for high-dimensional grouped variable selection that controls error rates like k-FWER and FDP, with proven guarantees and improved power.

Contribution

It develops the Group Stepdown SLOPE framework embedding stepdown rules into SLOPE, extending error control to grouped variables and non-orthogonal designs with scalable algorithms.

Findings

Achieves finite-sample error control for k-FWER and FDP.

Demonstrates higher power than existing procedures in simulations.

Provides scalable algorithms for complex design matrices.

Abstract

High-dimensional feature selection is routinely required to balance statistical power with strict control of multiple-error metrics such as the k-Family-Wise Error Rate (k-FWER) and the False Discovery Proportion (FDP), yet some existing frameworks are confined to the narrower goal of controlling the expected False Discovery Rate (FDR) and can not exploit the group-structure of the covariates, such as Sorted L-One Penalized Estimation (SLOPE). We introduce the Group Stepdown SLOPE, a unified optimization procedure which is capable of embedding the Lehmann-Romano stepdown rules into SLOPE to achieve finite-sample guarantees under k-FWER and FDP thresholds. Specifically, we derive closed-form regularization sequences under orthogonal designs that provably bound k-FWER and FDP at user-specified levels, and extend these results to grouped settings via gk-SLOPE and gF-SLOPE, which control the analogous group-level errors gk-FWER and gFDP. For non-orthogonal general designs, we provide a calibrated data-driven sequence inspired by Gaussian approximation and Monte-Carlo correction, preserving convexity and scalability. Extensive simulations are conducted across sparse, correlated, and group-structured regimes. Empirical results corroborate our theoretical findings that the proposed methods achieve nominal error control, while yielding markedly higher power than competing stepdown procedures, thereby confirming the practical value of the theoretical advances.