# The Strong Screening Rule for SLOPE

SLOPE
Screening Rules
Authors

Małgorzata Bogdan

Jonas Wallin

Published

6 December 2020

Details

Advances in Neural Information Processing Systems 33, Virtual

Abstract

Extracting relevant features from data sets where the number of observations (n) is much smaller then the number of predictors (p) is a major challenge in modern statistics. Sorted L-One Penalized Estimation (SLOPE)—a generalization of the lasso—is a promising method within this setting. Current numerical procedures for SLOPE, however, lack the efficiency that respective tools for the lasso enjoy, particularly in the context of estimating a complete regularization path. A key component in the efficiency of the lasso is predictor screening rules: rules that allow predictors to be discarded before estimating the model. This is the first paper to establish such a rule for SLOPE. We develop a screening rule for SLOPE by examining its subdifferential and show that this rule is a generalization of the strong rule for the lasso. Our rule is heuristic, which means that it may discard predictors erroneously. In our paper, however, we show that such situations are rare and easily safeguarded against by a simple check of the optimality conditions. Our numerical experiments show that the rule performs well in practice, leading to improvements by orders of magnitude for data in the (p \gg n) domain, as well as incurring no additional computational overhead when (n > p).

## Citation

BibTeX citation:
@inproceedings{larsson2020,
author = {Johan Larsson and Małgorzata Bogdan and Jonas Wallin},
editor = {Hugo Larochelle and Marc’Aurelio Ranzato and Raia Hadsell
and Maria-Florina Balcan and Hsuan-Tien Lin},
publisher = {Curran Associates, Inc.},
title = {The Strong Screening Rule for SLOPE},
booktitle = {Advances in Neural Information Processing Systems 33},
volume = {33},
pages = {14592–14603},
date = {2020-12-06},
address = {Red Hook, NY, USA},
url = {https://proceedings.neurips.cc/paper/2020/hash/a7d8ae4569120b5bec12e7b6e9648b86-Abstract.html},
langid = {English},
abstract = {Extracting relevant features from data sets where the
number of observations (\$n\$) is much smaller then the number of
predictors (\$p\$) is a major challenge in modern statistics. Sorted
L-One Penalized Estimation (SLOPE)—a generalization of the lasso—is
a promising method within this setting. Current numerical procedures
for SLOPE, however, lack the efficiency that respective tools for
the lasso enjoy, particularly in the context of estimating a
complete regularization path. A key component in the efficiency of
the lasso is predictor screening rules: rules that allow predictors
to be discarded before estimating the model. This is the first paper
to establish such a rule for SLOPE. We develop a screening rule for
SLOPE by examining its subdifferential and show that this rule is a
generalization of the strong rule for the lasso. Our rule is
heuristic, which means that it may discard predictors erroneously.
In our paper, however, we show that such situations are rare and
easily safeguarded against by a simple check of the optimality
conditions. Our numerical experiments show that the rule performs
well in practice, leading to improvements by orders of magnitude for
data in the (\$p \textbackslash gg n\$) domain, as well as incurring
no additional computational overhead when (\$n \textgreater{} p\$).}
}