Does SLOPE outperform bridge regression?

TL;DR

This paper analyzes the performance of the SLOPE estimator in high-dimensional sparse linear regression, revealing its limitations and optimality conditions compared to LASSO and bridge regression methods.

Contribution

It characterizes SLOPE's estimation error in a new regime where sparsity and sample size scale linearly with dimension, and compares its performance to LASSO and bridge regression.

Findings

LASSO is optimal for low noise, non-tied sparse signals.

SLOPE estimators are sub-optimal in high noise scenarios.

Provides a concentration inequality for SLOPE's mean square error.

Abstract

A recently proposed SLOPE estimator (arXiv:1407.3824) has been shown to adaptively achieve the minimax $\ell_2$ estimation rate under high-dimensional sparse linear regression models (arXiv:1503.08393). Such minimax optimality holds in the regime where the sparsity level $k$, sample size $n$, and dimension $p$ satisfy $k/p \rightarrow 0$, $k\log p/n \rightarrow 0$. In this paper, we characterize the estimation error of SLOPE under the complementary regime where both $k$ and $n$ scale linearly with $p$, and provide new insights into the performance of SLOPE estimators. We first derive a concentration inequality for the finite sample mean square error (MSE) of SLOPE. The quantity that MSE concentrates around takes a complicated and implicit form. With delicate analysis of the quantity, we prove that among all SLOPE estimators, LASSO is optimal for estimating $k$-sparse parameter vectors that do not have tied non-zero components in the low noise scenario. On the other hand, in the large noise scenario, the family of SLOPE estimators are sub-optimal compared with bridge regression such as the Ridge estimator.