Abstract

A recently proposed SLOPE estimator [6] has been shown to adaptively achieve the minimax $\ell_{2}$ estimation rate under high-dimensional sparse linear regression models [25]. 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. Concentration inequality, LASSO, mean square error, noise sensitivity, Ridge, SLOPE 2000 Math Subject C