# Does SLOPE outperform bridge regression?

arXiv: https://arxiv.org/abs/1909.09345 · HTML: https://mlsanity.com/paper/1909.09345/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 Sep 2019.

## Built on by (most cited first, 8 of 8)

- [Label-Imbalanced and Group-Sensitive Classification under Overparameterization](https://mlsanity.com/paper/2103.01550/) (2021)
- [Sharp global convergence guarantees for iterative nonconvex optimization: A Gaussian process perspective](https://mlsanity.com/paper/2109.09859/) (2021)
- [Certified Data Removal Under High-dimensional Settings](https://mlsanity.com/paper/2505.07640/) (2025)
- [Theoretical Analysis of Leave-one-out Cross Validation for Non-differentiable Penalties under High-dimensional Settings](https://mlsanity.com/paper/2402.08543/) (2024)
- [Characterizing the SLOPE Trade-off: A Variational Perspective and the Donoho-Tanner Limit](https://mlsanity.com/paper/2105.13302/) (2021)
- [Efficient Designs of SLOPE Penalty Sequences in Finite Dimension](https://mlsanity.com/paper/2102.07211/) (2021)
- [Optimal Implicit Bias in Linear Regression](https://mlsanity.com/paper/2506.17187/) (2025)
- [Sharp Trade-Offs in High-Dimensional Inference via 2-Level SLOPE](https://mlsanity.com/paper/2507.09110/) (2025)

## Abstract (arXiv)

A recently proposed SLOPE estimator (arXiv:1407.3824) has been shown to adaptively achieve the minimax ℓ2 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, klog 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.