# Efficient Designs of SLOPE Penalty Sequences in Finite Dimension

arXiv: https://arxiv.org/abs/2102.07211 · HTML: https://mlsanity.com/paper/2102.07211/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 27 references that are in the map.
- First posted on arXiv 14 Feb 2021.

## Builds on (most cited first, 6 of 6)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [The Lasso with general Gaussian designs with applications to hypothesis testing](https://mlsanity.com/paper/2007.13716/) (2020)
- [SLOPE - Adaptive variable selection via convex optimization](https://mlsanity.com/paper/1407.3824/) (2014)
- [Algorithmic Analysis and Statistical Estimation of SLOPE via Approximate Message Passing](https://mlsanity.com/paper/1907.07502/) (2019)
- [Does SLOPE outperform bridge regression?](https://mlsanity.com/paper/1909.09345/) (2019)
- [Consistent Parameter Estimation for LASSO and Approximate Message Passing](https://mlsanity.com/paper/1511.01017/) (2015)

## Built on by (most cited first, 3 of 3)

- [Sparse Neural Additive Model: Interpretable Deep Learning with Feature Selection via Group Sparsity](https://mlsanity.com/paper/2202.12482/) (2022)
- [Characterizing the SLOPE Trade-off: A Variational Perspective and the Donoho-Tanner Limit](https://mlsanity.com/paper/2105.13302/) (2021)
- [Sharp Trade-Offs in High-Dimensional Inference via 2-Level SLOPE](https://mlsanity.com/paper/2507.09110/) (2025)

## Abstract (arXiv)

In linear regression, SLOPE is a new convex analysis method that generalizes the Lasso via the sorted L1 penalty: larger fitted coefficients are penalized more heavily. This magnitude-dependent regularization requires an input of penalty sequence λ, instead of a scalar penalty as in the Lasso case, thus making the design extremely expensive in computation. In this paper, we propose two efficient algorithms to design the possibly high-dimensional SLOPE penalty, in order to minimize the mean squared error. For Gaussian data matrices, we propose a first order Projected Gradient Descent (PGD) under the Approximate Message Passing regime. For general data matrices, we present a zero-th order Coordinate Descent (CD) to design a sub-class of SLOPE, referred to as the k-level SLOPE. Our CD allows a useful trade-off between the accuracy and the computation speed. We demonstrate the performance of SLOPE with our designs via extensive experiments on synthetic data and real-world datasets.