# Characterizing the SLOPE Trade-off: A Variational Perspective and the Donoho-Tanner Limit

arXiv: https://arxiv.org/abs/2105.13302 · HTML: https://mlsanity.com/paper/2105.13302/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 53 references that are in the map.
- First posted on arXiv 27 May 2021.

## Builds on (most cited first, 10 of 10)

- [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)
- [False Discoveries Occur Early on the Lasso Path](https://mlsanity.com/paper/1511.01957/) (2015)
- [Algorithmic Analysis and Statistical Estimation of SLOPE via Approximate Message Passing](https://mlsanity.com/paper/1907.07502/) (2019)
- [Group SLOPE - adaptive selection of groups of predictors](https://mlsanity.com/paper/1610.04960/) (2016)
- [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)
- [When Is the First Spurious Variable Selected by Sequential Regression Procedures?](https://mlsanity.com/paper/1708.03046/) (2017)
- [Efficient Designs of SLOPE Penalty Sequences in Finite Dimension](https://mlsanity.com/paper/2102.07211/) (2021)
- [False Variable Selection Rates in Regression](https://mlsanity.com/paper/1302.2303/) (2013)

## Built on by (most cited first, 4 of 4)

- [Sparse Neural Additive Model: Interpretable Deep Learning with Feature Selection via Group Sparsity](https://mlsanity.com/paper/2202.12482/) (2022)
- [Spectral Estimators for Structured Generalized Linear Models via Approximate Message Passing](https://mlsanity.com/paper/2308.14507/) (2023)
- [Near-optimal multiple testing in Bayesian linear models with finite-sample FDR control](https://mlsanity.com/paper/2211.02778/) (2022)
- [Sharp Trade-Offs in High-Dimensional Inference via 2-Level SLOPE](https://mlsanity.com/paper/2507.09110/) (2025)

## Abstract (arXiv)

Sorted l1 regularization has been incorporated into many methods for solving high-dimensional statistical estimation problems, including the SLOPE estimator in linear regression. In this paper, we study how this relatively new regularization technique improves variable selection by characterizing the optimal SLOPE trade-off between the false discovery proportion (FDP) and true positive proportion (TPP) or, equivalently, between measures of type I error and power. Assuming a regime of linear sparsity and working under Gaussian random designs, we obtain an upper bound on the optimal trade-off for SLOPE, showing its capability of breaking the Donoho-Tanner power limit. To put it into perspective, this limit is the highest possible power that the Lasso, which is perhaps the most popular l1-based method, can achieve even with arbitrarily strong effect sizes. Next, we derive a tight lower bound that delineates the fundamental limit of sorted l1 regularization in optimally trading the FDP off for the TPP. Finally, we show that on any problem instance, SLOPE with a certain regularization sequence outperforms the Lasso, in the sense of having a smaller FDP, larger TPP and smaller l2 estimation risk simultaneously. Our proofs are based on a novel technique that reduces a calculus of variations problem to a class of infinite-dimensional convex optimization problems and a very recent result from approximate message passing theory.