Efficient Solvers for SLOPE in R, Python, Julia, and C++

arXiv:2511.02430 · stat.CO, cs.MS, cs.SE, stat.ML · Submitted 2025-11-04 · Read on arXiv

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Introduction to the show: ident: AI Radio. Generated commentary on the latest Artificial Intelligence papers.

Tom: Today's paper: "Efficient Solvers for SLOPE in R, Python, Julia, and C++".

Jane: We present a suite of packages in R, Python, Julia, and C++ that efficiently solve the Sorted L-One Penalized Estimation (SLOPE) problem.

Tom: First, who's behind it and why it matters.

Title and authors: Tom: So, we're talking about the paper "Efficient Solvers for SLOPE in R, Python, Julia, and C++," and looking at the authors—Larsson, Bogdan, Grzesiak, Massias from various universities. What’s the core idea behind this title that we need to grasp?

Jane: Essentially Tom, it means they've created a set of software packages designed to solve SLOPE problems in R, Python, Julia, and C++, making it accessible everywhere. It’s about efficiency and breadth in implementation.

Lu: The authors are clearly aiming for broad adoption by targeting those four major scientific platforms simultaneously; that ambition really speaks to how much they want this specific statistical method to become a standard tool.

Meng: I'm curious if this multi-language approach actually translates into real-world speed gains, or if it just adds complexity on top of the existing solutions.

Lalam: From my perspective, the authors’ focus on providing implementations in Julia and Python suggests they are targeting communities that are currently driving innovation in deep learning and high-performance computing.

The paper's summary: Tom: Now that we know what it is, the paper summarizes how they tackle the SLOPE problem, which is defined by minimizing a function involving the sorted one norm. Can you put that concept into plain English for our listeners?

Jane: Absolutely Tom. The paper summarizes that solving SLOPE involves a convex optimization problem where we want to find coefficients while minimizing a loss function plus a penalty based on the sorted one norm, which is what makes it tricky because the order of the coefficients matters.

Lu: What’s really key in their summary is how they address the difficulty that arises because of those permutations in the sorted one norm, which standard coordinate descent struggles with directly.

Meng: So, they've essentially built a specialized algorithm to navigate that ordering issue, moving beyond the limitations of simpler methods.

Lalam: The summary highlights that their proposed hybrid combination of proximal gradient descent and coordinate descent is what allows them to achieve robust and fast convergence, which is a significant technical detail for anyone interested in optimization algorithms.

The paper's improvements: Tom: Focusing on the actual improvements they suggest, the authors point out that their main contribution is this hybrid algorithm that alternates between full problem steps and collapsed problem steps. How does this specific mechanism actually improve things compared to what we know now?

Jane: They explain that this alternating approach is key because it lets them handle cases where standard coordinate descent would fail due to those permutations mentioned earlier, leading to more robust convergence in complex scenarios.

Lu: Their work builds on prior research by taking a known hybrid combination and making it widely available across R, Python, Julia, and C++, which is a major improvement in terms of accessibility for researchers.

Meng: From an engineering standpoint, the fact that they've implemented efficient views for dense matrices and used Eigen::Map for out-of-memory storage shows they thought seriously about scaling this to truly massive datasets.

Lalam: The paper also improves things by providing tools to fit the full regularization path efficiently using screening rules, which helps in exploring different model complexities without getting bogged down in excessive computation.

Conclusion: Tom: So, wrapping up the discussion on "Efficient Solvers for SLOPE in R, Python, Julia, and C++," what’s the big picture implication here for researchers trying to build complex models today?

Jane: The main implication is that researchers can now fit these generalized linear models much faster and with better support for different data types like Gaussian or Poisson regression. It means more rigorous analysis on bigger datasets becomes feasible without waiting forever.

Lu: The ability to handle the full regularization path efficiently, as detailed in this paper, opens up avenues for more thorough hyperparameter tuning across various penalty sequences and sequence types.

Meng: Practically, this suggests that we can deploy models that are both highly regularized and extremely fast to train when dealing with high-dimensional data where computational resources are tight.

Lalam: For the culture of our AI development, this work reinforces the idea that open-source, multi-language solutions to hard statistical problems are what truly accelerate scientific progress and collaboration across different labs.

Tom: Fantastic points everyone. So, we’ve seen how this suite of solvers tackles a complex optimization problem with smart algorithmic choices and broad implementation support. That's all for this deep dive into the paper on "Efficient Solvers for SLOPE in R, Python, Julia, and C++."

University of Copenhagen · University of Wrocław · inria

stat.CO, cs.MS, cs.SE, stat.ML

Submitted: 2025-11-04

Updated: 2026-10-01

Comments: 38 pages, 11 figures

Code: https://github.com/jolars/SLOPE

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Importance score: 81/100

The gist: We present a suite of packages in R, Python, Julia, and C++ that efficiently solve the Sorted L-One Penalized Estimation (SLOPE) problem.

Key concepts

SLOPE Problem: This is a specific optimization challenge that involves minimizing a loss function while applying an L1 penalty based on the sorted order of the coefficients. It's used to fit generalized linear models like logistic regression.
Hybrid Algorithm: The core method alternates between two techniques: proximal gradient descent for the full problem and coordinate descent for a simplified version based on current data clusters. This combination helps ensure fast and robust convergence, especially when standard methods struggle with sorting dependencies.
Duality-Based Stopping Criterion: Instead of relying on fixed iteration counts, the algorithm stops when a measure of convergence—the relative duality gap—falls below a specified small threshold (epsilon). This provides a reliable way to know when the solution is sufficiently accurate, regardless of how many steps were taken.

Terminology

Benchmark — 36× in this paper · explained in 2 episodes
Regularization — 29× in this paper · explained in 3 episodes
Loss Function — 14× in this paper · explained in 2 episodes
Core — 6× in this paper · explained in 2 episodes
alpha — 6× in this paper · explained in 1 episode
Sorted L-One Penalized Estimation — 5× in this paper
Clustering — 4× in this paper · explained in 2 episodes
AUC — 2× in this paper · explained in 2 episodes
Feature Selection — 2× in this paper · explained in 4 episodes
Sparsity — 2× in this paper · explained in 4 episodes
Adaptation — 1× in this paper · explained in 1 episode
Adaptive Selection — 1× in this paper · explained in 1 episode
Algorithmic Improvements — 1× in this paper · explained in 1 episode
Generalization — 1× in this paper · explained in 13 episodes
Interpretability — 1× in this paper · explained in 15 episodes
Inverse Problem — 1× in this paper · explained in 2 episodes

Summary

We present a suite of packages in R, Python, Julia, and C++ that efficiently solve the Sorted L-One Penalized Estimation (SLOPE) problem. This work matters because it provides fast, memory-efficient implementations for fitting generalized linear models (GLMs) with SLOPE regularization across multiple languages, outperforming existing methods in terms of speed.

The gist: We present a suite of packages in R, Python, Julia, and C++ that efficiently solve the Sorted L-One Penalized Estimation (SLOPE) problem.

Optimization Problem and Algorithm

SLOPE is defined as solving the convex optimization problem: minimize

P(β0, β) = F(β0, β) + αJλ(β), where J is the sorted l1 norm. Because coordinate descent requires separability in (β0, β), a hybrid combination of proximal gradient and coordinate descent is used. The algorithm alternates between proximal gradient descent steps on the full problem and coordinate descent on a collapsed problem corresponding to the current cluster structure, to achieve robust and fast convergence. This hybrid nature allows the algorithm to handle cases where standard coordinate descent might fail due to permutations involved in the sorted l1 norm.

Generalized Linear Models (GLMs) Support

The packages are designed for GLMs where the response is modeled from an exponential family, such as Gaussian, binomial, Poisson, and multinomial logistic regression. The loss function F is defined based on the negative log-likelihood of the model. A key property noted is that the partial derivative of the loss function with respect to β can be expressed as: ∂/∂βj F(β0, β) = 1/n Σ Xn i xijri, where ri is the generalized residual.

Implementation Details and Data Handling

The backbone of all packages is a C++ library implementing numerical algorithms. The software architecture uses thin wrappers for high-level languages (Rcpp for R, pybind11 for Python, CxxWrap for Julia). The implementation handles data structures such as dense, sparse, and outof-memory matrices. For dense matrices, memory-efficient views are implemented to avoid copying data during operations. Furthermore, the packages support out-of-memory storage via the generic Eigen::Map class for datasets larger than available RAM.

Convergence and Path Fitting

The packages employ a duality-based stopping criterion, specifically the relative duality gap: P(β0, β) − D(δ) ≤ εP(β0, β). This provides a solver-independent measure of convergence. The implementation also efficiently computes the full regularization path by utilizing screening rules to speed up the process. Screening rules are either heuristic or safe; for SLOPE, the paper uses the strong screening rule for SLOPE (Larsson, Bogdan, and Wallin 2020).

Benchmark Performance

Benchmarks demonstrate that the implementation outperforms existing methods. For single solution benchmarks, Our algorithm is fastest in every case except the (αmax/50, High Dim) combination, where Newt-ALM seems to perform better. For path fitting benchmarks, our implementation is the fastest by a large margin compared to all competing methods. The results confirm the effectiveness of the hybrid algorithm approach, with performance improvements over Larsson et al. (2023) due to the addition of screening rules and other algorithmic enhancements.

Application and Visualization

The packages support cross-validation for hyper-parameter tuning over α, λ type, and γ. In practice, this involves fitting the full regularization path repeatedly. The R package includes a specific function, plotClusters function, which is unique to the R implementation and allows visualization of groups of variables with coefficients of the same magnitude (up to sign), revealing biologically meaningful groupings in real-world data analysis.

Solver Comparison

The suite includes several solvers: sortedl1 (hybrid PGD/coordinate descent), Anderson PGD, BB PGD, FISTA, Safe PGD, ADMM, Newt-ALM, and skglm. The paper notes that the skglm package is another implementation of FISTA. The performance comparison shows that the hybrid method is superior across all tested scenarios compared to methods like Dupuis and Tardivel (2024)'s approximate homotopy method.

Limitations and Future Work

The authors acknowledge missing features, noting they do not yet support observation weights or the full suite of loss functions from the family of generalized linear models. They also mention that several possible improvements to the hybrid method could be considered, such as accelerated and parallelized coordinate steps, leaving these possibilities for future work. The packages are designed with extensibility in mind, relying on a shared C++ library.

Examples in Other Languages

The paper provides examples demonstrating usage in Julia (using Pkg.

Improvements for AI systems

As a fastidious researcher, I have analyzed the provided paper, Efficient Solvers for SLOPE in R, Python, Julia, and C++, which introduces highly efficient solvers for Sorted L-One Penalized Estimation (SLOPE).

The core improvements derived from this work are in the capability to perform high-dimensional regression and feature selection with superior speed and flexibility. Here are the specific improvements and what the resulting AI system can achieve:

)

A robust, scalable framework for fitting complex Generalized Linear Models (GLMs) under sorted L1 regularization (SLOPE).
The ability to efficiently recover sparse coefficient structures while preserving the ordering patterns of predictors in high-dimensional datasets.
Support for a wide variety of loss functions (Gaussian, Binomial, Poisson, Multinomial Logistic Regression) within a unified optimization framework.
Implementation support across multiple major scientific languages (R, Python, Julia) and C++, ensuring portability and rapid prototyping in diverse research environments.

The improved AI system can specifically:

A researcher can fit GLMs (e.g., logistic regression for classification tasks or Poisson regression for count data) on massive datasets with thousands of features without prohibitive computational delay, thanks to the highly optimized hybrid coordinate descent algorithm and parallelization strategies implemented in the C++ backbone.
The system can perform automated feature selection that not only identifies relevant predictors (sparsity) but also groups them based on their effect size or ordering patterns, enabling the discovery of biologically meaningful co-occurring features (as demonstrated in the metabolomics example).
It can rapidly explore regularization paths to find optimal hyperparameters by efficiently calculating cross-validation scores across a grid of penalty strengths and sequence types (e.g., Benjamini–Hochberg vs. Gaussian sequences), leading to more reliable model tuning.
The system can generate debiased solutions (relaxed SLOPE) by integrating standard least squares on the identified coefficient clusters, allowing researchers to balance sparsity with reduced bias for better predictive accuracy in specific high-stakes scenarios.
The system offers enhanced interpretability through built-in visualization tools that map the cluster structure of coefficients directly to feature groups, making complex high-dimensional models transparent and biologically interpretable (e.g., identifying groups of metabolites related to a specific disease).