# An efficient Hessian based algorithm for solving large-scale sparse group Lasso problems

arXiv: https://arxiv.org/abs/1712.05910 · HTML: https://mlsanity.com/paper/1712.05910/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Dec 2017.

## Built on by (most cited first, 13 of 13)

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- [A dual semismooth Newton based augmented Lagrangian method for large-scale linearly constrained sparse group square-root Lasso problems](https://mlsanity.com/paper/2111.13878/) (2021)
- [Multi-task Learning for Gaussian Graphical Regressions with High Dimensional Covariates](https://mlsanity.com/paper/2205.10672/) (2022)
- [Self-concordant smoothing in proximal quasi-Newton algorithms for large-scale convex composite optimization](https://mlsanity.com/paper/2309.01781/) (2023)
- [Learning the hub graphical Lasso model with the structured sparsity via an efficient algorithm](https://mlsanity.com/paper/2308.08852/) (2023)
- [Fast Computation of Superquantile-Constrained Optimization Through Implicit Scenario Reduction](https://mlsanity.com/paper/2405.07965/) (2024)
- [Convex Estimation of Gaussian Graphical Regression Models with Covariates](https://mlsanity.com/paper/2410.06326/) (2024)
- [Wasserstein Distributionally Robust Optimization Through the Lens of Structural Causal Models and Individual Fairness](https://mlsanity.com/paper/2509.26275/) (2025)
- [Efficient Group Lasso Regularized Rank Regression with Simulation-Based Tuning](https://mlsanity.com/paper/2510.11546/) (2025)

## Abstract (arXiv)

The sparse group Lasso is a widely used statistical model which encourages the sparsity both on a group and within the group level. In this paper, we develop an efficient augmented Lagrangian method for large-scale non-overlapping sparse group Lasso problems with each subproblem being solved by a superlinearly convergent inexact semismooth Newton method. Theoretically, we prove that, if the penalty parameter is chosen sufficiently large, the augmented Lagrangian method converges globally at an arbitrarily fast linear rate for the primal iterative sequence, the dual infeasibility, and the duality gap of the primal and dual objective functions. Computationally, we derive explicitly the generalized Jacobian of the proximal mapping associated with the sparse group Lasso regularizer and exploit fully the underlying second order sparsity through the semismooth Newton method. The efficiency and robustness of our proposed algorithm are demonstrated by numerical experiments on both the synthetic and real data sets.