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

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 su...

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Bibliographic Details
Published in:Mathematical programming Vol. 179; no. 1-2; pp. 223 - 263
Main Authors: Zhang, Yangjing, Zhang, Ning, Sun, Defeng, Toh, Kim-Chuan
Format: Journal Article
Language:English
Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.01.2020
Springer Nature B.V
Subjects:
Algorithms
Calculus of Variations and Optimal Control; Optimization
Combinatorics
Convergence
Full Length Paper
Iterative methods
Lagrange multiplier
Mapping
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Mathematics of Computing
Newton methods
Nonlinear programming
Numerical Analysis
Robustness (mathematics)
Sparsity
Statistical models
Theoretical
Augmented Lagrangian method
62J05
Semismooth Newton method
90C25
Sparse group Lasso
90C06
Generalized Jacobian
ISSN:0025-5610, 1436-4646
Online Access:Get full text
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Summary: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.
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ISSN:0025-5610
1436-4646
DOI:10.1007/s10107-018-1329-6