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

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Abstract

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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Notes

  1. http://www.public.asu.edu/~jye02/Software/SLEP.

  2. The source codes can be found in https://github.com/EugeneNdiaye/GAPSAFE_SGL.

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Acknowledgements

The authors would like to thank Dr. Xudong Li and Ms. Meixia Lin for their help in the numerical implementations. We also thank the referees for their valuable suggestions which have helped to improve quality of this paper.

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Authors and Affiliations

  1. Department of Mathematics, National University of Singapore, 10 Lower Kent Ridge Road, Singapore, 119076, Singapore

    Yangjing Zhang

  2. Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Hong Kong

    Ning Zhang & Defeng Sun

  3. Department of Mathematics, and Institute of Operations Research and Analytics, National University of Singapore, 10 Lower Kent Ridge Road, Singapore, 119076, Singapore

    Kim-Chuan Toh

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  1. Yangjing Zhang
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  2. Ning Zhang
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  3. Defeng Sun
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  4. Kim-Chuan Toh
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Corresponding author

Correspondence to Defeng Sun.

Additional information

N. Zhang: This author’s research is supported in part by the Singapore-ETH Centre (SEC), which was established as a collaboration between ETH Zurich and National Research Foundation (NRF) Singapore (FI 370074011) under the auspices of the NRF’s Campus for Research Excellence and Technological Enterprise (CREATE) programme.

D. Sun: This author’s research is supported in part by a start-up research grant from the Hong Kong Polytechnic University

K.-C. Toh: This author’s research is supported in part by an Academic Research Fund under Grant number R-146-000-257-112 from the Ministry of Education, Singapore.

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Zhang, Y., Zhang, N., Sun, D. et al. An efficient Hessian based algorithm for solving large-scale sparse group Lasso problems. Math. Program. 179, 223–263 (2020). https://doi.org/10.1007/s10107-018-1329-6

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