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Newton-Type Methods with the Proximal Gradient Step for Sparse Estimation

Author

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  • Ryosuke Shimmura

    (Osaka University)

  • Joe Suzuki

    (Osaka University)

Abstract

In this paper, we propose new methods to efficiently solve convex optimization problems encountered in sparse estimation. These methods include a new quasi-Newton method that avoids computing the Hessian matrix and improves efficiency, and we prove its fast convergence. We also prove the local convergence of the Newton method under the assumption of strong convexity. Our proposed methods offer a more efficient and effective approach, particularly for $$L_1$$ L 1 regularization and group regularization problems, as they incorporate variable selection with each update. Through numerical experiments, we demonstrate the efficiency of our methods in solving problems encountered in sparse estimation. Our contributions include theoretical guarantees and practical applications for various kinds of problems.

Suggested Citation

  • Ryosuke Shimmura & Joe Suzuki, 2024. "Newton-Type Methods with the Proximal Gradient Step for Sparse Estimation," SN Operations Research Forum, Springer, vol. 5(2), pages 1-27, June.
    • Handle: RePEc:spr:snopef:v:5:y:2024:i:2:d:10.1007_s43069-024-00307-x
    DOI: 10.1007/s43069-024-00307-x
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    References listed on IDEAS

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    1. Lorenzo Stella & Andreas Themelis & Panagiotis Patrinos, 2017. "Forward–backward quasi-Newton methods for nonsmooth optimization problems," Computational Optimization and Applications, Springer, vol. 67(3), pages 443-487, July.
    2. Ming Yuan & Yi Lin, 2006. "Model selection and estimation in regression with grouped variables," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 68(1), pages 49-67, February.
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