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A modified local quadratic approximation algorithm for penalized optimization problems

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  • Lee, Sangin
  • Kwon, Sunghoon
  • Kim, Yongdai

Abstract

In this paper, we propose an optimization algorithm called the modified local quadratic approximation algorithm for minimizing various ℓ1-penalized convex loss functions. The proposed algorithm iteratively solves ℓ1-penalized local quadratic approximations of the loss function, and then modifies the solution whenever it fails to decrease the original ℓ1-penalized loss function. As an extension, we construct an algorithm for minimizing various nonconvex penalized convex loss functions by combining the proposed algorithm and convex concave procedure, which can be applied to most nonconvex penalty functions such as the smoothly clipped absolute deviation and minimax concave penalty functions. Numerical studies show that the algorithm is stable and fast for solving high dimensional penalized optimization problems.

Suggested Citation

  • Lee, Sangin & Kwon, Sunghoon & Kim, Yongdai, 2016. "A modified local quadratic approximation algorithm for penalized optimization problems," Computational Statistics & Data Analysis, Elsevier, vol. 94(C), pages 275-286.
    • Handle: RePEc:eee:csdana:v:94:y:2016:i:c:p:275-286
    DOI: 10.1016/j.csda.2015.08.019
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    References listed on IDEAS

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    1. Hao Helen Zhang & Grace Wahba & Yi Lin & Meta Voelker & Michael Ferris & Ronald Klein & Barbara Klein, 2004. "Variable Selection and Model Building via Likelihood Basis Pursuit," Journal of the American Statistical Association, American Statistical Association, vol. 99, pages 659-672, January.
    2. Dankmar Böhning & Bruce Lindsay, 1988. "Monotonicity of quadratic-approximation algorithms," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 40(4), pages 641-663, December.
    3. Friedman, Jerome H. & Hastie, Trevor & Tibshirani, Rob, 2010. "Regularization Paths for Generalized Linear Models via Coordinate Descent," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 33(i01).
    4. Kim, Yongdai & Choi, Hosik & Oh, Hee-Seok, 2008. "Smoothly Clipped Absolute Deviation on High Dimensions," Journal of the American Statistical Association, American Statistical Association, vol. 103(484), pages 1665-1673.
    5. Kim, Yongdai & Kwon, Sunghoon & Heun Song, Seuck, 2006. "Multiclass sparse logistic regression for classification of multiple cancer types using gene expression data," Computational Statistics & Data Analysis, Elsevier, vol. 51(3), pages 1643-1655, December.
    6. Fan J. & Li R., 2001. "Variable Selection via Nonconcave Penalized Likelihood and its Oracle Properties," Journal of the American Statistical Association, American Statistical Association, vol. 96, pages 1348-1360, December.
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    Cited by:

    1. Xia Zheng & Yaohua Rong & Ling Liu & Weihu Cheng, 2021. "A More Accurate Estimation of Semiparametric Logistic Regression," Mathematics, MDPI, vol. 9(19), pages 1-12, September.
    2. Margherita Giuzio, 2017. "Genetic algorithm versus classical methods in sparse index tracking," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 40(1), pages 243-256, November.
    3. Young‐Geun Choi & Lawrence P. Hanrahan & Derek Norton & Ying‐Qi Zhao, 2022. "Simultaneous spatial smoothing and outlier detection using penalized regression, with application to childhood obesity surveillance from electronic health records," Biometrics, The International Biometric Society, vol. 78(1), pages 324-336, March.
    4. Majnu John & Sujit Vettam & Yihren Wu, 2025. "A novel nonconvex, smooth-at-origin penalty for statistical learning," Computational Statistics, Springer, vol. 40(3), pages 1397-1422, March.
    5. Lee, Sangin & Lee, Youngjo & Pawitan, Yudi, 2018. "Sparse pathway-based prediction models for high-throughput molecular data," Computational Statistics & Data Analysis, Elsevier, vol. 126(C), pages 125-135.

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