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High-dimensional penalized Bernstein support vector classifier

Author

Listed:
  • Rachid Kharoubi

    (Université du Québec à Montréal)

  • Abdallah Mkhadri

    (University of Cadi Ayyad)

  • Karim Oualkacha

    (Université du Québec à Montréal)

Abstract

The support vector machine (SVM) is a powerful classifier used for binary classification to improve the prediction accuracy. However, the nondifferentiability of the SVM hinge loss function can lead to computational difficulties in high-dimensional settings. To overcome this problem, we rely on the Bernstein polynomial and propose a new smoothed version of the SVM hinge loss called the Bernstein support vector machine (BernSVC). This extension is suitable for the high dimension regime. As the BernSVC objective loss function is twice differentiable everywhere, we propose two efficient algorithms for computing the solution of the penalized BernSVC. The first algorithm is based on coordinate descent with the maximization-majorization principle and the second algorithm is the iterative reweighted least squares-type algorithm. Under standard assumptions, we derive a cone condition and a restricted strong convexity to establish an upper bound for the weighted lasso BernSVC estimator. By using a local linear approximation, we extend the latter result to the penalized BernSVC with nonconvex penalties SCAD and MCP. Our bound holds with high probability and achieves the so-called fast rate under mild conditions on the design matrix. Simulation studies are considered to illustrate the prediction accuracy of BernSVC relative to its competitors and also to compare the performance of the two algorithms in terms of computational timing and error estimation. The use of the proposed method is illustrated through analysis of three large-scale real data examples.

Suggested Citation

  • Rachid Kharoubi & Abdallah Mkhadri & Karim Oualkacha, 2024. "High-dimensional penalized Bernstein support vector classifier," Computational Statistics, Springer, vol. 39(4), pages 1909-1936, June.
  • Handle: RePEc:spr:compst:v:39:y:2024:i:4:d:10.1007_s00180-023-01448-z
    DOI: 10.1007/s00180-023-01448-z
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    References listed on IDEAS

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    1. Marron, J.S. & Todd, Michael J. & Ahn, Jeongyoun, 2007. "Distance-Weighted Discrimination," Journal of the American Statistical Association, American Statistical Association, vol. 102, pages 1267-1271, December.
    2. Christmann, Andreas & Hable, Robert, 2012. "Consistency of support vector machines using additive kernels for additive models," Computational Statistics & Data Analysis, Elsevier, vol. 56(4), pages 854-873.
    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. Xiang Zhang & Yichao Wu & Lan Wang & Runze Li, 2016. "Variable selection for support vector machines in moderately high dimensions," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 78(1), pages 53-76, January.
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