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Regression tasks in machine learning via Fenchel duality

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  • Radu Boţ
  • André Heinrich

Abstract

Supervised learning methods are powerful techniques to learn a function from a given set of labeled data, the so-called training data. In this paper the support vector machines approach for regression is investigated under a theoretical point of view that makes use of convex analysis and Fenchel duality. Starting with the corresponding Tikhonov regularization problem, reformulated as a convex optimization problem, we introduce a conjugate dual problem to it and prove that, whenever strong duality holds, the function to be learned can be expressed via the optimal solutions of the dual problem. Corresponding dual problems are then derived for different loss functions. The theoretical results are applied by numerically solving the regression task for two data sets and the accuracy of the regression when choosing different loss functions is investigated. Copyright Springer Science+Business Media New York 2014

Suggested Citation

  • Radu Boţ & André Heinrich, 2014. "Regression tasks in machine learning via Fenchel duality," Annals of Operations Research, Springer, vol. 222(1), pages 197-211, November.
    • Handle: RePEc:spr:annopr:v:222:y:2014:i:1:p:197-211:10.1007/s10479-012-1304-1
    DOI: 10.1007/s10479-012-1304-1
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    References listed on IDEAS

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    1. Bot, Radu Ioan & Lorenz, Nicole, 2011. "Optimization problems in statistical learning: Duality and optimality conditions," European Journal of Operational Research, Elsevier, vol. 213(2), pages 395-404, September.
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    Cited by:

    1. Qingjie Hu & Jiguang Wang & Yu Chen, 2020. "New dualities for mathematical programs with vanishing constraints," Annals of Operations Research, Springer, vol. 287(1), pages 233-255, April.
    2. S. K. Mishra & Vinay Singh & Vivek Laha, 2016. "On duality for mathematical programs with vanishing constraints," Annals of Operations Research, Springer, vol. 243(1), pages 249-272, August.

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