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Optimization problems in statistical learning: Duality and optimality conditions

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  • Bot, Radu Ioan
  • Lorenz, Nicole

Abstract

Regularization methods are techniques for learning functions from given data. We consider regularization problems the objective function of which consisting of a cost function and a regularization term with the aim of selecting a prediction function f with a finite representation which minimizes the error of prediction. Here the role of the regularizer is to avoid overfitting. In general these are convex optimization problems with not necessarily differentiable objective functions. Thus in order to provide optimality conditions for this class of problems one needs to appeal on some specific techniques from the convex analysis. In this paper we provide a general approach for deriving necessary and sufficient optimality conditions for the regularized problem via the so-called conjugate duality theory. Afterwards we employ the obtained results to the Support Vector Machines problem and Support Vector Regression problem formulated for different cost functions.

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  • 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.
    • Handle: RePEc:eee:ejores:v:213:y:2011:i:2:p:395-404
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    References listed on IDEAS

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    1. R. I. Boţ & S. M. Grad & G. Wanka, 2007. "New Constraint Qualification and Conjugate Duality for Composed Convex Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 135(2), pages 241-255, November.
    2. NESTEROV, Yu., 2005. "Smooth minimization of non-smooth functions," LIDAM Reprints CORE 1819, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
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    Cited by:

    1. Radu Boţ & Christopher Hendrich, 2015. "A variable smoothing algorithm for solving convex optimization problems," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 23(1), pages 124-150, April.
    2. 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.
    3. Toriello, Alejandro & Vielma, Juan Pablo, 2012. "Fitting piecewise linear continuous functions," European Journal of Operational Research, Elsevier, vol. 219(1), pages 86-95.
    4. Corne, David & Dhaenens, Clarisse & Jourdan, Laetitia, 2012. "Synergies between operations research and data mining: The emerging use of multi-objective approaches," European Journal of Operational Research, Elsevier, vol. 221(3), pages 469-479.
    5. Brandner, Hubertus & Lessmann, Stefan & Voß, Stefan, 2013. "A memetic approach to construct transductive discrete support vector machines," European Journal of Operational Research, Elsevier, vol. 230(3), pages 581-595.
    6. Gambella, Claudio & Ghaddar, Bissan & Naoum-Sawaya, Joe, 2021. "Optimization problems for machine learning: A survey," European Journal of Operational Research, Elsevier, vol. 290(3), pages 807-828.

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