Regression tasks in machine learning via Fenchel duality

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.

Closedness Type Regularity Conditions in Convex Optimization and Beyond

“…Closedness type regularity conditions were employed by different authors in other related research fields, too, like subdifferential calculus (e.g., by [10,18,29,[50][51][52]), DC programming (e.g., by [51,[53][54][55][56][57]), generalized convex optimization (e.g., by [58][59][60]), semiinfinite programming (e.g., by [15,16,50,53,[61][62][63][64]), semidefinite programming (e.g., by [43,45,46,65]), robust optimization (e.g., by [63,66,67]), location optimization (e.g., by [68]), vector optimization (e.g., by [10,29,63,64,69]), monotone operators ( [17,[70][71][72][73][74][75][76][77][78]), machine learning ( [79]) or variati...…”