Local Duality for Sparse Support Vector Machines

TL;DR

This paper develops a local duality theory for sparse support vector machines (SSVMs), establishing their connection with 0/1-loss SVMs and providing insights into hyperparameter selection and solution convergence.

Contribution

It introduces a theoretical framework for SSVMs, linking them to 0/1-loss SVMs and proving properties like the linear representer theorem for local solutions.

Findings

SSVM is the dual of 0/1-loss SVM.

Local solutions of SSVM satisfy the linear representer theorem.

Global solutions of hSVM converge to local solutions of 0/1-loss SVM.

Abstract

Due to the rise of cardinality minimization in optimization, sparse support vector machines (SSVMs) have attracted much attention lately and show certain empirical advantages over convex SVMs. A common way to derive an SSVM is to add a cardinality function such as $\ell_0$-norm to the dual problem of a convex SVM. However, this process lacks theoretical justification. This paper fills the gap by developing a local duality theory for such an SSVM formulation and exploring its relationship with the hinge-loss SVM (hSVM) and the ramp-loss SVM (rSVM). In particular, we prove that the derived SSVM is exactly the dual problem of the 0/1-loss SVM, and the linear representer theorem holds for their local solutions. The local solution of SSVM also provides guidelines on selecting hyperparameters of hSVM and rSVM. {Under specific conditions, we show that a sequence of global solutions of hSVM converges to a local solution of 0/1-loss SVM. Moreover, a local minimizer of 0/1-loss SVM is a local minimizer of rSVM.} This explains why a local solution induced by SSVM outperforms hSVM and rSVM in the prior empirical study. We further conduct numerical tests on real datasets and demonstrate potential advantages of SSVM by working with locally nice solutions proposed in this paper.