An Equivalence between the Lasso and Support Vector Machines

TL;DR

This paper establishes a fundamental equivalence between support vector machines and Lasso regression, enabling the transfer of algorithms, theoretical insights, and kernel methods between the two, and revealing new connections in sparsity and regularization paths.

Contribution

The paper proves the equivalence of SVM and Lasso optimization problems, allowing cross-application of algorithms and insights, and introduces a kernelized Lasso and new sublinear algorithms.

Findings

Lasso solutions' sparsity equals the number of SVM support vectors.

Kernelized version of Lasso analogous to SVM kernels.

New sublinear time algorithm for Lasso.

Abstract

We investigate the relation of two fundamental tools in machine learning and signal processing, that is the support vector machine (SVM) for classification, and the Lasso technique used in regression. We show that the resulting optimization problems are equivalent, in the following sense. Given any instance of an $\ell_2$-loss soft-margin (or hard-margin) SVM, we construct a Lasso instance having the same optimal solutions, and vice versa. As a consequence, many existing optimization algorithms for both SVMs and Lasso can also be applied to the respective other problem instances. Also, the equivalence allows for many known theoretical insights for SVM and Lasso to be translated between the two settings. One such implication gives a simple kernelized version of the Lasso, analogous to the kernels used in the SVM setting. Another consequence is that the sparsity of a Lasso solution is equal to the number of support vectors for the corresponding SVM instance, and that one can use screening rules to prune the set of support vectors. Furthermore, we can relate sublinear time algorithms for the two problems, and give a new such algorithm variant for the Lasso. We also study the regularization paths for both methods.