Statistical performance of support vector machines

TL;DR

This paper analyzes the statistical properties of support vector machines (SVMs) using concentration theory and empirical processes, revealing conditions for optimal penalty selection and fast convergence rates.

Contribution

It interprets SVMs as a model selection procedure and derives oracle inequalities, providing new insights into penalty choices and convergence rates.

Findings

SVMs can be viewed as a regularization and model selection method.

Fast convergence rates for SVMs are achievable under certain conditions.

The study compares the actual penalty used in SVMs to the minimal penalty suggested by theory.

Abstract

The support vector machine (SVM) algorithm is well known to the computer learning community for its very good practical results. The goal of the present paper is to study this algorithm from a statistical perspective, using tools of concentration theory and empirical processes. Our main result builds on the observation made by other authors that the SVM can be viewed as a statistical regularization procedure. From this point of view, it can also be interpreted as a model selection principle using a penalized criterion. It is then possible to adapt general methods related to model selection in this framework to study two important points: (1) what is the minimum penalty and how does it compare to the penalty actually used in the SVM algorithm; (2) is it possible to obtain ``oracle inequalities'' in that setting, for the specific loss function used in the SVM algorithm? We show that the answer to the latter question is positive and provides relevant insight to the former. Our result shows that it is possible to obtain fast rates of convergence for SVMs.