Fast rates for support vector machines using Gaussian kernels

TL;DR

This paper proves that support vector machines with Gaussian kernels can achieve near-optimal learning rates of about n^{-1} for binary classification, under specific noise conditions without requiring smoothness assumptions.

Contribution

The paper introduces a new geometric noise condition that bounds approximation error without smoothness assumptions, enabling faster learning rates for SVMs with Gaussian kernels.

Findings

Achieves learning rates close to n^{-1} for SVMs with Gaussian kernels.

Introduces a novel geometric noise condition for bounding approximation error.

Demonstrates that smoothness assumptions are not necessary for fast rates.

Abstract

For binary classification we establish learning rates up to the order of $n^{-1}$ for support vector machines (SVMs) with hinge loss and Gaussian RBF kernels. These rates are in terms of two assumptions on the considered distributions: Tsybakov's noise assumption to establish a small estimation error, and a new geometric noise condition which is used to bound the approximation error. Unlike previously proposed concepts for bounding the approximation error, the geometric noise assumption does not employ any smoothness assumption.