Fast rates for support vector machines using Gaussian kernels

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.

💡 Research Summary

**
The paper “Fast rates for support vector machines using Gaussian kernels” investigates the statistical learning rates achievable by support vector machines (SVMs) equipped with the hinge loss and Gaussian radial basis function (RBF) kernels in binary classification. The authors aim to answer the long‑standing question of under which distributional conditions SVMs can attain learning rates faster than the classical $n^{-1/2}$ barrier, and whether rates approaching $n^{-1}$ are possible without imposing smoothness assumptions on the regression function $\eta(x)=P(Y=1\mid X=x)$.

Two central distributional assumptions are introduced. The first is the well‑known Tsybakov noise condition, which quantifies the concentration of the label noise near the decision boundary. Formally, for a noise exponent $q\ge0$ there exists $C>0$ such that
\