# Learning rates for classification with Gaussian kernels

## TL;DR

This paper provides refined error analysis for SVM with Gaussian kernels in binary classification, showing near-optimal and optimal learning rates under certain smoothness and noise conditions.

## Contribution

It demonstrates that SVM with Gaussian kernels can achieve near-optimal and optimal learning rates for specific loss functions and smoothness assumptions.

## Key findings

- SVM with Gaussian kernel reaches almost optimal rates for certain losses with smooth regression functions.
- Under Tsybakov noise, infinitely smooth regression functions enable SVM to attain a $m^{-1}$ learning rate.
- The analysis refines understanding of SVM performance in classification tasks with Gaussian kernels.

## Abstract

This paper aims at refined error analysis for binary classification using support vector machine (SVM) with Gaussian kernel and convex loss. Our first result shows that for some loss functions such as the truncated quadratic loss and quadratic loss, SVM with Gaussian kernel can reach the almost optimal learning rate, provided the regression function is smooth. Our second result shows that, for a large number of loss functions, under some Tsybakov noise assumption, if the regression function is infinitely smooth, then SVM with Gaussian kernel can achieve the learning rate of order $m^{-1}$, where $m$ is the number of samples.