# Geometric Rates of Convergence for Kernel-based Sampling Algorithms

## TL;DR

This paper analyzes the convergence rates of kernel-based sampling algorithms, showing near-geometric convergence under certain conditions and demonstrating their effectiveness through theoretical and empirical results.

## Contribution

It provides the first near-geometric convergence analysis for weighted kernel herding and sequential Bayesian quadrature, including distributed settings.

## Key findings

- Algorithms achieve near-geometric convergence for nearly atomic measures.
- Performance matches the theoretical optimum under maximum mean discrepancy.
- Empirical results validate theoretical convergence rates on real and simulated data.

## Abstract

The rate of convergence of weighted kernel herding (WKH) and sequential Bayesian quadrature (SBQ), two kernel-based sampling algorithms for estimating integrals with respect to some target probability measure, is investigated. Under verifiable conditions on the chosen kernel and target measure, we establish a near-geometric rate of convergence for target measures that are nearly atomic. Furthermore, we show these algorithms perform comparably to the theoretical best possible sampling algorithm under the maximum mean discrepancy. An analysis is also conducted in a distributed setting. Our theoretical developments are supported by empirical observations on simulated data as well as a real world application.