Ballistic Convergence in Hit-and-Run Monte Carlo and a Coordinate-free Randomized Kaczmarz Algorithm

TL;DR

This paper provides theoretical analysis of Hit-and-Run Monte Carlo and a coordinate-free randomized Kaczmarz algorithm, revealing their convergence rates and advantages in sampling and optimization.

Contribution

It offers sharp Wasserstein contraction estimates for Hit-and-Run and extends these results to a coordinate-free randomized Kaczmarz method, highlighting their convergence properties.

Findings

Uncovered ballistic and superdiffusive convergence rates.

Provided sharp Wasserstein contraction bounds.

Extended analysis to coordinate-free randomized Kaczmarz.

Abstract

Hit-and-Run is a coordinate-free Gibbs sampler, yet the quantitative advantages of its coordinate-free property remain largely unexplored beyond empirical studies. In this paper, we prove sharp estimates for the Wasserstein contraction of Hit-and-Run in Gaussian target measures via coupling methods and conclude mixing time bounds. Our results uncover ballistic and superdiffusive convergence rates in certain settings. Furthermore, we extend these insights to a coordinate-free variant of the randomized Kaczmarz algorithm, an iterative method for linear systems, and demonstrate analogous convergence rates. These findings offer new insights into the advantages and limitations of coordinate-free methods for both sampling and optimization.