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

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.

28 pages

References in corpus (4)

• Quantitative bounds for Markov chain convergence: Wasserstein and total variation distances
• Mixing Time Guarantees for Unadjusted Hamiltonian Monte Carlo
• Mixing of Metropolis-Adjusted Markov Chains via Couplings: The High Acceptance Regime
• Non-reversible lifts of reversible diffusion processes and relaxation times