Weak Poincar\'e Inequalities for Markov chains: theory and applications
Probability
2023-12-20 v1 Computation
Abstract
We investigate the application of Weak Poincar\'e Inequalities (WPI) to Markov chains to study their rates of convergence and to derive complexity bounds. At a theoretical level we investigate the necessity of the existence of WPIs to ensure \mathrm{L}^{2}-convergence, in particular by establishing equivalence with the Resolvent Uniform Positivity-Improving (RUPI) condition and providing a counterexample. From a more practical perspective, we extend the celebrated Cheeger's inequalities to the subgeometric setting, and further apply these techniques to study random-walk Metropolis algorithms for heavy-tailed target distributions and to obtain lower bounds on pseudo-marginal algorithms.
Cite
@article{arxiv.2312.11689,
title = {Weak Poincar\'e Inequalities for Markov chains: theory and applications},
author = {Christophe Andrieu and Anthony Lee and Sam Power and Andi Q. Wang},
journal= {arXiv preprint arXiv:2312.11689},
year = {2023}
}