On the convergence time of some non-reversible Markov chain Monte Carlo methods

It is commonly admitted that non-reversible Markov chain Monte Carlo (MCMC) algorithms usually yield more accurate MCMC estimators than their reversible counterparts. In this note, we show that in addition to their variance reduction effect, some non-reversible MCMC algorithms have also the undesirable property to slow down the convergence of the Markov chain. This point, which has been overlooked by the literature, has obvious practical implications. We illustrate this phenomenon for different non-reversible versions of the Metropolis-Hastings algorithm on several discrete state space examples and discuss ways to mitigate the risk of a small asymptotic variance/slow convergence scenario.