Convergence rates for Metropolis-Hastings algorithms in the Wasserstein distance

We develop necessary conditions for geometrically fast convergence in the Wasserstein distance for Metropolis-Hastings algorithms on $\mathbb{R}^d$ when the metric used is a norm. This is accomplished through a lower bound which is of independent interest. We show exact convergence expressions in more general Wasserstein distances (e.g. total variation) can be achieved for a large class of distributions by centering an independent Gaussian proposal, that is, matching the optimal points of the proposal and target densities. This approach has applications for sampling posteriors of many popular Bayesian generalized linear models. In the case of Bayesian binary response regression, we show when the sample size $n$ and the dimension $d$ grow in such a way that the ratio $d/n \to \gamma \in (0, +\infty)$, the exact convergence rate can be upper bounded asymptotically.