Explicit convergence bounds for Metropolis Markov chains: Isoperimetry, spectral gaps and profiles

Abstract

We derive the first explicit bounds for the spectral gap of a random walk Metropolis algorithm on ${\mathbb{R}}^{\mathit{d}}$ for any value of the proposal variance, which when scaled appropriately recovers the correct ${\mathit{d}}^{-1}$ dependence on dimension for suitably regular invariant distributions. We also obtain explicit bounds on the ${\mathrm{L}}^2$-mixing time for a broad class of models. In obtaining these results, we refine the use of isoperimetric profile inequalities to obtain conductance profile bounds, which also enable the derivation of explicit bounds in a much broader class of models. We also obtain similar results for the preconditioned Crank–Nicolson Markov chain, obtaining dimension-independent bounds under suitable assumptions.