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

Christophe Andrieu, Anthony Lee, Samuel Power, Andi Q. Wang

Abstract:

We derive the first explicit bounds for the spectral gap of a random walk Metropolis algorithm
on $R^d$ for any value of the proposal variance, which when scaled appropriately recovers the correct
d−1 dependence on dimension for suitably regular invariant distributions. We also obtain explicit
bounds on the L2-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