A new and asymptotically optimally contracting coupling for the random walk Metropolis

The reflection-maximal coupling of the random walk Metropolis (RWM) algorithm was recently proposed for use within unbiased MCMC. Numerically, when the target is spherical this coupling has been shown to perform well even in high dimensions. We derive high-dimensional ODE limits for Gaussian targets, which confirm this behaviour in the spherical case. However, we then extend our theory to the elliptical case and find that as the dimension increases the reflection coupling performs increasingly poorly relative to the mixing of the underlying RWM chains. To overcome this obstacle, we introduce gradient common random number (GCRN) couplings, which leverage gradient information. We show that the behaviour of GCRN couplings does not break down with the ellipticity or dimension. Moreover, we show that GCRN couplings are asymptotically optimal for contraction, in a sense which we make precise, and scale in proportion to the mixing of the underling RWM chains. Numerically, we apply GCRN couplings for convergence and bias quantification, and demonstrate that our theoretical findings extend beyond the Gaussian case.