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The reflection-maximal coupling of the random walk Metropolis (RWM) algorithmwas recently proposed for use within unbiased MCMC. Numerically, when thetarget is spherical this coupling has been shown to perform well even in highdimensions. We derive high-dimensional ODE limits for Gaussian targets, whichconfirm this behaviour in the spherical case. However, we then extend ourtheory to the elliptical case and find that as the dimension increases thereflection coupling performs increasingly poorly relative to the mixing of theunderlying RWM chains. To overcome this obstacle, we introduce gradient commonrandom number (GCRN) couplings, which leverage gradient information. We showthat the behaviour of GCRN couplings does not break down with the ellipticityor dimension. Moreover, we show that GCRN couplings are asymptotically optimalfor contraction, in a sense which we make precise, and scale in proportion tothe mixing of the underling RWM chains. Numerically, we apply GCRN couplingsfor convergence and bias quantification, and demonstrate that our theoreticalfindings extend beyond the Gaussian case.

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