Optimal Scaling of the Mala Algorithm with Irreversible Proposals for Gaussian Targets

Abstract. It is known that reversible Langevin diffusions in confining potentials converge to equilibrium exponentially fast. Adding a divergence free component to the drift of a Langevin diffusion accelerates its convergence to stationarity. However such a stochastic differential equation (SDE) is no longer reversible. In this paper, we analyze the optimal scaling of MCMC algorithms constructed from discretizations of irreversible Langevin dynamics for high dimensional Gaussian target measures. In particular, we make use of Metropolis-Hastings algorithms where the proposal move is a time-step Euler discretization of an irreversible SDE. We call the resulting algorithm the ipMALA , in comparison to the classical MALA algorithm. We stress that the usual Metropolis-Hastings accept-reject mechanism makes the ipMALA chain reversible; thus, it is of interest to study the effect of irreversible proposals in these algorithms. In order to quantify how the cost of the algorithm scales with the dimension N , we prove invariance principles for the appropriately rescaled chain. In contrast to the usual MALA algorithm, we show that there could be three regimes asymptotically: (i) a diffusive regime, as in the MALA algorithm, (ii) a “transitive” regime and (iii) a “fluid” regime where the limit is an ordinary differential equation. Numerical results are also given corroborating the theory.