Nonreversibie Perturbations Accelerate Convergence*

To sample from distributions in high dimensional spaces or finite large sets di-rectly is not feasible in practice, especially when the corresponding densities are known up to normalizing constants only. One has to resort to approximations. A Markov process with the underlying distribution as its equilibrium is often used to generate an approximation ( ” MCMC ”). How good the approximation is depends on the approximating Markov process and on the specific criterion used for com-parison. One may investigate the convergence properties of some particular Monte Carlo Markov processes, or compare the convergence rate within a family of Markov processes (with the same equilibrium) w.r.t. different criteria, or even try to find optimal solutions in that family. Mathematical problems arising from this approach are challenging. By simply adding a weighted divergence-free drift to a reversible diffusion, the convergence to equilibrium is accelerated. In other words, from an algorithm ic point of view the nonreversibie algorithm performs better. Note that two criteria are considered. The analysis is related to the study of antisymmetric perturbations of self-adjoint infinitesimal generators. The optimal solution is still open. However on torus the rate could be pushed to infinity. As for finite sample space nonreversibie perturbations reduce variance.