  It has been shown that the nonreversible overdamped Langevin dynamics enjoy
better convergence properties in terms of spectral gap and asymptotic variance
than the reversible one. In this article we propose a variance reduction method
for the Metropolis-Hastings Adjusted Langevin Algorithm (MALA) that makes use
of the good behaviour of the these nonreversible dynamics. It consists in
constructing a nonreversible Markov chain (with respect to the target invariant
measure) by using a Generalized Metropolis-Hastings adjustment on a lifted
state space. We present two variations of this method and we discuss the
importance of a well-chosen proposal distribution in terms of average rejection
probability. We conclude with numerical experimentations to compare our
algorithms with the MALA, and show variance reduction of several order of
magnitude in some favourable toy cases.
