  Langevin dynamics (LD) has been proven to be a powerful technique for
optimizing a non-convex objective as an efficient algorithm to find local
minima while eventually visiting a global minimum on longer time-scales. LD is
based on the first-order Langevin diffusion which is reversible in time. We
study two variants that are based on non-reversible Langevin diffusions: the
underdamped Langevin dynamics (ULD) and the Langevin dynamics with a
non-symmetric drift (NLD). Adopting the techniques of Tzen, Liang and Raginsky
(2018) for LD to non-reversible diffusions, we show that for a given local
minimum that is within an arbitrary distance from the initialization, with high
probability, either the ULD trajectory ends up somewhere outside a small
neighborhood of this local minimum within a recurrence time which depends on
the smallest eigenvalue of the Hessian at the local minimum or they enter this
neighborhood by the recurrence time and stay there for a potentially
exponentially long escape time. The ULD algorithms improve upon the recurrence
time obtained for LD in Tzen, Liang and Raginsky (2018) with respect to the
dependency on the smallest eigenvalue of the Hessian at the local minimum.
Similar result and improvement are obtained for the NLD algorithm. We also show
that non-reversible variants can exit the basin of attraction of a local
minimum faster in discrete time when the objective has two local minima
separated by a saddle point and quantify the amount of improvement. Our
analysis suggests that non-reversible Langevin algorithms are more efficient to
locate a local minimum as well as exploring the state space. Our analysis is
based on the quadratic approximation of the objective around a local minimum.
As a by-product of our analysis, we obtain optimal mixing rates for quadratic
objectives in the 2-Wasserstein distance for two non-reversible Langevin
algorithms we consider.
