Xuefeng GAO, Mert Gurbuzbalaban, Lingjiong Zhu
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 et al. (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 algorithm improves upon the recurrence time obtained for LD in Tzen et al. (2018) with respect to the dependency on the smallest eigenvalue of the Hessian at the local minimum. Similar results and improvements 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.