On the Generalized Langevin Equation for Simulated Annealing

In this paper, we consider the generalized (higher order) Langevin equation for the purpose of simu-lated annealing and optimization of nonconvex functions. Our approach modifies the underdamped Langevin equation by replacing the Brownian noise with an appropriate Ornstein-Uhlenbeck process to account for memory in the system. Under reasonable conditions on the loss function and the an-nealing schedule, we establish convergence of the continuous time dynamics to a global minimum. In addition, we investigate the performance numerically and show better performance and higher exploration of the state space compared to the underdamped Langevin dynamics with the same annealing schedule.