Accelerated Flow for Probability Distributions

This paper presents a methodology and numerical algorithms for constructing accelerated gradient flows on the space of probability distributions. In particular, we extend the recent variational formulation of accelerated methods in (Wibisono et al., 2016) from vector valued variables to probability distributions. The variational problem is modeled as a mean-field optimal control problem. A quantitative estimate on the asymptotic convergence rate is provided based on a Lyapunov function construction, when the objective functional is displacement convex. An important special case is considered where the objective functional is the relative entropy. For this case, two numerical approximations are presented to implement the Hamilton's equations as a system of N interacting particles. The algorithm is numerically illustrated and compared with the MCMC and Hamiltonian MCMC algorithms.