Acceleration of gradient-based optimization methods is an issue of
significant practical and theoretical interest, particularly in machine
learning applications. Most research has focused on optimization over Euclidean
spaces, but given the need to optimize over spaces of probability measures in
many machine learning problems, it is of interest to investigate accelerated
gradient methods in this context too. To this end, we introduce a
Hamiltonian-flow approach that is analogous to moment-based approaches in
Euclidean space. We demonstrate that algorithms based on this approach can
achieve convergence rates of arbitrarily high order. Numerical examples
illustrate our claim