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.

在机器学习应用中，梯度优化方法的加速是一个实际和理论上感兴趣的问题。大多数研究关注欧几里得空间上的优化，但鉴于在许多机器学习问题中需要在概率测度空间上进行优化，研究在这个背景下的加速梯度方法很有意义。为此，我们介绍了与欧几里得空间中基于矩的方法类似的哈密顿流方法。我们证明了基于这种方法的算法可以实现任意高阶的收敛速度。数值例子证明了我们的论断。