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.

翻译：基于梯度的优化方法的加速问题在实际和理论上具有重要研究价值，尤其在机器学习应用中。现有研究主要聚焦于欧氏空间上的优化，但鉴于许多机器学习问题需要在概率测度空间上进行优化，因此在此背景下研究加速梯度方法具有现实意义。为此，我们提出一种哈密顿流方法，该方法与欧氏空间中的矩方法类似。研究表明，基于该方法的算法能够实现任意高阶的收敛速率。数值实验验证了我们的结论。