Accelerating optimization over the space of probability measures
This work addresses the need for faster optimization in machine learning applications involving probability measures, representing an incremental advancement by extending momentum-based methods to non-Euclidean spaces.
The paper tackles the problem of accelerating gradient-based optimization over probability measure spaces, introducing a Hamiltonian-flow approach that achieves arbitrarily high-order convergence rates in continuous-time settings.
The acceleration of gradient-based optimization methods is a subject of significant practical and theoretical importance, particularly within machine learning applications. While much attention has been directed towards optimizing within Euclidean space, the need to optimize over spaces of probability measures in machine learning motivates exploration of accelerated gradient methods in this context too. To this end, we introduce a Hamiltonian-flow approach analogous to momentum-based approaches in Euclidean space. We demonstrate that, in the continuous-time setting, algorithms based on this approach can achieve convergence rates of arbitrarily high order. We complement our findings with numerical examples.