Accelerating optimization over the space of probability measures

TL;DR

This paper introduces a Hamiltonian-flow approach for accelerating gradient-based optimization over probability measure spaces, achieving high-order convergence rates and demonstrating practical effectiveness through numerical examples.

Contribution

It proposes a novel Hamiltonian-flow method for optimization over probability measures, extending acceleration techniques beyond Euclidean spaces.

Findings

Achieves arbitrarily high-order convergence rates in continuous-time settings.

Introduces a Hamiltonian-flow approach analogous to momentum methods.

Provides numerical examples demonstrating effectiveness.

Abstract

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.