Nesterov acceleration for the Wasserstein minimization of displacement-convex free energies

TL;DR

This paper demonstrates that the mean-field underdamped Langevin process accelerates convergence in Wasserstein gradient flows of displacement-convex free energies, achieving an optimal rate inspired by recent entropy improvement results.

Contribution

It extends Nesterov acceleration to Wasserstein gradient flows of displacement-convex free energies using recent entropy improvement insights.

Findings

Achieves Nesterov acceleration in Wasserstein gradient flows.

Convergence rate matches the optimal Polyak-Łojasiewicz constant.

Builds on recent diffusive-to-ballistic entropy improvements.

Abstract

We show that the mean-field underdamped Langevin process (associated to the non-linear Vlasov-Fokker-Planck equation) achieves a Nesterov acceleration with respect to the Wasserstein gradient flow of a displacement-convex free energy, in the sense that it converges at a rate of order given by the square-root of the Polyak-{\L}ojasiewicz constant of the free energy (which is the optimal convergence rate for the corresponding gradient flow). This result has been made possible by the recent breakthrough [42] by Jianfeng Lu, which establishes such a \emph{diffusive-to-ballistic} improvement in term of entropy in the linear case.