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

Pierre Monmarché

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arXiv · 2605.13186

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

Pierre Monmarché

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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.

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Pierre Monmarché. 2026-05-13. Nesterov acceleration for the Wasserstein minimization of displacement-convex free energies. https://arxiv.org/abs/2605.13186

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