{"paper":{"title":"Nesterov acceleration for the Wasserstein minimization of displacement-convex free energies","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"The mean-field underdamped Langevin process achieves Nesterov acceleration for Wasserstein minimization of displacement-convex free energies.","cross_cats":["math.OC","math.PR"],"primary_cat":"math.AP","authors_text":"Pierre Monmarch\\'e","submitted_at":"2026-05-13T08:43:27Z","abstract_excerpt":"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 ter"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"The mean-field underdamped Langevin process 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-Łojasiewicz constant of the free energy.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The free energy must be displacement-convex and satisfy a Polyak-Łojasiewicz inequality; the nonlinear extension relies on the linear-case breakthrough in [42] carrying over without additional obstructions.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Mean-field underdamped Langevin dynamics achieve Nesterov acceleration for Wasserstein gradient flows of displacement-convex free energies.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The mean-field underdamped Langevin process achieves Nesterov acceleration for Wasserstein minimization of displacement-convex free energies.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"0e8dfba01fda89ce475c95d89ee4e07722af4771dc11fb9e5b8e9398db31973c"},"source":{"id":"2605.13186","kind":"arxiv","version":1},"verdict":{"id":"6727a739-577f-4f22-acfb-592dfc0d2403","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T18:26:29.144885Z","strongest_claim":"The mean-field underdamped Langevin process 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-Łojasiewicz constant of the free energy.","one_line_summary":"Mean-field underdamped Langevin dynamics achieve Nesterov acceleration for Wasserstein gradient flows of displacement-convex free energies.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The free energy must be displacement-convex and satisfy a Polyak-Łojasiewicz inequality; the nonlinear extension relies on the linear-case breakthrough in [42] carrying over without additional obstructions.","pith_extraction_headline":"The mean-field underdamped Langevin process achieves Nesterov acceleration for Wasserstein minimization of displacement-convex free energies."},"references":{"count":67,"sample":[{"doi":"","year":1953,"title":"Vari- ational methods for the kinetic fokker–planck equation.Analysis & PDE, 17(6):1953– 2010, 2024","work_id":"9453c431-9dc7-4a86-b99f-929f08bc300e","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2025,"title":"Shifted composition iv: toward ballistic acceleration for log-concave sampling.arXiv preprint arXiv:2506.23062, 2025","work_id":"7d5e93ba-bdfa-4474-a36b-765f792a5589","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2005,"title":"Springer Science & Business Media, 2005","work_id":"3bab6832-6534-4341-b544-60eadfd5473d","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2021,"title":"Hypocoercivity of piecewise deterministic markov process-monte carlo.The Annals of Applied Probability, 31(5):2478–2517, 2021","work_id":"0d8ac4dc-d9fd-4e6b-97b7-e93f274a2a1e","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2019,"title":"Maximum mean discrep- ancy gradient flow.Advances in neural information processing systems, 32, 2019","work_id":"7e7fef4c-6252-4cef-ac33-2bbff723e79b","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":67,"snapshot_sha256":"5dcf5884262b8dfbece89f77729e5060ee9d6d37441b7d6f06429e9424203cc6","internal_anchors":3},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}