{"ID":2879503,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2508.16485","arxiv_id":"2508.16485","title":"Underdamped Langevin MCMC with third order convergence","abstract":"In this paper, we propose a new numerical method for the underdamped Langevin diffusion (ULD) and present a non-asymptotic analysis of its sampling error in the 2-Wasserstein distance when the $d$-dimensional target distribution $p(x)\\propto e^{-f(x)}$ is strongly log-concave and has varying degrees of smoothness. Precisely, under the assumptions that the gradient and Hessian of $f$ are Lipschitz continuous, our algorithm achieves a 2-Wasserstein error of $\\varepsilon$ in $\\mathcal{O}(\\sqrt{d}/\\varepsilon)$ and $\\mathcal{O}(\\sqrt{d}/\\sqrt{\\varepsilon})$ steps respectively. Therefore, our algorithm has a similar complexity as other popular Langevin MCMC algorithms under matching assumptions. However, if we additionally assume that the third derivative of $f$ is Lipschitz continuous, then our algorithm achieves a 2-Wasserstein error of $\\varepsilon$ in $\\mathcal{O}(\\sqrt{d}/\\varepsilon^{\\frac{1}{3}})$ steps. To the best of our knowledge, this is the first gradient-only method for ULD with third order convergence. To support our theory, we perform Bayesian logistic regression across a range of real-world datasets, where our algorithm achieves competitive performance compared to an existing underdamped Langevin MCMC algorithm and the popular No U-Turn Sampler (NUTS).","short_abstract":"In this paper, we propose a new numerical method for the underdamped Langevin diffusion (ULD) and present a non-asymptotic analysis of its sampling error in the 2-Wasserstein distance when the $d$-dimensional target distribution $p(x)\\propto e^{-f(x)}$ is strongly log-concave and has varying degrees of smoothness. Prec...","url_abs":"https://arxiv.org/abs/2508.16485","url_pdf":"https://arxiv.org/pdf/2508.16485v1","authors":"[\"Maximilian Scott\",\"Dáire O'Kane\",\"Andraž Jelinčič\",\"James Foster\"]","published":"2025-08-22T16:00:01Z","proceeding":"stat.ML","tasks":"[\"stat.ML\",\"cs.LG\",\"math.NA\",\"math.PR\",\"math.ST\"]","methods":"[\"Diffusion Model\"]","has_code":false}