Underdamped Langevin MCMC with third order convergence

• URL: http://arxiv.org/abs/2508.16485v1
• Date: Fri, 22 Aug 2025 16:00:01 GMT
• Title: Underdamped Langevin MCMC with third order convergence
• Authors: Maximilian Scott, Dáire O'Kane, Andraž Jelinčič, James Foster,
• Abstract summary: We present a new numerical method for the underdamped Langevin diffusion (ULD)<n>Under the assumptions that the gradient and Hessian of $f$ are Lipschitz continuous, our algorithm achieves a 2-Wasserstein error of $varepsilon$ in $mathcalO(sqrtd/varepsilon)$ steps.<n>This is the first gradient-only method for ULD with third order convergence.
• Score: 1.8374319565577153
• License: http://creativecommons.org/licenses/by/4.0/
• 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).

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