{"id":6319,"date":"2025-08-25T07:03:24","date_gmt":"2025-08-25T07:03:24","guid":{"rendered":"https:\/\/mailitics.com\/index.php\/2025\/08\/25\/2508-16485\/"},"modified":"2025-08-25T07:03:24","modified_gmt":"2025-08-25T07:03:24","slug":"2508-16485","status":"publish","type":"post","link":"https:\/\/mailitics.com\/index.php\/2025\/08\/25\/2508-16485\/","title":{"rendered":"Underdamped Langevin MCMC with third order convergence"},"content":{"rendered":"

Underdamped Langevin MCMC with third order convergence
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arXiv:2508.16485v1 Announce Type: new
\nAbstract: 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).<\/div>\n

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\n Maximilian Scott, D’aire O’Kane, Andrav{z} Jelinv{c}iv{c}, James Foster
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Underdamped Langevin MCMC with third order convergence arXiv:2508.16485v1 Announce Type: new 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 […]<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[62,113,450,451,420,190,112,191],"tags":[778,718,835],"class_list":["post-6319","post","type-post","status-publish","format-standard","hentry","category-aimldsaimlds","category-cs-lg","category-cs-na","category-math-na","category-math-pr","category-math-st","category-stat-ml","category-stat-th","tag-algorithm","tag-langevin","tag-our"],"_links":{"self":[{"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/posts\/6319"}],"collection":[{"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/comments?post=6319"}],"version-history":[{"count":0,"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/posts\/6319\/revisions"}],"wp:attachment":[{"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/media?parent=6319"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/categories?post=6319"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/tags?post=6319"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}