{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/convergence-of-sparse-variational-inference","title":"Convergence of Sparse Variational Inference in Gaussian Processes Regression","arxiv_id":"2008.00323","date":"2020-08-01","proceeding":null,"authors":["David R. Burt","Carl Edward Rasmussen","Mark van der Wilk"],"abstract":"Gaussian processes are distributions over functions that are versatile and mathematically convenient priors in Bayesian modelling. However, their use is often impeded for data with large numbers of observations, $N$, due to the cubic (in $N$) cost of matrix operations used in exact inference. Many solutions have been proposed that rely on $M \\ll N$ inducing variables to form an approximation at a cost of $\\mathcal{O}(NM^2)$. While the computational cost appears linear in $N$, the true complexity depends on how $M$ must scale with $N$ to ensure a certain quality of the approximation. In this work, we investigate upper and lower bounds on how $M$ needs to grow with $N$ to ensure high quality approximations. We show that we can make the KL-divergence between the approximate model and the exact posterior arbitrarily small for a Gaussian-noise regression model with $M\\ll N$. Specifically, for the popular squared exponential kernel and $D$-dimensional Gaussian distributed covariates, $M=\\mathcal{O}((\\log N)^D)$ suffice and a method with an overall computational cost of $\\mathcal{O}(N(\\log N)^{2D}(\\log\\log N)^2)$ can be used to perform inference.","url_abs":"https://arxiv.org/abs/2008.00323v1","url_pdf":"https://arxiv.org/pdf/2008.00323v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"abstracts"},"code_links":[{"paper_slug":"convergence-of-sparse-variational-inference","repo_url":"https://github.com/markvdw/RobustGP","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"tf","reach":null}],"tasks":[{"task_slug":"gaussian-processes","task_name":"Gaussian Processes"},{"task_slug":"variational-inference","task_name":"Variational Inference"},{"task_slug":"regression-1","task_name":"regression"}],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=2008.00323","mcp":{"get_harvested_code_for_paper":{"arxiv_id":"2008.00323"}},"developers":"https://syntology.ai/developers","read_at":"2026-09-24T18:15:14+00:00","read_at_is":"when the build read Syntology's graph, not when any sample ran","claim":"Per-sample execution status on synthesized fixtures; not a correctness claim about the paper. 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