{"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/a-variational-formula-for-renyi-divergences","title":"Variational Representations and Neural Network Estimation of Rényi Divergences","arxiv_id":"2007.03814","date":"2020-07-07","proceeding":null,"authors":["Jeremiah Birrell","Paul Dupuis","Markos A. Katsoulakis","Luc Rey-Bellet","Jie Wang"],"abstract":"We derive a new variational formula for the R\\'enyi family of divergences, $R_\\alpha(Q\\|P)$, between probability measures $Q$ and $P$. Our result generalizes the classical Donsker-Varadhan variational formula for the Kullback-Leibler divergence. We further show that this R\\'enyi variational formula holds over a range of function spaces; this leads to a formula for the optimizer under very weak assumptions and is also key in our development of a consistency theory for R\\'enyi divergence estimators. By applying this theory to neural-network estimators, we show that if a neural network family satisfies one of several strengthened versions of the universal approximation property then the corresponding R\\'enyi divergence estimator is consistent. In contrast to density-estimator based methods, our estimators involve only expectations under $Q$ and $P$ and hence are more effective in high dimensional systems. We illustrate this via several numerical examples of neural network estimation in systems of up to 5000 dimensions.","url_abs":"https://arxiv.org/abs/2007.03814v4","url_pdf":"https://arxiv.org/pdf/2007.03814v4.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":"a-variational-formula-for-renyi-divergences","repo_url":"https://github.com/jvbirrell/RenyiNeuralEstimation","is_official":1,"mentioned_in_paper":0,"mentioned_in_github":0,"framework":"tf","reach":{"status":"ok","spdx":"MIT"}}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=2007.03814","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}