{"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/the-cramer-distance-as-a-solution-to-biased","title":"The Cramer Distance as a Solution to Biased Wasserstein Gradients","arxiv_id":"1705.10743","date":"2017-05-30","proceeding":"ICLR 2018 1","authors":["Marc G. Bellemare","Ivo Danihelka","Will Dabney","Shakir Mohamed","Balaji Lakshminarayanan","Stephan Hoyer","Rémi Munos"],"abstract":"The Wasserstein probability metric has received much attention from the\nmachine learning community. Unlike the Kullback-Leibler divergence, which\nstrictly measures change in probability, the Wasserstein metric reflects the\nunderlying geometry between outcomes. The value of being sensitive to this\ngeometry has been demonstrated, among others, in ordinal regression and\ngenerative modelling. In this paper we describe three natural properties of\nprobability divergences that reflect requirements from machine learning: sum\ninvariance, scale sensitivity, and unbiased sample gradients. The Wasserstein\nmetric possesses the first two properties but, unlike the Kullback-Leibler\ndivergence, does not possess the third. We provide empirical evidence\nsuggesting that this is a serious issue in practice. Leveraging insights from\nprobabilistic forecasting we propose an alternative to the Wasserstein metric,\nthe Cram\\'er distance. We show that the Cram\\'er distance possesses all three\ndesired properties, combining the best of the Wasserstein and Kullback-Leibler\ndivergences. To illustrate the relevance of the Cram\\'er distance in practice\nwe design a new algorithm, the Cram\\'er Generative Adversarial Network (GAN),\nand show that it performs significantly better than the related Wasserstein\nGAN.","url_abs":"http://arxiv.org/abs/1705.10743v1","url_pdf":"http://arxiv.org/pdf/1705.10743v1.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":"the-cramer-distance-as-a-solution-to-biased","repo_url":"https://github.com/Mintas/diving-deep-learning","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"pytorch","reach":{"status":"unanswered"}},{"paper_slug":"the-cramer-distance-as-a-solution-to-biased","repo_url":"https://github.com/mbarbetti/tf-gen-models","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"tf","reach":null}],"tasks":[{"task_slug":"machine-learning","task_name":"BIG-bench Machine Learning"},{"task_slug":null,"task_name":"Generative Adversarial Network"}],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=1705.10743","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}