dc.contributor.author Genevay, Aude dc.contributor.author Chizat, Lenaic dc.contributor.author Bach, Francis dc.contributor.author Cuturi, Marco dc.contributor.author Peyré, Gabriel dc.date.accessioned 2020-06-09T13:40:51Z dc.date.available 2020-06-09T13:40:51Z dc.date.issued 2019 dc.identifier.uri https://basepub.dauphine.fr/handle/123456789/20859 dc.language.iso en en dc.subject Sinkhorn divergences en dc.subject.ddc 515 en dc.title Sample Complexity of Sinkhorn divergences en dc.type Communication / Conférence dc.description.abstracten Optimal transport (OT) and maximum mean discrepancies (MMD) are now routinely used in machine learning to compare probability measures. We focus in this paper on \emph{Sinkhorn divergences} (SDs), a regularized variant of OT distances which can interpolate, depending on the regularization strength ε, between OT (ε=0) and MMD (ε=∞). Although the tradeoff induced by that regularization is now well understood computationally (OT, SDs and MMD require respectively O(n3logn), O(n2) and n2 operations given a sample size n), much less is known in terms of their \emph{sample complexity}, namely the gap between these quantities, when evaluated using finite samples \emph{vs.} their respective densities. Indeed, while the sample complexity of OT and MMD stand at two extremes, 1/n1/d for OT in dimension d and 1/n−−√ for MMD, that for SDs has only been studied empirically. In this paper, we \emph{(i)} derive a bound on the approximation error made with SDs when approximating OT as a function of the regularizer ε, \emph{(ii)} prove that the optimizers of regularized OT are bounded in a Sobolev (RKHS) ball independent of the two measures and \emph{(iii)} provide the first sample complexity bound for SDs, obtained,by reformulating SDs as a maximization problem in a RKHS. We thus obtain a scaling in 1/n−−√ (as in MMD), with a constant that depends however on ε, making the bridge between OT and MMD complete. en dc.identifier.citationpages 11 en dc.subject.ddclabel Analyse en dc.relation.conftitle AISTATS'19 - 22nd International Conference on Artificial Intelligence and Statistics en dc.relation.confdate 2019-04 dc.relation.confcity Okinawa en dc.relation.confcountry Japan en dc.relation.forthcoming non en dc.description.ssrncandidate non en dc.description.halcandidate non en dc.description.readership recherche en dc.description.audience International en dc.relation.Isversionofjnlpeerreviewed non en dc.relation.Isversionofjnlpeerreviewed non en dc.date.updated 2020-06-09T13:36:10Z hal.person.labIds 60 hal.person.labIds 60 hal.person.labIds 25027 hal.person.labIds 409737 hal.person.labIds 66
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