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Sample Complexity of Sinkhorn divergences

hal.structure.identifierCEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
dc.contributor.authorGenevay, Aude
hal.structure.identifierCEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
dc.contributor.authorChizat, Lenaic
HAL ID: 19586
hal.structure.identifierDépartement d'informatique - ENS Paris [DI-ENS]
dc.contributor.authorBach, Francis
hal.structure.identifierGraduate School of Informatics [Kyoto]
dc.contributor.authorCuturi, Marco
HAL ID: 3354
hal.structure.identifierDépartement de Mathématiques et Applications - ENS Paris [DMA]
dc.contributor.authorPeyré, Gabriel
HAL ID: 1211
dc.date.accessioned2020-06-09T13:40:51Z
dc.date.available2020-06-09T13:40:51Z
dc.date.issued2019
dc.identifier.urihttps://basepub.dauphine.fr/handle/123456789/20859
dc.language.isoenen
dc.subjectSinkhorn divergencesen
dc.subject.ddc515en
dc.titleSample Complexity of Sinkhorn divergencesen
dc.typeCommunication / Conférence
dc.description.abstractenOptimal 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.citationpages11en
dc.subject.ddclabelAnalyseen
dc.relation.conftitleAISTATS'19 - 22nd International Conference on Artificial Intelligence and Statisticsen
dc.relation.confdate2019-04
dc.relation.confcityOkinawaen
dc.relation.confcountryJapanen
dc.relation.forthcomingnonen
dc.description.ssrncandidatenonen
dc.description.halcandidatenonen
dc.description.readershiprechercheen
dc.description.audienceInternationalen
dc.relation.Isversionofjnlpeerreviewednonen
dc.relation.Isversionofjnlpeerreviewednonen
dc.date.updated2020-06-09T13:36:10Z
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