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Learning with minibatch Wasserstein : asymptotic and gradient properties

Kilian Fatras 1, 2 Younes Zine 2 Rémi Flamary 3, 4 Rémi Gribonval 5 Nicolas Courty 1
1 OBELIX - Observation de l’environnement par imagerie complexe
IRISA-D5 - SIGNAUX ET IMAGES NUMÉRIQUES, ROBOTIQUE
2 PANAMA - Parcimonie et Nouveaux Algorithmes pour le Signal et la Modélisation Audio
Inria Rennes – Bretagne Atlantique , IRISA-D5 - SIGNAUX ET IMAGES NUMÉRIQUES, ROBOTIQUE
3 LITIS - Laboratoire d'Informatique, de Traitement de l'Information et des Systèmes
4 LAGRANGE - Joseph Louis LAGRANGE
5 DANTE - Dynamic Networks : Temporal and Structural Capture Approach
Inria Grenoble - Rhône-Alpes, LIP - Laboratoire de l'Informatique du Parallélisme, IXXI - Institut Rhône-Alpin des systèmes complexes
Abstract : Optimal transport distances are powerful tools to compare probability distributions and have found many applications in machine learning. Yet their algorithmic complexity prevents their direct use on large scale datasets. To overcome this challenge , practitioners compute these distances on minibatches i.e. they average the outcome of several smaller optimal transport problems. We propose in this paper an analysis of this practice, which effects are not well understood so far. We notably argue that it is equivalent to an implicit regularization of the original problem, with appealing properties such as unbiased estimators, gradients and a concentration bound around the expectation, but also with defects such as loss of distance property. Along with this theoretical analysis, we also conduct empirical experiments on gradient flows, GANs or color transfer that highlight the practical interest of this strategy.
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https://hal.archives-ouvertes.fr/hal-02502329
Contributor : Kilian Fatras <>
Submitted on : Monday, March 9, 2020 - 11:23:46 AM
Last modification on : Friday, March 20, 2020 - 11:17:57 AM

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  • HAL Id : hal-02502329, version 1
  • ARXIV : 1910.04091

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UNIV-LYON3 | INSMI | OCA | UGA | LITIS | UNIV-RENNES1 | ENS-LYON | INRIA | UNIV-LYON1 | INSA-LYON | COMUE-NORMANDIE | UNICE | INSA-ROUEN | UNIV-COTEDAZUR | LIP | UNIV-RENNES | UNIROUEN | UNILEHAVRE | UNIV-LYON2 | INSA-GROUPE | IRISA | CNRS | UDL | LAGRANGE | CENTRALESUPELEC | IRISA_UBS | UBS | UR1-UFR-ISTIC

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Kilian Fatras, Younes Zine, Rémi Flamary, Rémi Gribonval, Nicolas Courty. Learning with minibatch Wasserstein : asymptotic and gradient properties. the 23nd International Conference on Artificial Intelligence and Statistics, Jun 2020, Palermo, Italy. ⟨hal-02502329⟩

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