BERCU, Bernard; BIGOT, Jérémie; GADAT, Sébastien; SIVIERO, Emilia

doi:10.1093/imaiai/iaac014

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BERCU, Bernard
Institut de Mathématiques de Bordeaux [IMB]

BIGOT, Jérémie
Institut de Mathématiques de Bordeaux [IMB]

GADAT, Sébastien
Toulouse School of Economics [TSE-R]

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Article de revue

Ce document a été publié dans

Information and Inference: A Journal of the IMA. 2022-05-19p. 1-56

Résumé en anglais

We introduce a new second order stochastic algorithm to estimate the entropically regularized optimal transport cost between two probability measures. The source measure can be arbitrary chosen, either absolutely continuous or discrete, while the target measure is assumed to be discrete. To solve the semi-dual formulation of such a regularized and semi-discrete optimal transportation problem, we propose to consider a stochastic Gauss-Newton algorithm that uses a sequence of data sampled from the source measure. This algorithm is shown to be adaptive to the geometry of the underlying convex optimization problem with no important hyperparameter to be accurately tuned. We establish the almost sure convergence and the asymptotic normality of various estimators of interest that are constructed from this stochastic Gauss-Newton algorithm. We also analyze their non-asymptotic rates of convergence for the expected quadratic risk in the absence of strong convexity of the underlying objective function. The results of numerical experiments from simulated data are also reported to illustrate the nite sample properties of this Gauss-Newton algorithm for stochasticregularized optimal transport, and to show its advantages over the use of the stochastic gradient descent, stochastic Newton and ADAM algorithms.< Réduire

Mots clés en anglais

Stochastic optimization

Stochastic Gauss-Newton algorithm

Optimal transport

Entropic regularization

Convergence of random variables.

URI

https://oskar-bordeaux.fr/handle/20.500.12278/191055

DOI

http://dx.doi.org/10.1093/imaiai/iaac014

Project ANR

Mathématiques de l'optimisation déterministe et stochastique liées à l'apprentissage profond - ANR-19-CE23-0017

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Importé de hal

Unités de recherche

Institut de Mathématiques de Bordeaux (IMB) - UMR 5251