A stochastic Gauss–Newton algorithm for regularized semi-discrete optimal transport

Author:

Bercu Bernard1,Bigot Jérémie1,Gadat Sébastien2,Siviero Emilia3

Affiliation:

1. Institut de Mathématiques de Bordeaux et CNRS (UMR 5251) , Université de Bordeaux

2. Toulouse School of Economics , Université Toulouse 1 Capitole

3. LTCI , Télécom Paris, Institut Polytechnique de Paris

Abstract

Abstract We introduce a new second order stochastic algorithm to estimate the entropically regularized optimal transport (OT) cost between two probability measures. The source measure can be arbitrary chosen, either absolutely continuous or discrete, whereas 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 (SGN) 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 SGN 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 finite sample properties of this Gauss–Newton algorithm for stochastic regularized OT and to show its advantages over the use of the stochastic gradient descent, stochastic Newton and ADAM algorithms.

Funder

Agence Nationale de la Recherche

Publisher

Oxford University Press (OUP)

Subject

Applied Mathematics,Computational Theory and Mathematics,Numerical Analysis,Statistics and Probability,Analysis

Reference50 articles.

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4. Asymptotic distribution and convergence rates of stochastic algorithms for entropic optimal transportation between probability measures;Bercu;Ann. Stat.,2021

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