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April 2021 Asymptotic distribution and convergence rates of stochastic algorithms for entropic optimal transportation between probability measures
Bernard Bercu, Jérémie Bigot
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Ann. Statist. 49(2): 968-987 (April 2021). DOI: 10.1214/20-AOS1987

Abstract

This paper is devoted to the stochastic approximation of entropically regularized Wasserstein distances between two probability measures, also known as Sinkhorn divergences. The semi-dual formulation of such regularized optimal transportation problems can be rewritten as a nonstrongly concave optimisation problem. It allows to implement a Robbins–Monro stochastic algorithm to estimate the Sinkhorn divergence using a sequence of data sampled from one of the two distributions. Our main contribution is to establish the almost sure convergence and the asymptotic normality of a new recursive estimator of the Sinkhorn divergence between two probability measures in the discrete and semi-discrete settings. We also study the rate of convergence of the expected excess risk of this estimator in the absence of strong concavity of the objective function. Numerical experiments on synthetic and real datasets are also provided to illustrate the usefulness of our approach for data analysis.

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Bernard Bercu. Jérémie Bigot. "Asymptotic distribution and convergence rates of stochastic algorithms for entropic optimal transportation between probability measures." Ann. Statist. 49 (2) 968 - 987, April 2021. https://doi.org/10.1214/20-AOS1987

Information

Received: 1 October 2019; Revised: 1 June 2020; Published: April 2021
First available in Project Euclid: 2 April 2021

Digital Object Identifier: 10.1214/20-AOS1987

Subjects:
Primary: 62G05
Secondary: 62G20

Rights: Copyright © 2021 Institute of Mathematical Statistics

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Vol.49 • No. 2 • April 2021
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