Asymptotic distribution and convergence rates of stochastic algorithms for entropic optimal transportation between probability measures

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 non-strongly 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.

Statistical bounds for entropic optimal transport: sample complexity and the central limit theorem

“…(Q) can be computed sequentially with each operation requiring O(n) memory, in the most naive implementation (used here) both ĥpaired (Q), ĥind (Q) demand O(n 2 ) space for storing the matrix D i,j = e −||x i −y j || 2 /2σ 2 g , to which the Sinkhorn algorithm is applied. This memory requirement might be alleviated with the use of stochastic methods (Bercu and Bigot, 2018;Genevay et al, 2016).…”

Statistical data analysis in the Wasserstein space

“…Techniques based on optimal transport for data science have thus recently received an increasing interest in mathematical and computational statistics [8,[10][11][12][13][14][15]28,29,44,45,47,49,51,54,58,62,66,67], machine learning [4,6,22,23,32,[36][37][38]40,55,57], image processing and computer vision [7,17,24,30,31,41,52,59,60] or computational biology [56].…”