  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 finite sample properties of
this Gauss-Newton algorithm for stochastic regularized optimal transport, and
to show its advantages over the use of the stochastic gradient descent,
stochastic Newton and ADAM algorithms.
