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.

After some general views on stochastic algorithms and some recent advances (pre-conditionners, adaptive and Bregman-like adaptations), we specify our talk to the Gauss-Newton algorithm used for solving the semi-dual formulation of a regularized and semi-discrete optimal transportation problem. 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.