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

TL;DR

This paper presents a novel stochastic Gauss-Newton algorithm for efficiently estimating regularized semi-discrete optimal transport costs, demonstrating strong theoretical guarantees and superior practical performance over existing methods.

Contribution

The paper introduces a second order stochastic Gauss-Newton algorithm tailored for semi-discrete optimal transport, with proven convergence and improved finite sample properties.

Findings

Algorithm is adaptive to the problem geometry.

Almost sure convergence and asymptotic normality established.

Numerical experiments show advantages over SGD, Newton, and ADAM.

Abstract

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.