2021/07/12 by Bernard Bercu, Bercu, Bernard, Jérémie Bigot +5 · 1 citation
Economics, Econometrics and Finance · Engineering · Mathematics · #62G05 #62G20 #Economic and Environmental Valuation #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Statistics Theory (math.ST) #Water resources management and optimization
paper · pdf · doi:10.48550/arxiv.2107.05291
openalex publication_date 2021/07/12 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28
We introduce a new second order stochastic algorithm to estimate the\nentropically regularized optimal transport cost between two probability\nmeasures. The source measure can be arbitrary chosen, either absolutely\ncontinuous or discrete, while the target measure is assumed to be discrete. To\nsolve the semi-dual formulation of such a regularized and semi-discrete optimal\ntransportation problem, we propose to consider a stochastic Gauss-Newton\nalgorithm that uses a sequence of data sampled from the source measure. This\nalgorithm is shown to be adaptive to the geometry of the underlying convex\noptimization problem with no important hyperparameter to be accurately tuned.\nWe establish the almost sure convergence and the asymptotic normality of\nvarious estimators of interest that are constructed from this stochastic\nGauss-Newton algorithm. We also analyze their non-asymptotic rates of\nconvergence for the expected quadratic risk in the absence of strong convexity\nof the underlying objective function. The results of numerical experiments from\nsimulated data are also reported to illustrate the finite sample properties of\nthis Gauss-Newton algorithm for stochastic regularized optimal transport, and\nto show its advantages over the use of the stochastic gradient descent,\nstochastic Newton and ADAM algorithms.\n